# Assistant conversation Q8EH2J - Canonical URL: - Resource IRI: - Type: assistant conversation - Mode: chat ## Transcript ### 1. User Import the attached PDF documents. [Uploaded PDFs] - on-implication-and-negation-in-partition-logic.pdf: Sheaf file #8LATV8 ### 2. Assistant tool calls #### Tool call: `document_import` ```json { "action": "stage", "file_ids": [ "8LATV8" ], "name": "Import on implication and negation in partition logic" } ``` ### 3. Tool result: document_import { "status": { "counts": { "pending": 1 }, "files": [ { "error": null, "status": "pending", "file_id": "8LATV8", "execution_id": null } ] }, "sources": [], "action": "stage", "run_id": "ANC7G5", "file_ids": [ "8LATV8" ], "run_iri": "https://m.sheaf.less.rest/ANC7G5" } ### 4. Assistant tool calls #### Tool call: `document_import` ```json { "action": "extract", "run_id": "ANC7G5" } ``` ### 5. Tool result: document_import { "status": { "counts": { "completed": 1 }, "files": [ { "error": null, "status": "completed", "file_id": "8LATV8", "execution_id": "pex_vUwHGj9hoTqh" } ] }, "action": "extract", "completed": 1, "run_id": "ANC7G5", "submitted": 1 } ### 6. Assistant tool calls #### Tool call: `document_import` ```json { "action": "inspect", "run_id": "ANC7G5" } ``` ### 7. Tool result: document_import { "status": { "counts": { "completed": 1 }, "files": [ { "error": null, "status": "completed", "file_id": "8LATV8", "execution_id": "pex_vUwHGj9hoTqh" } ] }, "action": "inspect", "run_id": "ANC7G5", "files": [ { "status": "completed", "file_id": "8LATV8", "quality": { "blocks": 124, "pages": 9, "math_expressions": 454, "equation_blocks": 13, "pages_with_math": 9, "empty_equation_blocks": 0, "page_continuations": 2, "block_types": { "Caption": 3, "Equation": 13, "Figure": 3, "Footnote": 4, "ListGroup": 1, "PageFooter": 3, "PageHeader": 18, "Picture": 1, "SectionHeader": 12, "Text": 66 } }, "output_path": "/home/mbrock/sheaf.mikael/var/datalab/ANC7G5/8LATV8.datalab.json" } ] } ### 8. Assistant tool calls #### Tool call: `document_import` ```json { "action": "import", "run_id": "ANC7G5" } ``` ### 9. Tool result: document_import { "next": "Resolve metadata, rebuild indexes, and validate reader pages.", "action": "import", "documents": [ { "status": "imported", "title": null, "file_id": "8LATV8", "document_id": "C8FHDZ", "document_iri": "https://m.sheaf.less.rest/C8FHDZ" } ], "run_id": "ANC7G5" } ### 10. Assistant tool calls #### Tool call: `document_import` ```json { "action": "metadata", "run_id": "ANC7G5" } ``` ### 11. Tool result: document_import { "action": "metadata", "documents": [ { "match": { "reason": "title match accepted for Crossref type journal-article", "source": "doi", "identifier": "10.30538/oms2025.0250", "score": 1.0, "doi": "10.30538/oms2025.0250", "accept?": true, "crossref_title": "On implication and negation in partition logic", "crossref_type": "journal-article" }, "metadata": { "title": "On implication and negation in partition logic", "usage": { "cost": { "total": 6.39e-4, "tokens": 6.39e-4, "tools": 0.0, "storage": 0.0, "images": 0.0, "input_cost": 4.44e-4, "output_cost": 1.95e-4, "reasoning_cost": 0.0, "line_items": [ { "count": 1777, "id": "token.input", "kind": "tokens", "component": "token.input", "cost": 4.44e-4, "quantity": 1777 }, { "count": 130, "id": "token.output", "kind": "tokens", "component": "token.output", "cost": 1.95e-4, "quantity": 130 } ] }, "output_tokens": 130, "input_tokens": 1777, "total_cost": 6.39e-4, "reasoning_tokens": 0, "input_cost": 4.44e-4, "output_cost": 1.95e-4, "reasoning_cost": 0.0, "cached_tokens": 0, "total_tokens": 1907, "tool_usage": {}, "image_usage": {}, "cache_creation_tokens": 0 }, "year": "2025", "model": "google:gemini-3.1-flash-lite-preview", "notes": null, "issue": null, "pages": "149-157", "authors": [ "David Ellerman" ], "doi": "10.30538/oms2025.0250", "volume": "9", "isbn": null, "publication": "Open Journal of Mathematical Sciences", "confidence": "high", "source_filename": "f2ea600a98daac1d83553cb1064f0f3dd5c3af9c144d730b8173257b79729e69.pdf" }, "document_id": "C8FHDZ", "wrote": true } ], "run_id": "ANC7G5" } ### 12. Assistant tool calls #### Tool call: `document_import` ```json { "action": "validate", "run_id": "ANC7G5" } ``` ### 13. Tool result: document_import { "action": "validate", "documents": [ { "title": "On implication and negation in partition logic", "valid": true, "document_id": "C8FHDZ", "readable_chunks": 101, "source_pages": 9, "reader_path": "/C8FHDZ" } ], "run_id": "ANC7G5", "embedding_status": "completed", "search_rows": 3564, "embedding_errors": 0 } ### 14. Assistant Imported and validated: - **David Ellerman (2025), “On implication and negation in partition logic”** — #C8FHDZ - DOI: `10.30538/oms2025.0250` - 9 pages; extraction and search indexing completed successfully. ### 15. User can you read it, describe it somewhat conversationally, and then generate a nice cover artwork for it? ### 16. Assistant tool calls #### Tool call: `get_document` ```json { "id": "C8FHDZ" } ``` ### 17. Tool result: get_document DOCUMENT #C8FHDZ Title: On implication and negation in partition logic Outline: - 1 #L8JZAG On implication and negation in partition logic - 1.1 #ZZPBEJ 1. Introduction: The Boolean logic of subsets and the logic of partitions - 1.2 #7WB5RZ 2. The logic of partitions - 1.3 #9DDUBK 3. Implication and negation in partition logic - 1.4 #FEMQRA 4. Three more equivalent ways to define implication for partitions - 1.4.1 #8P96YU 4.1. The adjunctive definition - 1.4.2 #WUWW5V 4.2. The graph-theoretic definition - 1.4.3 #H2YZ4E 4.3. The ditset definition - 1.5 #KS4RBJ 5. Relative negation in partition logic - 1.6 #LEZCCF 6. Valid formulas - 1.7 #GNBDTD 7. Concluding remarks - 1.8 #87DAFA References ### 18. Assistant tool calls #### Tool call: `read` ```json { "blocks": [ "ZZPBEJ", "7WB5RZ", "9DDUBK", "FEMQRA", "KS4RBJ", "LEZCCF", "GNBDTD" ], "expand": true } ``` ### 19. Tool result: read DOCUMENT #C8FHDZ On implication and negation in partition logic SECTION #ZZPBEJ 1. Introduction: The Boolean logic of subsets and the logic of partitions EXCERPT #BLF82B p. 0 T oday, the ‘classical’ form of logic is seen as the Boolean logic of subsets usually presented as the special case of propositional logic (i.e., the logic of subsets 0 and 1 of the one element set 1). Other related logics, such as intuitionistic logic (e.g., the logic of the open subsets of a topological space) are considered as non-classical. But there is another recently developed and little-known logic that is at the same mathematical level of fundamentality as subset logic and is thus ‘classical’ in that nontemporal sense. Since the development of category theory starting in the middle of the twentieth century, it has been known that the concept of a subset has a category-theoretic dual in the notion of a quotient set (or, equivalently, a partition or equivalence relation). F. William Lawvere calls the generalization of a subset a “part” and “The dual notion (obtained by reversing the arrows) of ‘part’ is the notion of partition.” [1, p. 85] The simplest illustration of this is the fact that given a set function f : X \rightarrow Y , the image of f is a subset f(X) \subseteq Y of the codomain Y and the inverse-image \{f^{-1}(y) \neq \emptyset : y \in Y\} is a partition on the domain X . Hence, it should be expected that there is a logic of partitions ([2,3]) dual to the Boolean logic of subsets. And since subsets and quotient sets are at the same basic level from the mathematical point of view, partition logic is more of a dual sibling to subset logic rather than being another ‘non-classical’ off-shoot of the classical subset logic. DOCUMENT #C8FHDZ On implication and negation in partition logic SECTION #7WB5RZ 2. The logic of partitions EXCERPT #ZBQE96 p. 0 Our purpose here is briefly present the basics of partition logic that suffice to explore the role of negation and implication in that logic. A partition \pi = \{B, B', \dots\} on a set U is a set of non-empty subsets B, B', \dots (“blocks”) of U where the blocks are mutually exclusive (the intersection of distinct blocks is empty) and jointly exhaustive (the union of the blocks is U ). An equivalence relation is a binary relation E \subseteq U \times U that is reflexive, symmetric, and transitive. Every equivalence relation on a set U determines a partition on U where the equivalence classes are the mutually exclusive and jointly exhaustive blocks of the partition. Conversely, every partition on a set determines an equivalence relation on the set; two elements are equivalent if they are in the same block of the partition. The notions of a partition on a set and an equivalence relation on a set are thus interdefinable. Indeed, equivalence relations and partitions are often considered as the “same” as in the conventional practice (not used here) of defining the “lattice of partitions” as the lattice of equivalence relations [4]. EXCERPT #QEASWR p. 0 For the purposes of partition logic, it is important to consider the complementary binary relation to an equivalence relation. A partition relation (also called an apartness relation ) R \subseteq U \times U is irreflexive (i.e., (u, u) \notin R ). EXCERPT #7Z3MBV p. 0 EXCERPT #THTJ3S p. 0 EXCERPT #QKWJ9K p. 1 EXCERPT #XKGC29 p. 1 EXCERPT #YXGP5T p. 1 R for any u \in U ), symmetric (i.e., (u, u') \in R implies (u', u) \in R ), and anti-transitive in the sense that if (u, u') \in R , then for any a \in U , either (u, a) \in R or (a, u') \in R (i.e., U \times U - R = R^c is transitive). Thus as binary relations, equivalence relations and partition relations are complementary. That is, E \subseteq U \times U is an equivalence relation if and only if (iff) E^c \subseteq U \times U is a partition relation. EXCERPT #B7W87A p. 1 A distinction of a partition is an ordered pair (u, u') of elements of U in distinct blocks of the partition. The set of distinctions (abbreviated "dits") of a partition is the ditset EXCERPT #ES3YR9 p. 1 \text{dit}(\pi) = \{(u, u') : \exists B, B' \in \pi; B \neq B'; u \in B; u' \in B'\}. EXCERPT #PS79AA p. 1 Similarly an indistinction or indit of a partition is an ordered pair of elements in the same block of the partition so: EXCERPT #GMLPHZ p. 1 \text{indit}(\pi) = \{(u, u') : \exists B \in \pi; u, u' \in B\} = \bigcup_{B \in \pi} B \times B = U \times U - \text{dit}(\pi). EXCERPT #HTEQ5J p. 1 The indit set of a partition is the equivalence relation defined by the partition, and the ditset of a partition is the complementary partition relation defined by the partition. EXCERPT #Z6B58M p. 1 If \sigma = \{C, C', \dots\} is another partition on U , then the partial order of refinement is defined by: EXCERPT #ZVJVBC p. 1 \sigma \preceq \pi \text{ (read: } \pi \text{ refines } \sigma \text{ or } \sigma \text{ is refined by } \pi) \text{ if } \forall B \in \pi, \exists C \in \sigma \text{ such that } B \subseteq C. EXCERPT #JDV5KF p. 1 Note that if \sigma \preceq \pi , then for any C \in \sigma , there is a set of blocks of \pi whose union is C . The most refined partition on U is the discrete partition \mathbf{1} = \{\{u\}\}_{u \in U} whose blocks are all singletons. It is the top or maximal element in the refinement partial order. The least refined partition is the indiscrete partition (nicknamed the 'blob') \mathbf{0} = \{U\} whose only block is U itself. It is bottom or minimal element in the refinement partial order. The join \pi \vee \sigma (least upper bound) of \pi and \sigma is the partition whose blocks are the non-empty intersections of the blocks of \pi and \sigma : EXCERPT #NWSHDJ p. 1 \pi \vee \sigma = \{B \cap C \neq \emptyset : B \in \pi; C \in \sigma\}. EXCERPT #HG4C5G p. 1 To define the meet \pi \wedge \sigma (greatest lower bound) of \pi and \sigma , we define an equivalence relation on U that is generated by u \sim u' if u and u' are in the same block of \pi or \sigma . Thus if two blocks of \pi and \sigma overlap (non-empty intersection) then all the elements of the two blocks are equated and so forth for any finite sequence of overlapping blocks. Hence a block of the meet partition, i.e., an equivalence class of that equivalence relation, is a precise union of blocks of \pi and a union of blocks of \sigma , and is the smallest such union. These definitions of refinement, join, and meet turn the set \Pi(U) of partitions on U into a lattice. The notion of refinement between partitions is equivalent to inclusion between their corresponding ditsets or partition relations, i.e., \sigma \preceq \pi iff \text{dit}(\sigma) \subseteq \text{dit}(\pi) , so the lattice of partitions on U can be represented as the isomorphic lattice of partition relations on U \times U . But it should be carefully noted that what many textbooks call the "lattice of partitions" is really the opposite lattice of equivalence relations, e.g., Birkhoff [4] or Grätzer [5], where the join and meet are interchanged. EXCERPT #SKL3RS p. 1 The lattice of partitions (in either presentation) was known and studied in the nineteenth century by Richard Dedekind and others. But no other operations on partitions besides join and meet were defined throughout the twentieth century. EXCERPT #9XKV6Y p. 1 "Equivalence relations are so ubiquitous in everyday life that we often forget about their proactive existence. Much is still unknown about equivalence relations. Were this situation remedied, the theory of equivalence relations could initiate a chain reaction generating new insights and discoveries in many fields dependent upon it. EXCERPT #4J3GNC p. 1 This paper springs from a simple acknowledgement: the only operations on the family of equivalence relations fully studied, understood and deployed are the binary join \vee and meet \wedge operations" [6, p. 445]. EXCERPT #78JCG3 p. 1 Hence the development of partition logic depended on defining at least implication \sigma \Rightarrow \pi , and then all the other logical (i.e., Boolean) operations on partitions, e.g., [7]. EXCERPT #9T4244 p. 2 EXCERPT #22BSZE p. 2 DOCUMENT #C8FHDZ On implication and negation in partition logic SECTION #9DDUBK 3. Implication and negation in partition logic EXCERPT #FH9NKH p. 2 There are at least four equivalent ways to define the implication operation \sigma \Rightarrow \pi on partitions. The most intuitive and useful set-of-blocks definition will be used here. The implication partition \sigma \Rightarrow \pi is like the partition \pi except that every block B \in \pi that is contained in some block C \in \sigma is replaced by singletons of its elements. Such an ‘atomized’ or discretized block B might be denoted 1_B as the local B -version of the discrete partition 1 . If a block B \in \pi is not contained in any block of \sigma , then it remains the same which might be denoted 0_B as the local B -version of the indiscrete partition 0 . Hence the implication partition \sigma \Rightarrow \pi functions as an indicator or characteristic function with blocks 1_B or 0_B according to whether or not B was contained in a block of \sigma . With the implication operation, we could refer to \Pi(U) as the algebra of partitions on U instead of just the lattice of partitions. EXCERPT #QBT84C p. 2 Both subset logic and partition logic have their corresponding lattices with the join and meet operations based on their respective partial orders of inclusion and refinement. What is the intuition about the implication operation to go beyond the join and meet operations? The idea of the implication is that it is an operation on two elements in the lattice so that if the result is the top of the lattice, then those two elements are related by the partial order (the ‘antecedent’ is lower than or equal to the ‘consequent’ in the partial order). Thus in the case of the lattice of subsets, the implication or conditional operation takes subsets S, T \subseteq U to the subset S \supset T := S^c \cup T . And if that subset equals the top of the lattice U , then the partial order holds between those two subsets, i.e., S \supset T = U iff S \subseteq T . The same thing holds for the implication operation in the lattice of partitions. If all the blocks of \pi are atomized or discretized, i.e., \sigma \Rightarrow \pi = \mathbf{1} , then \sigma \preceq \pi and vice versa. Thus we might say intuitively that the implication of two elements in the lattice is another lattice element that indicates the extent to which those two elements stand in the partial order to one another. Thus even if \sigma \Rightarrow \pi is not the top of the lattice, if we ‘restrict’ the implication to the antecedent, i.e., \sigma \wedge (\sigma \Rightarrow \pi) , then we get a lattice element that does stand in the partial order to the consequent, i.e., [(\sigma \wedge (\sigma \Rightarrow \pi)) \Rightarrow \pi] = \mathbf{1} , and thus modus ponens is a validity in both logics. EXCERPT #FFVCXY p. 2 With the implication operation, the (absolute) negation of \sigma can be defined as \neg\sigma := \sigma \Rightarrow \mathbf{0} . But the more interesting (relative) \pi -negation of \sigma is defined as: {}^\pi\neg\sigma := \sigma \Rightarrow \pi , so the \pi -negation of \sigma is just another way of considering the implication \sigma \Rightarrow \pi . EXCERPT #ZTPFTV p. 2 The equivalence relation corresponding to the indiscrete partition 0 is the universal relation U \times U . For any two equivalence relations E, E' \subseteq U \times U , if E \cup E' = U \times U , then E = U \times U or E' = U \times U . This is essentially the standard result of graph theory that the complement of any disconnected graph is connected [8, p. 30]. Since the indiscrete partition has no distinctions, i.e., \text{dit}(\mathbf{0}) = \emptyset , the complementary form of that result is that for any two partitions \sigma, \pi , if \text{dit}(\sigma) \cap \text{dit}(\pi) = \emptyset , then \text{dit}(\sigma) = \emptyset or \text{dit}(\pi) = \emptyset , i.e., \sigma = \mathbf{0} or \pi = \mathbf{0} . An alternative form of the result is useful to understand the negation \sigma \Rightarrow \mathbf{0} . EXCERPT #C53C8A p. 2 Theorem 1 (Common-Dits Theorem). Any two non-empty ditsets overlap, i.e., have some dits in common. EXCERPT #3K4JXL p. 2 Proof. Let \pi and \sigma be any two partitions on U with non-empty dit sets, i.e., \pi \neq \mathbf{0} \neq \sigma . We need to show that \text{dit}(\pi) \cap \text{dit}(\sigma) \neq \emptyset . Since \sigma is not the blob 0 , consider two elements u and u' distinguished by \sigma but identified by \pi [otherwise (u, u') \in \text{dit}(\pi) \cap \text{dit}(\sigma) and we are finished]. Since \pi is also not the blob, there must be a third element u'' not in the same block of \pi as u and u' . EXCERPT #VBK6HW p. 2 Diagram illustrating the Common Dits Theorem. It shows two partitions, pi (solid line) and sigma (dashed line), on a set U. Partition pi has two blocks: one containing u and u' (labeled u, u') and another containing u'' (labeled u''). Partition sigma has two blocks: one containing u (labeled u) and another containing u' and u'' (labeled u', u''). The intersection of the ditsets of pi and sigma is non-empty, containing the pair (u, u'). EXCERPT #FPJB3E p. 2 Figure 1. Common dits to any two non-empty ditsets EXCERPT #FW3PAF p. 3 EXCERPT #DAWDWY p. 3 EXCERPT #K3PFAA p. 3 But since u and u' are in different blocks of \sigma , the third element u'' must be distinguished from one or the other or both in \sigma . Hence (u, u'') (as in Figure 1) or (u', u'') must be distinguished by both partitions and thus must be in \text{dit}(\pi) \cap \text{dit}(\sigma) . \square EXCERPT #2K99KN p. 3 This means that for any two non-blob partitions \pi and \sigma on U , there is always a pair of elements u, u' \in U that are in different blocks of both partitions. This result is perhaps particularly striking if we take \pi and \sigma to be atomic partitions, namely, partitions with only two blocks. For any two ways to divide the elements of U ( |U| \geq 2 ) into two parts, there is always a pair of elements separated by both divisions. EXCERPT #2TNJYT p. 3 Since intuitionistic logic is the most developed logic aside from Boolean logic, it is often suggestive to compare the ditsets of partition logic with the open sets in the topological representation of intuitionistic logic, i.e., of a Heyting algebra (also called a pseudo-Boolean algebra or Brouwer algebra). The negation of an open set is the largest open set disjoint from the given set. But now we see that there is no non-empty ditsets disjoint from any given non-empty ditset. Hence intuitively the negation of any partition \sigma \neq \mathbf{0} , is the partition \mathbf{0} with an empty ditset. The definition \neg\sigma := \sigma \Rightarrow \mathbf{0} gives the same result since the only block U in \mathbf{0} = \{U\} is not contained in any block of \sigma \neq \mathbf{0} . And when \sigma = \mathbf{0} , then \neg\mathbf{0} = \mathbf{0} \Rightarrow \mathbf{0} = \mathbf{1} since U \subseteq U so it is discretized in the implication. That is why the absolute negation \neg\sigma is of less interest than the relative negation \frac{\pi}{\sigma}\sigma = \sigma \Rightarrow \pi which is simply the partition implication. DOCUMENT #C8FHDZ On implication and negation in partition logic SECTION #FEMQRA 4. Three more equivalent ways to define implication for partitions SECTION #8P96YU 4.1. The adjunctive definition EXCERPT #K9HZ9A p. 3 For subsets R, S, T \subseteq U , the set implication or conditional S \supset T in subset logic can also be characterized by a category-theoretic adjunction: EXCERPT #9AWN79 p. 3 R \cap S \subseteq T \text{ iff } R \subseteq (S \supset T). EXCERPT #BF2EGN p. 3 Since S \supset T is clearly the maximal subset to satisfy that characterization, we could define: EXCERPT #8HFLPS p. 3 S \supset T := \cup \{R : R \cap S \subseteq T\}. EXCERPT #U4TAMP p. 3 The partition implication \sigma \Rightarrow \pi can be similarly characterized in partition logic substituting ditsets for subsets [2]. For a third partition \tau = \{D, D', \dots\} , the characterization is: EXCERPT #AM4WKF p. 3 \text{dit}(\tau) \cap \text{dit}(\sigma) \subseteq \text{dit}(\pi) \text{ iff } \tau \preceq \sigma \Rightarrow \pi. EXCERPT #LJDK5T p. 3 Since any intersection of equivalence relations is an equivalence relation, any union of their complements, the partition relations or ditsets, is also a ditset. Hence we have a second definition of the partition implication by: EXCERPT #556TTP p. 3 \text{dit}(\sigma \Rightarrow \pi) := \cup \{\text{dit}(\tau) : \text{dit}(\tau) \cap \text{dit}(\sigma) \subseteq \text{dit}(\pi)\}. SECTION #WUWW5V 4.2. The graph-theoretic definition EXCERPT #RM3FBS p. 3 Another way to define the partition implication or any Boolean operation on partitions is the graph-theoretic method [7]. Let K(U) be the complete undirected loop-free graph on U . The links u - u' corresponding to dits of a partition, i.e., (u, u') \in \text{dit}(\pi) , of a partition are labelled with the ‘truth value’ T_\pi and the links corresponding to indits (u, u') \in \text{indit}(\pi) are labelled with the ‘truth value’ F_\pi . Given two partitions \pi and \sigma , each link in the complete graph K(U) is labelled with a pair of truth values. Then to define any binary Boolean operation \pi \# \sigma , one evaluates those two truth values on each link of K(U) according to that binary operation to obtain either T_{\pi \# \sigma} or F_{\pi \# \sigma} on that link. Then we obtain the graph G(\pi \# \sigma) for that operation by deleting all the links with the truth value T_{\pi \# \sigma} so that only the F_{\pi \# \sigma} links remain. Those F_{\pi \# \sigma} links then generate an equivalence relation on U whose blocks are the connected components of the graph G(\pi \# \sigma) . Those connected components or equivalence classes are the partition \pi \# \sigma on U . EXCERPT #WR8JA4 p. 3 Specializing to the implication operation \sigma \Rightarrow \pi , the links retained in G(\sigma \Rightarrow \pi) are the links labelled with T_\sigma and F_\pi since that combination is the only one to be evaluated to F_{\sigma \Rightarrow \pi} in the truth table for implication (or conditional). The connected components in that graph G(\sigma \Rightarrow \pi) are the blocks in the partition implication \sigma \Rightarrow \pi . EXCERPT #JXALN7 p. 4 EXCERPT #MUZ5NT p. 4 EXCERPT #NFPGWW p. 4 Example 1. Let U = \{a, b, c, d\} so that K(U) = K_4 is the complete graph on four points. Let \sigma = \{\{a\}, \{b, c, d\}\} and \pi = \{\{a, b\}, \{c, d\}\} so we see immediately from the usual definition, that the \pi -block of \{c, d\} will be discretized while the \pi -block of \{a, b\} will remain whole so the partition implication is \sigma \Rightarrow \pi = \{\{a, b\}, \{c\}, \{d\}\} . After labelling the links in K(U) , we see that only the a - b link has the F_{\sigma \Rightarrow \pi} ‘truth value’ so the graph G(\sigma \Rightarrow \pi) has only that a - b link (thickened in Figure 2). Then the connected components of G(\sigma \Rightarrow \pi) give the same partition implication \sigma \Rightarrow \pi = \{\{a, b\}, \{c\}, \{d\}\} . EXCERPT #EYYMC2 p. 4 Figure 2: A complete graph K4 with vertices a, b, c, d. The edges are labeled with truth values for sigma and pi. The edge (a,b) is thickened and labeled T_sigma, F_pi. The edge (a,c) is labeled T_sigma, T_pi. The edge (a,d) is labeled T_sigma, T_pi. The edge (b,c) is labeled F_sigma, T_pi. The edge (b,d) is labeled F_sigma, T_pi. The edge (c,d) is labeled F_sigma, F_pi. To the left of the graph, the partitions are listed: sigma = {{a}, {b,c,d}}, pi = {{a,b}, {c,d}}, and sigma -> pi = {{a,b}, {c}, {d}}. EXCERPT #3YAU8U p. 4 Figure 2. Graph to define the partition implication SECTION #H2YZ4E 4.3. The ditset definition EXCERPT #LTGSSD p. 4 Another equivalent way to define partition implication is to mimic the subset definition using ditsets except for the fact that Boolean subset operations on ditsets do not necessarily lead to ditsets. By the analogy with the interior operation on subsets of a topological space, the interior \text{int}(S) of an arbitrary subset S \subseteq U , where U is a topological space, is the largest open set contained in S . The interior \text{int}(S) can also be defined as the complement of the (topological) closure of the complement, i.e., (\overline{S^c})^c . Similarly, we could start with any subset S \subseteq U \times U , and define the interior of S as the largest ditset contained in S . It could be constructed by first taking the complement S^c in U \times U and then its reflexive, symmetric, and transitive closure \overline{S^c} which is just the intersection of all the equivalence relations containing S^c . Then the complement is the interior : \text{int}(S) := (\overline{S^c})^c . This partition-theoretic closure operation is not a topological closure operation, e.g., since the intersection of two ditsets is not necessarily a ditset, whereas the intersection of two open sets is open. EXCERPT #2H58X9 p. 4 We can now use this interior operation to mimic the subset-logic definition of the implication: S \supset T := S^c \cup T . Hence to define the ditset \text{dit}(\sigma \Rightarrow \pi) , we first form \text{dit}(\sigma)^c \cup \text{dit}(\pi) but that is not a ditset, so the definition is: EXCERPT #CJP8VQ p. 4 \text{dit}(\sigma \Rightarrow \pi) = \text{int}[\text{dit}(\sigma)^c \cup \text{dit}(\pi)]. EXCERPT #MLTJJH p. 4 Like the graph-theoretic definition, this approach can also be used for the other operations. The interior operation isn't needed for the ditset treatment of the join since \text{dit}(\sigma \vee \pi) = \text{dit}(\sigma) \cup \text{dit}(\pi) , but the meet could be defined as \text{dit}(\sigma \wedge \pi) = \text{int}[\text{dit}(\sigma) \cap \text{dit}(\pi)] . EXCERPT #TZHU7C p. 4 Thus we have four definitions of the partition implication \sigma \Rightarrow \pi that are equivalent. DOCUMENT #C8FHDZ On implication and negation in partition logic SECTION #KS4RBJ 5. Relative negation in partition logic EXCERPT #TTFKUQ p. 4 To study relative negation, we take the ‘consequence’ \pi as fixed and then let the ‘antecedent’ \sigma vary in the \pi -negation \neg_\pi \sigma := \sigma \Rightarrow \pi . Another suggestion from intuitionistic logic is that the negated elements in a Heyting algebra form a Boolean algebra. In partition logic, this is trivially true for absolute negation since the negated elements form the two-element Boolean algebra. And it is also true for the relative \pi -negation as was suggested by viewing the implication \sigma \Rightarrow \pi as an indicator or characteristic function for the inclusion of the blocks of \pi in the blocks of \sigma . And the double \pi -negation \neg_\pi \neg_\pi \sigma = (\sigma \Rightarrow \pi) \Rightarrow \pi just interchanges the 0_B and 1_B so \pi -negation is like the usual negation of a subset represented by its indicator function (i.e., negation interchanges the zero-one values). 1 Thus the triple \pi -negation is the same as the single \pi -negation. For another partition \tau = \{D, D', \dots\} , the join \overset{\pi}{\neg}\sigma \vee \overset{\pi}{\neg}\tau = (\sigma \Rightarrow \pi) \vee (\tau \Rightarrow \pi) would have B discretized, i.e., turned into \mathbf{1}_B , iff B is contained in a block C \in \sigma or B is contained in a block D \in \tau , so it acts like the Boolean join or disjunction: \mathbf{0}_B \vee \mathbf{1}_B = \mathbf{1}_B \vee \mathbf{0}_B = \mathbf{1}_B \vee \mathbf{1}_B = \mathbf{1}_B and \mathbf{0}_B \vee \mathbf{0}_B = \mathbf{0}_B . Similarly, the meet \overset{\pi}{\neg}\sigma \wedge \overset{\pi}{\neg}\tau = (\sigma \Rightarrow \pi) \wedge (\tau \Rightarrow \pi) would have B discretized, i.e., turned into \mathbf{1}_B iff B is contained in a block C \in \sigma and B is contained in a block D \in \tau , so it acts like the Boolean conjunction. Thus all the partitions over U in the form of a \pi -negation \overset{\pi}{\neg}\sigma = \sigma \Rightarrow \pi form a Boolean algebra \mathcal{B}_\pi with \pi = \overset{\pi}{\neg}\mathbf{1} as the bottom element and \mathbf{1} = \pi \Rightarrow \pi as the top element. Since all the \pi -negated partitions (also called \pi -regular partitions) refine \pi , the Boolean algebra \mathcal{B}_\pi is contained in the upper segment [\pi, \mathbf{1}] and might be called the Boolean core \mathcal{B}_\pi of [\pi, \mathbf{1}] . EXCERPT #2ZGCCM p. 5 EXCERPT #HLYNAH p. 5 EXCERPT #NU42RQ p. 5 There is another construction of \mathcal{B}_\pi based on the fact that singleton blocks in \pi are already atomized so the implication \sigma \Rightarrow \pi essentially ignores the singletons of \pi . Those singletons are always contained in some block of \sigma so they should be discretized into singletons, but they are already singletons. If we let \pi_{ns} stand for the set of non-singleton blocks of \pi , then every \pi -negated formula \sigma \Rightarrow \pi is characterized by the set of non-singleton blocks B \in \pi_{ns} that were discretized, i.e., were assigned \mathbf{1}_B instead of \mathbf{0}_B by the implication \sigma \Rightarrow \pi viewed as an indicator function (for inclusion of blocks of \pi in blocks of \sigma ). Then it is easily seen that the powerset Boolean algebra \wp(\pi_{ns}) on the set of non-singleton blocks of \pi is (anti-)isomorphic to the Boolean core \mathcal{B}_\pi , i.e., EXCERPT #AFS2KN p. 5 \wp(\pi_{ns}) \cong \mathcal{B}_\pi. EXCERPT #95SCCV p. 5 Thus, we also have: \wp(\pi) \cong \mathcal{B}_\pi \times \prod_{\{u\} \in \pi} 2 , where 2 = \{0, 1\} . EXCERPT #MMGG3Y p. 5 The single \pi -negation \overset{\pi}{\neg}\sigma , the double \pi -negation \overset{\pi}{\neg}\overset{\pi}{\neg}\sigma , and the excluded middle \sigma \vee \overset{\pi}{\neg}\sigma are all partitions of special interest. The non-singleton blocks of \overset{\pi}{\neg}\sigma are the blocks B \in \pi that intersect two or more blocks of \sigma . Thus the non-singleton blocks of the double \pi -negation \overset{\pi}{\neg}\overset{\pi}{\neg}\sigma are the blocks B \in \pi that are contained in blocks of \sigma so \sigma \Rightarrow \overset{\pi}{\neg}\overset{\pi}{\neg}\sigma is a partition tautology and \sigma \lesssim \overset{\pi}{\neg}\overset{\pi}{\neg}\sigma . The double \pi -negation \overset{\pi}{\neg}\overset{\pi}{\neg}\sigma can be thought of as the \pi -closure of any \sigma inside of \mathcal{B}_\pi . Since \pi \lesssim \overset{\pi}{\neg}\overset{\pi}{\neg}\sigma , we also have that \sigma \vee \pi \lesssim \overset{\pi}{\neg}\overset{\pi}{\neg}\sigma . EXCERPT #VQ5SJ2 p. 5 Moreover, since the non-singleton blocks of \overset{\pi}{\neg}\sigma intersect two or more blocks of \sigma , the blocks (always non-singleton unless otherwise specified) in the excluded middle partition \sigma \vee \overset{\pi}{\neg}\sigma are all (strictly) smaller than the blocks of \pi so \pi \lesssim \sigma \vee \overset{\pi}{\neg}\sigma and thus \sigma \vee \pi \lesssim \sigma \vee \overset{\pi}{\neg}\sigma . And since the blocks of \sigma \vee \overset{\pi}{\neg}\sigma are strictly smaller than the blocks of \pi , no blocks of \pi are discretized in its \pi -negation, i.e., \overset{\pi}{\neg}(\sigma \vee \overset{\pi}{\neg}\sigma) = \pi . Thus the double \pi -negation of the excluded-middle partition is \mathbf{1} , i.e., \overset{\pi}{\neg}\overset{\pi}{\neg}(\sigma \vee \overset{\pi}{\neg}\sigma) is a partition tautology. While the excluded middle partition \sigma \vee \overset{\pi}{\neg}\sigma is not (in general) equal to \mathbf{1} (i.e., is not in general a partition tautology) and is not even in \mathcal{B}_\pi , it could be said to be \pi -dense in \mathbf{1} since its \pi -closure is \mathbf{1} . EXCERPT #7FLLBB p. 5 Since both the excluded middle partition \sigma \vee \overset{\pi}{\neg}\sigma and the double \pi -negation partition \overset{\pi}{\neg}\overset{\pi}{\neg}\sigma refine \sigma \vee \pi , they meet (greatest lower bound) (\sigma \vee \overset{\pi}{\neg}\sigma) \wedge \overset{\pi}{\neg}\overset{\pi}{\neg}\sigma must also refine \sigma \vee \pi . Moreover, that is an equality since the blocks of \sigma \vee \overset{\pi}{\neg}\sigma are the non-empty intersections C \cap B for C \in \sigma and B \in \pi where B is not contained in any C \in \sigma , and the blocks of \overset{\pi}{\neg}\overset{\pi}{\neg}\sigma are the blocks B contained in some C \in \sigma . Those non-singleton blocks are all disjoint, so there are no overlaps in the meet operation. Hence those blocks remain the same in the meet and they are precisely the blocks of the join \sigma \vee \pi , i.e., EXCERPT #TTBGV9 p. 5 \sigma \vee \pi = (\sigma \vee \overset{\pi}{\neg}\sigma) \wedge \overset{\pi}{\neg}\overset{\pi}{\neg}\sigma. DOCUMENT #C8FHDZ On implication and negation in partition logic SECTION #LEZCCF 6. Valid formulas EXCERPT #QLQFLJ p. 5 Since there are partition operations corresponding to all the Boolean operations [7], we can just write logical formulas using those logical operations without first specifying whether the variables stand for subsets EXCERPT #CBELAF p. 5 1 This is also true of the relative negated elements b \Rightarrow a in a Heyting algebra H since they are the negated elements in the upper segment \uparrow a = \{b \in H : a \leq b\} which is also a Heyting algebra [9, p. 15]. EXCERPT #44V9AG p. 6 EXCERPT #PVCNA2 p. 6 EXCERPT #BKPKY2 p. 6 or partitions (or open subsets as in the intuitionistic case) with the corresponding operations. Thus we can directly compare the valid formulas in the different logics. EXCERPT #YPKBDS p. 6 In the Boolean logic of subsets, a valid formula (or subset tautology ) is a formula (N.B., a formula, not a proposition) where no matter what subsets of the universe set U (where |U| \geq 1 ) are substituted for the variables, the whole formula evaluates to U , the top of the Boolean algebra of subsets on U . The fact that the same set of valid formulas is obtained if one only considers the two subsets of the one-element universe U = 1 is a theorem of subset logic (which was known to Boole). But today most, if not all, textbooks unfortunately ignore subset logic and present only that special case where U = 1 , "propositional logic," and then define a valid formula as a proposition that is a truth-table tautology. That common misconception that Boolean logic is just about propositions (or zero-one entities) rather than subsets seems to have retarded the development of the dual logic of partitions (since subsets have a dual, i.e., quotient sets or partitions, while propositions do not). Valid formulas in intuitionistic logic may be defined similarly with open subsets substituted for arbitrary subsets. EXCERPT #YHR7SC p. 6 In the logic of partitions on U , a valid formula or partition tautology is a formula where no matter what partitions on the universe U (where |U| \geq 2 ) are substituted for the variables, the whole formula evaluates to the discrete partition \mathbf{1} , the top of the algebra of partitions on U . 2 The usual focus on the propositional interpretation of logic leads to the usual assumption that a tautology is a valid statement or proposition. But in subset or partition logic, a tautology is a formula so that no matter what subsets or partitions respectively are substituted for the variables, then the result is the subset or partition that is the top of the respective lattice. The variables and formulas constructed from the variables do not stand for propositions but for subsets or partitions. EXCERPT #XWG2AT p. 6 There is a simple way to see that all partition tautologies are also subset tautologies, i.e., valid formulas of subset logic. Consider the partition algebra \Pi(2) on the two-element set 2 = \{0, 1\} . It has only two partitions, the discrete partition \mathbf{1} = \{\{0\}, \{1\}\} where 0 and 1 are distinguished, and the indiscrete partition \mathbf{0} = \{\{0, 1\}\} where they are not distinguished. The partition operations, such as join, meet, and implication, applied to those two partitions could be described in "truth tables" since \mathbf{0} and \mathbf{1} are the only partitions on 2. And those truth tables are the same as the Boolean subset operations on the two subsets of the one-element set. Hence we have an isomorphism between the partition algebra \Pi(2) on 2 and the power-set Boolean algebra \wp(1) = 2 for 1 as the one-element set. 3 Now consider any formula that is a valid formula in partition logic. Since it evaluates to \mathbf{1} for all partitions on any U where |U| \geq 2 , it does that for U = 2 , but then the isomorphism \Pi(2) \cong \wp(1) means that the same formula will be a truth table tautology in \wp(1) and thus it is a valid formula for subset logic in general. Hence all partition tautologies are subset tautologies. But the inclusion is strict. For instance, the law of excluded middle \sigma \vee \neg\sigma = \sigma \vee (\sigma \Rightarrow \mathbf{0}) is not a partition tautology since for any \sigma \neq \mathbf{0}, \mathbf{1} , \sigma \Rightarrow \mathbf{0} = \mathbf{0} , and \sigma \vee \neg\sigma = \sigma \vee \mathbf{0} = \sigma \neq \mathbf{1} . EXCERPT #5AWNV7 p. 6 The Boolean core \mathcal{B}_\pi of the upper segment [\pi, \mathbf{1}] for any partition \pi , provides a way to 'automatically' generate partition tautologies. Since \mathcal{B}_\pi is a Boolean algebra, any Boolean tautology comprised of \pi -negated partitions will also be a partition tautology. For instance, the law of excluded middle in \mathcal{B}_\pi has the form \frac{\pi}{\neg\pi} \vee \frac{\pi}{\neg\pi} \sigma which is the "weak law of excluded middle" in partition logic. It is a partition validity since it is a Boolean tautology that evaluates to \mathbf{1} no matter what partitions on U are substituted for \pi and \sigma . EXCERPT #J8KTXB p. 6 Conversely, given any formula using the connectives of \vee, \wedge, \Rightarrow , and the constants of \mathbf{0} and \mathbf{1} , its single \pi -negation transform is obtained by replacing each atomic variable \sigma by its single \pi -negation \frac{\pi}{\neg\sigma} = \sigma \Rightarrow \pi and by replacing the constant \mathbf{0} by \pi . The binary operations \vee, \wedge , and \Rightarrow as well as the constant \mathbf{1} all remain the same. For instance, the single \pi -negation transform of the excluded middle formula \sigma \vee \neg\sigma = \sigma \vee (\sigma \Rightarrow \mathbf{0}) is the partition tautology of the weak law of excluded middle for \pi -negation: EXCERPT #3F5G54 p. 6 (\sigma \Rightarrow \pi) \vee ((\sigma \Rightarrow \pi) \Rightarrow \pi) = \frac{\pi}{\neg\sigma} \vee \frac{\pi}{\neg\pi} \sigma. EXCERPT #ERL4RP p. 6 2 A system of semantic tableaux for partition logic is given in [2] but there is today no known Hilbert-style axiom system for the partition tautologies. EXCERPT #E9U2QU p. 6 3 There is essentially only one distinction in \Pi(2) (between \mathbf{0} and \mathbf{1} ). This is why when Spencer Brown [10] develops, in a rather esoteric way, the primary algebra of "the distinction," it turns out to be the two element Boolean algebra \wp(1) [11]. EXCERPT #QKC3ML p. 7 EXCERPT #P35UYX p. 7 EXCERPT #L4CR23 p. 7 Then the single \pi -negation transform of any classical tautology will still be a tautology but now expressed in \mathcal{B}_\pi and thus it is also a partition tautology. Thus all partition tautologies are ordinary Boolean logic tautologies, and any ordinary subset tautology transforms into a partition tautology via the single \pi -negation transform. EXCERPT #JPCRQL p. 7 The weak law of excluded middle \pi\sigma \vee \neg\pi\sigma is also an example of a partition tautology that is not an intuitionistic validity. And accumulation \pi \Rightarrow (\sigma \Rightarrow (\pi \wedge \sigma)) is valid in intuitionistic logic but not in partition logic. Since the lattice of partitions is the standard example of a non-distributive lattice while intuitionistic logic or Heyting algebras are distributive, the distributive laws are also examples of formulas that are valid in intuitionistic logic but not in partition logic. Thus there is no inclusion either way between partition and intuitionistic tautologies. EXCERPT #PF9BKN p. 7 Since partition lattices and their non-distributivity were known to Dedekind and some other European mathematicians such as Ernest Schröder, it was an embarrassing moment in American mathematics when the philosopher-mathematician, Charles Saunders Peirce, claimed to prove the distributivity of all lattices [12] but omitted the ‘proof’ as being too tedious. Europeans soon besieged him with examples of partition lattices including the simplest non-trivial one on a three-element set U = \{a, b, c\} . EXCERPT #FD6BRX p. 7 Figure 3: Partition lattice on U = {a, b, c}. The lattice has four levels. The top level is a single node labeled {{a},{b},{c}} = 1. The second level has three nodes: {{a,b},{c}}, {{a},{b,c}}, and {{b},{a,c}}. The third level has a single node labeled {{a,b,c}} = 0. Lines connect the top node to each of the three middle nodes, and each of the three middle nodes to the bottom node. EXCERPT #XMNHXF p. 7 Figure 3. Partition lattice on U = \{a, b, c\} EXCERPT #2GGKXT p. 7 Taking the three middle partitions, \pi = \{\{a, b\}, \{c\}\} , \sigma = \{\{a\}, \{b, c\}\} , and \tau = \{\{b\}, \{a, c\}\} , then: \pi \vee (\sigma \wedge \tau) = \pi \vee \mathbf{0} = \pi and (\pi \vee \sigma) \wedge (\pi \vee \tau) = \mathbf{1} \wedge \mathbf{1} = \mathbf{1} . If Peirce had known about the partition implication and the Boolean core \mathcal{B}_\pi , then he could at least have pointed out that the Boolean core \mathcal{B}_\pi for any \pi is distributive, and moreover any partition \varphi \in [\pi, \mathbf{1}] distributes across the Boolean core in the sense that: EXCERPT #GG2LEV p. 7 \begin{aligned}\varphi \vee \left( \pi\sigma \wedge \pi\tau \right) &= \left( \varphi \vee \pi\sigma \right) \wedge \left( \varphi \vee \pi\tau \right) \\ \varphi \wedge \left( \pi\sigma \vee \pi\tau \right) &= \left( \varphi \wedge \pi\sigma \right) \vee \left( \varphi \wedge \pi\tau \right).\end{aligned} DOCUMENT #C8FHDZ On implication and negation in partition logic SECTION #GNBDTD 7. Concluding remarks EXCERPT #LFDVDT p. 7 Our purpose has been to develop the notions of negation and implication (relative negation) in the logic of partitions. Since partition relations (ditsets) and equivalence relations (indit sets) are complementary in U \times U , every result in the logic of partitions has a complementary-dual result in the logic of equivalence relations so the latter is not really a different logic but a complementary way to view partition logic. There is a similar complementary-duality in intuitionistic logic between Heyting algebras (modelled by the open subsets of a topological space) and co-Heyting algebras [13] (modelled by the closed subsets). 4 Intuitionistic logic makes the symmetry-breaking choice to deal with Heyting algebras rather than co-Heyting algebras, and we have made the similar choice to develop the logic of partitions rather than the (‘anti-isomorphic’) logic of equivalence relations. For instance, the complementary-dual to the implication operation on partitions is the difference or subtraction operation on equivalence relations. The partition logic tautology of modus ponens has the customary form: (\sigma \wedge (\sigma \Rightarrow \pi)) \Rightarrow \pi , whereas the corresponding formula in the dual logic of EXCERPT #BQU56T p. 7 4 Our approach here is entirely semantic (i.e., no axiom system for partition logic) so there would seem to be no analogue of bi-intuitionistic or Heyting-Brouwer logic [14] since no partition relations are also equivalence relations—except in the ‘waste case’ of the empty partition on the empty set where both the ditset and indit set are empty. EXCERPT #E5CCYQ p. 8 EXCERPT #NTSXK2 p. 8 EXCERPT #6YDBUF p. 8 equivalence relations is the unfamiliar \pi - (\sigma \vee (\pi - \sigma)) . Hence we have made no independent development of the equivalence relation notions of difference or of negation as "difference from U \times U ." EXCERPT #T6E7D9 p. 8 No new logical operations on partitions, aside from join and meet, were defined throughout the twentieth century. The definition of the partition implication (or relative negation) in any of the many equivalent ways was the key to the development of the full logic of partitions. Why the delay? One reason is perhaps the fact that partition lattices are so general that any partition tautology or identity involving just the lattice operations and top and bottom, e.g., \mathbf{1} \wedge \pi = \pi or \mathbf{0} \vee \pi = \pi , are in fact identities that hold on all such lattices [15]. Thus the logic of general set partitions only becomes of independent interest by moving beyond the lattice operations on partitions. Alternatively, one could develop a 'logic' of equivalence relations sticking with only the lattice operations but specializing to certain types of equivalence relations [16]. But the main reason for the delay seems to be that the Boolean logic of subsets was and is presented in only the special case of the logic of propositions. And propositions, unlike subsets, do not have a category-theoretic dual concept. Hence twentieth century mathematical logicians were not even looking for the dual logic of quotient sets, equivalence relations, or partitions. EXCERPT #XPGYMY p. 8 Much of the modern development in "logic" is focused on the propositional interpretation of the variables which is a special case of the general logic of subsets of some given universe U (where U = 2 = \{0,1\} or \{F,T\} in the propositional case). Then attention has been focused on the many variations and sublogics of the Boolean logic of subsets. Partition logic is not about propositions at all and it is not a variation or sublogic of the Boolean logic of subsets. Partition logic is the category-theoretic dual to subset logic, and it is coequal to subset logic from the purely mathematical point of view (see, for instance, [17]). Hence our focus has been to address the topic of implication and negation in this little-known form of logic that is the mathematical coequal twin or dual to the Boolean logic of subsets. ### 20. Assistant tool calls #### Tool call: `generate_image` ```json { "prompt": "Create an elegant vertical cover artwork for a scholarly mathematics paper on partition logic, implication, and negation. Dark midnight-blue to black background with a luminous abstract finite set visualized as a constellation of small ivory and cyan points. Enclose the points in several overlapping translucent partition regions: some points grouped inside smooth amber contours, others split into individual glowing singleton cells, suggesting refinement from an indiscrete whole into discrete elements. Across the composition, a subtle implication arrow emerges through the geometry rather than as a literal typographic symbol: on the left, clustered blocks; toward the right, one selected block atomizes into separate points while another remains intact. Include a delicate complete-graph network in the background, with only a few edges brightly retained to suggest connected components and ditsets. Add a restrained visual motif of complementary positive/negative space to evoke negation. Sophisticated mathematical editorial design, precise clean geometry, soft glow, generous negative space, balanced composition, modern academic journal aesthetic, no equations, no words, no letters, no title, no logos, no watermark." } ``` ### 21. Tool result: generate_image GENERATED IMAGE #LBHXDZ URL: /images/LBHXDZ Model: gpt-image-2 Prompt: Create an elegant vertical cover artwork for a scholarly mathematics paper on partition logic, implication, and negation. Dark midnight-blue to black background with a luminous abstract finite set visualized as a constellation of small ivory and cyan points. Enclose the points in several overlapping translucent partition regions: some points grouped inside smooth amber contours, others split into individual glowing singleton cells, suggesting refinement from an indiscrete whole into discrete elements. Across the composition, a subtle implication arrow emerges through the geometry rather than as a literal typographic symbol: on the left, clustered blocks; toward the right, one selected block atomizes into separate points while another remains intact. Include a delicate complete-graph network in the background, with only a few edges brightly retained to suggest connected components and ditsets. Add a restrained visual motif of complementary positive/negative space to evoke negation. Sophisticated mathematical editorial design, precise clean geometry, soft glow, generous negative space, balanced composition, modern academic journal aesthetic, no equations, no words, no letters, no title, no logos, no watermark. ### 22. Assistant tool calls #### Tool call: `update_document_metadata` ```json { "cover_image_id": "LBHXDZ", "document_id": "C8FHDZ" } ``` ### 23. Tool result: update_document_metadata Updated metadata for #C8FHDZ: cover_image_id ### 24. Assistant ### What the paper is doing Ellerman’s starting point is surprisingly intuitive: ordinary Boolean logic is really the logic of **subsets**, but subsets have a mathematical dual—**quotient sets**, equivalently partitions. So partition logic should be regarded not as an eccentric alternative to classical logic, but as its lesser-known “dual sibling” (#BLF82B). A partition describes which elements are grouped together. Its logical content can instead be expressed through its **distinctions** or “dits”: ordered pairs lying in different blocks (#B7W87A, #ES3YR9). Refining a partition means making more distinctions. The paper’s central operation is partition implication, $\sigma\Rightarrow\pi$. The easiest way to picture it is to inspect each block of $\pi$: - if that block fits wholly inside a block of $\sigma$, split it into singletons; - otherwise, leave it intact. So the result records, block by block, how far $\sigma$ stands in the refinement relation to $\pi$. In particular, $$ \sigma\Rightarrow\pi=\mathbf 1 \quad\text{iff}\quad \sigma\preceq\pi. $$ That makes implication a kind of **refinement indicator** (#FH9NKH, #QBT84C). Ellerman then shows that the same operation can be defined in four equivalent ways: directly through blocks, by an adjunction, through graph connectivity, or through the interior of a relation on $U\times U$ (#AM4WKF, #RM3FBS, #CJP8VQ). The interesting twist concerns negation. Absolute negation is rather blunt: every nontrivial partition negates to the indiscrete “blob” $\mathbf 0$. This follows from the striking **Common-Dits Theorem**, according to which any two nonempty sets of distinctions overlap (#C53C8A, #2K99KN). Consequently, the richer notion is **relative negation**: $$ \neg_\pi\sigma=\sigma\Rightarrow\pi. $$ For each fixed $\pi$, these relatively negated partitions form a Boolean algebra—the **Boolean core** $\mathcal B_\pi$ inside the generally non-distributive partition lattice (#TTFKUQ, #AFS2KN). This is probably the paper’s nicest structural result: ordinary Boolean behavior has not disappeared; it survives locally inside partition logic. The final sections compare valid formulas. Every partition tautology is a Boolean tautology, but not conversely: ordinary excluded middle generally fails for partitions (#XWG2AT). Nevertheless, Boolean tautologies can be converted into partition tautologies by replacing variables with their relative negations (#J8KTXB, #L4CR23). Partition logic also differs genuinely from intuitionistic logic; neither system’s validities contain all of the other’s (#JPCRQL). Overall, it is a compact, largely semantic paper making a conceptual case: logic need not be only about truth-valued propositions. Once logic is understood more broadly as the algebra of subsets, its dual logic of partitions becomes visible—and implication is the operation that turns the old partition lattice into a full logical system (#T6E7D9, #XPGYMY). ### Cover artwork I generated a cover that translates the paper’s central construction into an abstract progression from grouped blocks to atomized singleton points, with graph edges and complementary negative space evoking implication, ditsets, and negation. **Artwork:** #LBHXDZ It is now attached as the cover of #C8FHDZ. ### 25. User let's also give it a good microabstract, and then let's look through the corpus for notions that can be thought of in terms of partitions... maybe for example the catchment basin partition of a landscape... ### 26. Assistant tool calls #### Tool call: `update_document_metadata` ```json { "document_id": "C8FHDZ", "micro_abstract": "Develops implication as a refinement-sensitive operation on set partitions, showing how relative negation yields local Boolean cores within the non-distributive algebra of partitions." } ``` #### Tool call: `list_documents` ```json {} ``` ### 27. Tool result: update_document_metadata Updated metadata for #C8FHDZ: micro_abstract ### 28. Tool result: list_documents FOLDERS - Living Structure, Wholeness & Beauty - Procedural Generation & Co-Creation - Roads, Trails & Movement - Terrain, Hydrology & Erosion - Water Simulation & Rendering Living Structure, Wholeness & Beauty (8) - #MH5J8D Beautimeter: Harnessing GPT for Assessing Architectural and Urban Beauty Based on the 15 Properties of Living Structure - 2025 | Bin Jiang | AI | 12 pp. | doi:10.3390/ai6040074 Micro abstract: Presents Beautimeter, a GPT-based tool that scores buildings and urban scenes against Christopher Alexander’s 15 properties of living structure to assess their coherence and beauty. - #XW22YY Generative Codes: The Path to Building Welcoming, Beautiful, Sustainable Neighborhoods - 2005 | Brian Hanson, Christopher Alexander, Maggie Moore Alexander, Michael Mehaffy, Randall Schmidt | Center for Environmental Structure | 21 pp. Micro abstract: Argues that living neighborhoods arise from generative codes: ordered, participatory steps that let buildings and public spaces unfold from local people, land, and context. - #SKRF4C Geography as a Science of the Earth’s Surface Founded on the Third View of Space - 2022 | Bin Jiang | Annals of GIS | 14 pp. | doi:10.1080/19475683.2021.1966502 Micro abstract: Recasts geography around an organismic view of space, using scaling and spatial dependence to understand—and deliberately create—places with greater living structure. - #PXG56P Harmony-Seeking Computations: A Science of Non-Classical Dynamics Based on the Progressive Evolution of the Larger Whole - 2009 | Christopher Alexander | Unpublished manuscript | 66 pp. Micro abstract: Proposes harmony-seeking computation as a creative process that repeatedly strengthens latent centers in a configuration while preserving and deepening the larger whole. - #MJKTBB Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space - 2023 | Bin Jiang, Chris de Rijke | Annals of the American Association of Geographers | 19 pp. | doi:10.1080/24694452.2023.2178376 Micro abstract: Measures an image’s structural beauty by recursively extracting its nested substructures, revealing a compact hierarchy that also captures visual saliency. - #3XSLTA Structural Beauty: A Structure-Based Computational Approach to Quantifying the Beauty of an Image - 2021 | Bin Jiang, Chris de Rijke | Journal of Imaging | 15 pp. | doi:10.3390/jimaging7050078 Micro abstract: Proposes a quantitative measure of structural beauty based on how many substructures an image contains and how strongly they form a hierarchy across scales. - #ZU8GZV Structure-Preserving Transformations - 2002 | Christopher Alexander | The Nature of Order, Book Two: The Process of Creating Life | 4 pp. | doi:10.2307/j.ctv27ftw6c.5 Micro abstract: Explains structure-preserving transformations: incremental changes that extend the centers and relationships already present in a place rather than weakening its wholeness. - #BYG3BQ Wholeness as a Hierarchical Graph to Capture the Nature of Space - 2015 | Bin Jiang | International Journal of Geographical Information Science | 14 pp. | doi:10.1080/13658816.2015.1038542 Micro abstract: Models spatial wholeness as a hierarchical graph of mutually reinforcing centers, using PageRank and scaling depth to quantify the life of parts and wholes. Procedural Generation & Co-Creation (6) - #4TH488 Explainable AI for Designers: A Human-Centered Perspective on Mixed-Initiative Co-Creation - 2018 | Antonios Liapis, G. Michael Youngblood, Jichen Zhu, Rafael Bidarra, Sebastian Risi | 2018 IEEE Conference on Computational Intelligence and Games (CIG) | 8 pp. | doi:10.1109/CIG.2018.8490433 Micro abstract: Defines explainable AI for game designers, mapping co-creative systems by their explainability, initiative, and domain overlap so explanations serve concrete design tasks. - #9NQ94D Extracting Physics from Blended Platformer Game Levels - 2020 | Adam Summerville, Anurag Sarkar, Joseph C. Osborn, Sam Snodgrass | Joint Proceedings of the AIIDE 2020 Workshops (CEUR Workshop Proceedings, Vol. 2862) | 7 pp. Micro abstract: Infers playable jump physics from generated platformer levels, including hybrid physics models for levels that blend the geometry and style of multiple games. - #7GR3AQ Procedural Content Generation through Quality Diversity - 2019 | Ahmed Khalifa, Antonios Liapis, Daniele Gravina, Georgios N. Yannakakis, Julian Togelius | 2019 IEEE Conference on Games (CoG) | 8 pp. | doi:10.1109/CIG.2019.8848053 Micro abstract: Argues for quality-diversity algorithms in procedural generation, producing broad collections of varied, playable content while exposing the design space for exploration and co-creation. - #CQBDX4 Procedural Content Generation via Machine Learning (PCGML) - 2018 | Aaron Isaksen, Adam Summerville, Amy K. Hoover, Andy Nealen, Christoffer Holmgård, Julian Togelius, Matthew Guzdial, Sam Snodgrass | IEEE Transactions on Games | 15 pp. | doi:10.1109/TG.2018.2846639 Micro abstract: Defines and surveys PCGML: generating functional game content directly from models trained on existing examples, with uses spanning creation, completion, repair, critique, and compression. - #WZ8DHP Runtime Evaluation of Procedural Content Generation in an Endless Runner Game Using Autonomous Agents - 2026 | Rishabh Kar | arXiv | 25 pp. | doi:10.48550/arXiv.2605.01783 Micro abstract: Integrates procedural generation and validation in an endless runner, using aerial and ground agents to detect blocked or unnavigable content before the player reaches it. - #NRBMD5 Towards Friendly Mixed Initiative Procedural Content Generation: Three Pillars of Industry - 2020 | Frederic Fol Leymarie, Gorm Lai, William Latham | Proceedings of the International Conference on the Foundations of Digital Games (FDG '20) | 4 pp. | doi:10.1145/3402942.3402946 Micro abstract: Distills three requirements for industry-friendly co-creative PCG tools: preserve designer control, keep feedback loops short, and fit into existing production pipelines. Roads, Trails & Movement (7) - #G3TBNG A Sequential Two-Step Algorithm for Fast Generation of Vehicle Racing Trajectories - 2016 | J. Christian Gerdes, John Subosits, Nitin R. Kapania | Journal of Dynamic Systems, Measurement, and Control | 12 pp. | doi:10.1115/1.4033311 Micro abstract: Generates near-optimal racing trajectories quickly by alternating between a minimum-time speed profile and a convex path update that reduces curvature. - #B6P8L4 Active walker model for the formation of human and animal trail systems - 1997 | Dirk Helbing, Frank Schweitzer, Joachim Keltsch, Péter Molnár | Physical Review E | 34 pp. | doi:10.1103/physreve.56.2527 Micro abstract: Models trail systems as self-organization: walkers reinforce attractive routes while unused traces fade, producing dendritic ant trails and low-detour pedestrian networks. - #V4TQYB Interactive procedural street modeling - 2008 | Eugene Zhang, Gregory Esch, Guoning Chen, Pascal Müller, Peter Wonka | ACM Transactions on Graphics | 10 pp. | doi:10.1145/1360612.1360702 Micro abstract: Lets designers generate and edit large street networks through tensor fields, combining procedural speed with brush-like global and local control over street patterns. - #UYLTYJ Modelling the Evolution of Human Trail Systems - 1997 | Dirk Helbing, Joachim Keltsch, Péter Molnár | Nature | 11 pp. | doi:10.1038/40353 Micro abstract: Shows how pedestrian trails emerge through feedback between destination-seeking walkers, existing paths, and vegetation recovery, yielding a compromise between directness and shared infrastructure. - #GY93FG Mountain Trail Formation and the Active Walker Model - 2009 | J. P. Hague, S. J. Gilks | International Journal of Modern Physics C | 22 pp. | doi:10.1142/S0129183109014059 Micro abstract: Extends the active-walker model to steep terrain, explaining zigzag mountain trails through slope avoidance, directional persistence, and mutual reinforcement by ascending and descending walkers. - #LXV9AT Principles of Trail Layout and Design - 2019 | California State Parks | California State Parks Trails Handbook | 64 pp. Micro abstract: A field-oriented guide to durable trail design, emphasizing curvilinear alignment, natural drainage, sustainable grades, control points, and close reading of landform and soils. - #XDEFZS Procedural Generation of Roads - 2010 | A. Peytavie, E. Galin, E. Guérin, N. Maréchal | Computer Graphics Forum | 10 pp. | doi:10.1111/j.1467-8659.2009.01612.x Micro abstract: Automatically routes and constructs roads with an anisotropic shortest-path method that weighs slope and obstacles while treating surface segments, bridges, and tunnels consistently. Terrain, Hydrology & Erosion (6) - #NV2YRW FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation - 2024 | Aryamaan Jain, Bernhard Kerbl, Brandon Finley, Guillaume Cordonnier, James Gain | Computer Graphics Forum | 13 pp. | doi:10.1111/cgf.15243 Micro abstract: A GPU framework for routing surface flow through terrain and its depressions fast enough to make erosion, river, lake, and ecosystem simulations interactive. - #96ZMGK Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion - 2016 | Adrien Peytavie, Bedrich Benes, Guillaume Cordonnier, Jean Braun, Marie-Paule Cani, Éric Galin, Éric Guérin | Computer Graphics Forum | 11 pp. | doi:10.1111/cgf.12820 Micro abstract: Generates large, controllable mountain terrains by coupling user-painted tectonic uplift with fluvial erosion, then turning the resulting stream graph into detailed landforms. - #DWXKYQ Physically-based analytical erosion for fast terrain generation - 2024 | Boris Gailleton, Guillaume Cordonnier, Petros Tzathas, Philippe Steer | Computer Graphics Forum | 14 pp. | doi:10.1111/cgf.15033 Micro abstract: Turns the stream power law into an interactive terrain tool, replacing thousands of erosion time steps with analytical solutions and a direct control for landscape age. - #MTDKDE Priority-Flood: An Optimal Depression-Filling and Watershed-Labeling Algorithm for Digital Elevation Models - 2014 | Clarence Lehman, David Mulla, Richard Barnes | Computers & Geosciences | 17 pp. | doi:10.1016/j.cageo.2013.04.024 Micro abstract: Introduces Priority-Flood, a simple, optimal algorithm that removes drainage-blocking depressions from elevation models and can also derive watersheds and flow directions. - #AK7NGE Procedural Riverscapes - 2019 | A. Peytavie, B. Benes, E. Galin, E. Guérin, J. Gain, T. Dupont, Y. Cortial | Computer Graphics Forum | 12 pp. | doi:10.1111/cgf.13814 Micro abstract: Builds editable, animated riverscapes from bare terrain by carving hydrologically plausible channels and blending real-time procedural water primitives instead of simulating fluids. - #DMTA8Y Terrain Generation Using Procedural Models Based on Hydrology - 2013 | Adrien Peytavie, Bedřich Beneš, Jean-David Génevaux, Éric Galin, Éric Guérin | ACM Transactions on Graphics | 10 pp. | doi:10.1145/2461912.2461996 Micro abstract: Generates controllable, multiscale terrain from a sketched drainage network, representing rivers and landforms as an editable hierarchy of continuous procedural primitives. Water Simulation & Rendering (12) - #RBS5K6 A Layered Particle-Based Fluid Model for Real-Time Rendering of Water - 2010 | Daniel Scherzer, Florian Bagar, Michael Wimmer | Computer Graphics Forum | 7 pp. | doi:10.1111/j.1467-8659.2010.01734.x Micro abstract: Renders particle-based water and volumetric foam in real time using perspective-aware surface smoothing, physically guided foam formation, and layered depth compositing. - #C4AY2M A Survey of Ocean Simulation and Rendering Techniques in Computer Graphics - 2011 | B. Crespin, D. Ghazanfarpour, E. Darles, J.-C. Gonzato | Computer Graphics Forum | 17 pp. | doi:10.1111/j.1467-8659.2010.01828.x Micro abstract: Surveys ocean graphics from spectral deep-water models to near-shore fluid simulation, then covers the foam, spray, and light transport needed for convincing rendering. - #WZMZGY Advected river textures - 2009 | Dirk Arnold, Stephen Brooks, Tim Burrell | Computer Animation and Virtual Worlds | 11 pp. | doi:10.1002/cav.288 Micro abstract: Combines a 2D Navier–Stokes solver, hydrostatic pressure columns, and advected procedural textures to render detailed, terrain-responsive rivers at real-time frame rates. - #92XRH7 Lagrangian Texture Advection: Preserving both Spectrum and Velocity Field - 2011 | Qizhi Yu, E. Bruneton, F. Neyret, N. Holzschuch | IEEE Transactions on Visualization and Computer Graphics | 13 pp. | doi:10.1109/tvcg.2010.263 Micro abstract: Advects fluid textures with deformable particle grids, preserving both the input texture’s visual spectrum and exact motion along the velocity field without cumulative stretching. - #8SERGP Real-time Breaking Waves for Shallow Water Simulations - 2007 | Markus Gross, Matthias Müller-Fischer, Nils Thürey, Simon Schirm | 15th Pacific Conference on Computer Graphics and Applications (Pacific Graphics 2007) | 8 pp. | doi:10.1109/PG.2007.33 Micro abstract: Adds real-time overturning waves to shallow-water heightfields by detecting steep fronts and spawning connected particle sheets that collapse into splashes and foam. - #CWC7H9 Real-time Rendering of Enhanced Shallow Water Fluid Simulations - 2013 | Antonio Susín, Jesús Ojeda | Computers & Graphics | 9 pp. Micro abstract: Builds a real-time rendering pipeline for shallow-water simulations, adding fine surface detail, advected foam, photon-based caustics, and screen-space reflection and refraction. - #MVUJ8Z Real-time Rendering of River Networks - 2010 | Quintijn Hendrickx, Rafael Bidarra, Ruben M. Smelik | Proceedings of the ACM SIGGRAPH Symposium on Interactive 3D Graphics and Games | 1 pp. Micro abstract: Renders branching river networks efficiently with quadratic Bézier curves, GPU distance fields, and streaming normal maps instead of dense geometry or particle simulation. - #5MGCZ5 Real-time River Representation by Dynamic Control of Data on Waves - 2008 | Makoto Kosugi, Nobuhiko Mukai, Yasuhiro Kato | 4 pp. | doi:10.3169/itej.62.2063 Micro abstract: Dynamically switches river-wave models by viewing distance, preserving nearby reflection and wave detail while retaining wind-driven motion across the full landscape. - #XDESU9 Scalable real‐time animation of rivers - 2009 | Eric Bruneton, Fabrice Neyret, Nicolas Holzschuch, Qizhi Yu | Computer Graphics Forum | 11 pp. | doi:10.1111/j.1467-8659.2009.01363.x Micro abstract: Animates rivers across vast terrains by computing steady flow locally and advecting fine surface detail only where visible, at a screen-space sampling density. - #869NHK Very Fast Real-Time Ocean Wave Foam Rendering Using Halftoning - 2011 | Ian Parberry, Jennifer R. Alford, Mary Yingst | Proceedings of the 6th International North American Conference on Intelligent Games and Simulation (GAMEON-NA) | 7 pp. Micro abstract: Uses a spectrally controlled halftone mask to make texture-based ocean foam dissipate as clumped bubbles that pop, rather than merely fading transparently. - #A2QB8L Water Flow in Portal 2 - 2010 | Alex Vlachos | ACM SIGGRAPH 2010 Course: Advances in Real-Time Rendering in 3D Graphics and Games | 54 pp. Micro abstract: Explains Portal 2’s production-friendly water shader: artist-painted flow maps advect layered normal maps, while offsets and noise suppress repetition and pulsing on limited hardware. - #PBM2TC Water surface wavelets - 2018 | Chris Wojtan, Matthias Müller-Fischer, Miles Macklin, Nuttapong Chentanez, Stefan Jeschke, Tomáš Skřivan | ACM Transactions on Graphics | 13 pp. | doi:10.1145/3197517.3201336 Micro abstract: Represents water waves as spatially varying directional amplitudes, enabling vast interactive simulations with fine detail, moving obstacles, solid coupling, and direct artistic control. Unfiled (1) - #C8FHDZ On implication and negation in partition logic - 2025 | , David Ellerman | Open Journal of Mathematical Sciences | 9 pp. | doi:10.30538/oms2025.0250 ### 29. Assistant tool calls #### Tool call: `search_text` ```json { "limit": 10, "query": "watershed catchment basin drainage divide watershed labeling" } ``` #### Tool call: `search_text` ```json { "limit": 10, "query": "partition segmentation regions zones classification clusters" } ``` #### Tool call: `search_text` ```json { "limit": 10, "query": "flow direction each cell outlet basin label" } ``` #### Tool call: `search_text` ```json { "limit": 10, "query": "substructures centers hierarchy image segmentation recursively" } ``` #### Tool call: `search_text` ```json { "limit": 10, "query": "road network regions blocks districts tensor field" } ``` #### Tool call: `search_text` ```json { "limit": 10, "query": "river network drainage network tributary hierarchy" } ``` ### 30. Tool result: search_text Exact matches 1. Source: Procedural Riverscapes (#AK7NGE), A. Peytavie, B. Benes, E. Galin, E. Guérin, J. Gain, T. Dupont, Y. Cortial, p. 3 Context: #JJE8HN Procedural Riverscapes #Z6DVCN 4. River Network Graph Matching excerpt #Y3HUSK: The labeling of graph edges is more involved, since classification by Rosgen type [Ros94] requires a computation of the following river properties: segment-based river flow, stream power, and the Horton-Strahler number. First, river flow \phi_{ij} is averaged over the cells occupied by the edge. Then, the stream power, which captures the erosive action of water flowing in the river [CBC*16], is calculated based on slope S_{ij} and drainage area A_{ij} per edge cell as: 2. Source: Physically-based analytical erosion for fast terrain generation (#DWXKYQ), Boris Gailleton, Guillaume Cordonnier, Petros Tzathas, Philippe Steer, p. 2 Context: #JDHNVB Physically-based analytical erosion for fast terrain generation #QW6CT7 3. Background and overview #MYA2EJ 3.1. Landscape dynamics in geology Matching excerpt #ESC38J: A first common modeling approximation is to consider surface evolution as a detachment-limited process (as opposed to transport-limited) where the evolution of surface elevation z is directly related to the competition between rock uplift and erosion rate, and not to the capacity of the rivers to transport or deposit sediments [How94]. In this setting, erosion is generally described using the stream power incision model [HK83, How94, WT99, Lag14], where erosion rate is a power law of the surface slope \|\nabla z\| , and drainage area A that acts as a proxy for the river discharge. The drainage area A(\mathbf{x}) is defined at a position \mathbf{x} as the area of the drainage basin - or catchment - upstream of \mathbf{x} . Coupled with the uplift u , the Stream Power Law expresses the rate of change of surface elevation: 3. Source: Principles of Trail Layout and Design (#LXV9AT), California State Parks, p. 31 Context: #HE95FY Chapter 5. Principles of Trail Layout and Design #EEPQMJ 5.7. Trail Layout #XSQ2CN 5.7.3. Maximum Sustainable Linear Grades #74WYLZ 5.7.3.3. Rainfall Intensity Matching excerpt #UG2WS4: The intensity of rainfall can affect the performance of a trail's surface, especially where the runoff coefficient is high due to up slope conditions, such as the amount of exposed bed rock in the watershed, a lack of vegetative cover, road building, grazing, or recent fire activity in the watershed. High rainfall intensity can generate a significant runoff response up slope, which can impact drainage structures and trail surfaces. Drainage structures need to be designed and constructed to accommodate this runoff and the linear grades need to be adjusted to reduce the possibility of rilling caused by increased sheet flow. 4. Source: Priority-Flood: An Optimal Depression-Filling and Watershed-Labeling Algorithm for Digital Elevation Models (#MTDKDE), Clarence Lehman, David Mulla, Richard Barnes, p. 0 Matching excerpt #JK4NUD: Cite as Barnes, Lehman, Mulla. “Priority-Flood: An Optimal Depression-Filling and Watershed-Labeling Algorithm for Digital Elevation Models”. Computers & Geosciences . Vol 62, Jan 2014, pp 117–127. doi: “10.1016/j.cageo.2013.04.024”. 5. Source: Priority-Flood: An Optimal Depression-Filling and Watershed-Labeling Algorithm for Digital Elevation Models (#MTDKDE), Clarence Lehman, David Mulla, Richard Barnes, p. 2 Context: #XJ3LVD Priority-Flood: An Optimal Depression-Filling and Watershed-Labeling Algorithm for Digital Elevation Models #C5TMDU 3. The Priority-Flood Algorithm #CUHFR9 3.1. History Matching excerpt #ZZD5A8: The most accurate method for determining watershed boundaries involves placing a person familiar with the nuances of contour maps at a drafting table to manually interpret drainage basins. 6. Source: Priority-Flood: An Optimal Depression-Filling and Watershed-Labeling Algorithm for Digital Elevation Models (#MTDKDE), Clarence Lehman, David Mulla, Richard Barnes, p. 7 Context: #XJ3LVD Priority-Flood: An Optimal Depression-Filling and Watershed-Labeling Algorithm for Digital Elevation Models #ZVXSCH 4. Ordering Matching excerpt #CKT5LH: Fortunately, for depression filling with \epsilon = 0 , it does not matter whether the priority queue has a total order or a strict weak order, as both will produce the same results. However, if \epsilon \neq 0 , or if Priority-Flood is used for watershed labeling or to determine flow directions directly, then the ordering is important and a total order is the best choice. 7. Source: Priority-Flood: An Optimal Depression-Filling and Watershed-Labeling Algorithm for Digital Elevation Models (#MTDKDE), Clarence Lehman, David Mulla, Richard Barnes, p. 12 Context: #XJ3LVD Priority-Flood: An Optimal Depression-Filling and Watershed-Labeling Algorithm for Digital Elevation Models #4K4XN3 7. Variants #LFYVTV 7.3. Watershed Labeling Matching excerpt #NNC37P: Watershed labeling applies a common label—such as an integer number—to all cells which drain to a given outlet. The algorithm (Alg. 5) for this works in much the same way as the improved Priority-Flood (Alg. 2). The DEM is flooded inwards from its edges with the lowest cell always being processed first. Rather than filling cells in depressions, this algorithm merely prioritizes them to the level of their outlet. 8. Source: Priority-Flood: An Optimal Depression-Filling and Watershed-Labeling Algorithm for Digital Elevation Models (#MTDKDE), Clarence Lehman, David Mulla, Richard Barnes, p. 14 Context: #XJ3LVD Priority-Flood: An Optimal Depression-Filling and Watershed-Labeling Algorithm for Digital Elevation Models #9D8AS4 8. Coda Matching excerpt #477DMT: Algorithm 5 IMPROVED PRIORITY-FLOOD+WATERSHED LABELS: This variation of the IMPROVED PRIORITY-FLOOD follows the work of Beucher and Meyer [1992] and Beucher and Beucher [2011] . It applies a common label to all cells draining to an outlet. Line 21 should be interpreted as pushing a copy of the cell's coordinates into Pit with the copy's z -value set to c.z . If simultaneous watershed labeling and depression-filling is desired, change the original z value of n to c.z before making the copy. Upon entry , (1) DEM contains the elevations of every cell or the value NODATA for cells not part of the DEM. At exit , (1) Labels contains a label for every cell or the value NODATA for cells not part of the DEM. (2) All cells which drain to a common point at the edge of the DEM bear the same label. 9. Source: Priority-Flood: An Optimal Depression-Filling and Watershed-Labeling Algorithm for Digital Elevation Models (#MTDKDE), Clarence Lehman, David Mulla, Richard Barnes, p. 1 Context: #XJ3LVD Priority-Flood: An Optimal Depression-Filling and Watershed-Labeling Algorithm for Digital Elevation Models #Z3MQUT 2. Alternative Algorithms Matching excerpt #GDQUCW: Arge et al. [2003] describes a specialized O(n \log_2 n) procedure to perform watershed labeling, determine flow directions, and calculate flow accumulation on massive grids in situations where I/O must be minimized. Although parts of this procedure share sufficient similarities with Priority-Flood to be considered related, the combinations of algorithms used and the way in which they are specialized place it outside the scope of this paper. It is expected that the algorithm by Arge will run slower than that described here due to the greater overhead involved in explicit data management; however, this efficiency of memory may allow the algorithm by Arge to run better in situations where memory is limited. 10. Source: FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation (#NV2YRW), Aryamaan Jain, Bernhard Kerbl, Brandon Finley, Guillaume Cordonnier, James Gain, p. 3 Context: #6UY46T FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation #SD69QC 3. Overview #CPNMBK 3.2. Depression routing Matching excerpt #A6G5PJ: We are now in a position to define what a depression means in our context. As a preliminary, we use the term basin for the set of terrain cells that share the same stream tree, effectively all belonging to the same catchment area and channeling water to a single root cell. If this root cell is an outflow, then the basin is classified as an outflow basin, otherwise, it constitutes a depression (Figure 2 (c)). Approximate matches 1. Source: Terrain Generation Using Procedural Models Based on Hydrology (#DMTA8Y), Adrien Peytavie, Bedřich Beneš, Jean-David Génevaux, Éric Galin, Éric Guérin, p. 5 Context: #RULAFW Terrain Generation Using Procedural Models Based on Hydrology #T6FSBF 5 River Classification #TRR58W 5.1 Segmentation and Elevation of Crests Score: 0.028 Related excerpt #WL3SCW: Watersheds are associated with each water outlet s of a cell V_j and are defined as the set of upstream connected cells V_k \in V . 2. Source: Priority-Flood: An Optimal Depression-Filling and Watershed-Labeling Algorithm for Digital Elevation Models (#MTDKDE), Clarence Lehman, David Mulla, Richard Barnes, p. 12 Context: #XJ3LVD Priority-Flood: An Optimal Depression-Filling and Watershed-Labeling Algorithm for Digital Elevation Models #4K4XN3 7. Variants #LFYVTV 7.3. Watershed Labeling Score: 0.029 Related excerpt #NNC37P: Watershed labeling applies a common label—such as an integer number—to all cells which drain to a given outlet. The algorithm (Alg. 5) for this works in much the same way as the improved Priority-Flood (Alg. 2). The DEM is flooded inwards from its edges with the lowest cell always being processed first. Rather than filling cells in depressions, this algorithm merely prioritizes them to the level of their outlet. 3. Source: Priority-Flood: An Optimal Depression-Filling and Watershed-Labeling Algorithm for Digital Elevation Models (#MTDKDE), Clarence Lehman, David Mulla, Richard Barnes, p. 12 Context: #XJ3LVD Priority-Flood: An Optimal Depression-Filling and Watershed-Labeling Algorithm for Digital Elevation Models #4K4XN3 7. Variants #LFYVTV 7.3. Watershed Labeling Score: 0.029 Related excerpt #L6E5XU: Following execution, watershed boundaries may be identified by locating adjacent cells with differing labels. To mark the border cells, consistently choose either the cell with the lower label, the cell with the higher label, or both. 4. Source: Priority-Flood: An Optimal Depression-Filling and Watershed-Labeling Algorithm for Digital Elevation Models (#MTDKDE), Clarence Lehman, David Mulla, Richard Barnes, p. 12 Context: #XJ3LVD Priority-Flood: An Optimal Depression-Filling and Watershed-Labeling Algorithm for Digital Elevation Models #4K4XN3 7. Variants #LFYVTV 7.3. Watershed Labeling Score: 0.028 Related excerpt #HBWRR5: Watershed outlets are identified as being unlabeled cells adjacent to a NoDATA cell. These cells are given a unique label which then floods inwards to cover all the cells in the watershed. 5. Source: Priority-Flood: An Optimal Depression-Filling and Watershed-Labeling Algorithm for Digital Elevation Models (#MTDKDE), Clarence Lehman, David Mulla, Richard Barnes, p. 2 Context: #XJ3LVD Priority-Flood: An Optimal Depression-Filling and Watershed-Labeling Algorithm for Digital Elevation Models #C5TMDU 3. The Priority-Flood Algorithm #CUHFR9 3.1. History Score: 0.026 Related excerpt #ZZD5A8: The most accurate method for determining watershed boundaries involves placing a person familiar with the nuances of contour maps at a drafting table to manually interpret drainage basins. 6. Source: Priority-Flood: An Optimal Depression-Filling and Watershed-Labeling Algorithm for Digital Elevation Models (#MTDKDE), Clarence Lehman, David Mulla, Richard Barnes, p. 14 Context: #XJ3LVD Priority-Flood: An Optimal Depression-Filling and Watershed-Labeling Algorithm for Digital Elevation Models #9D8AS4 8. Coda Score: 0.025 Related excerpt #477DMT: Algorithm 5 IMPROVED PRIORITY-FLOOD+WATERSHED LABELS: This variation of the IMPROVED PRIORITY-FLOOD follows the work of Beucher and Meyer [1992] and Beucher and Beucher [2011] . It applies a common label to all cells draining to an outlet. Line 21 should be interpreted as pushing a copy of the cell's coordinates into Pit with the copy's z -value set to c.z . If simultaneous watershed labeling and depression-filling is desired, change the original z value of n to c.z before making the copy. Upon entry , (1) DEM contains the elevations of every cell or the value NODATA for cells not part of the DEM. At exit , (1) Labels contains a label for every cell or the value NODATA for cells not part of the DEM. (2) All cells which drain to a common point at the edge of the DEM bear the same label. 7. Source: Priority-Flood: An Optimal Depression-Filling and Watershed-Labeling Algorithm for Digital Elevation Models (#MTDKDE), Clarence Lehman, David Mulla, Richard Barnes, p. 13 Context: #XJ3LVD Priority-Flood: An Optimal Depression-Filling and Watershed-Labeling Algorithm for Digital Elevation Models #9D8AS4 8. Coda Score: 0.023 Related excerpt #HESQ5R: The Priority-Flood algorithm is also versatile. It can be used to label watersheds (Alg. 5) as well as to determine flow directions either by terrain increments (Alg. 3) or by carving depressions (Alg. 4). 8. Source: Priority-Flood: An Optimal Depression-Filling and Watershed-Labeling Algorithm for Digital Elevation Models (#MTDKDE), Clarence Lehman, David Mulla, Richard Barnes, p. 3 Context: #XJ3LVD Priority-Flood: An Optimal Depression-Filling and Watershed-Labeling Algorithm for Digital Elevation Models #C5TMDU 3. The Priority-Flood Algorithm #CUHFR9 3.1. History Score: 0.022 Related excerpt #UXZQG4: Meyer [1994] gives an overview of many existing methods to derive watershed boundaries. He identifies plateaus and regions of equal elevation as being potentially problematic, but solves the problem by using first-in, first-out (FIFO) queues. This is analogous to the total order priority queues discussed later in this paper. 9. Source: FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation (#NV2YRW), Aryamaan Jain, Bernhard Kerbl, Brandon Finley, Guillaume Cordonnier, James Gain, p. 5 Context: #6UY46T FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation #J5PECC 5. Depression routing #UJCLXC Algorithm 2: Propagation of basin identifiers Score: 0.028 Related excerpt #C9XNSJ: Basin identification. Our algorithm begins by segmenting out each stream tree and associating with its cells a unique basin identifier (hereafter referred to as a basin-id , or bid in the pseudo-code of Algorithm 2). To this end, we assign to each non-outflow local minima a unique identifier – leaving a common outflow identifier for all outflow basins – and use pointer jumping to copy this identifier in parallel to all upstream nodes of its stream tree (Algorithm 2). 10. Source: FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation (#NV2YRW), Aryamaan Jain, Bernhard Kerbl, Brandon Finley, Guillaume Cordonnier, James Gain, p. 3 Context: #6UY46T FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation #SD69QC 3. Overview #CPNMBK 3.2. Depression routing Score: 0.023 Related excerpt #A6G5PJ: We are now in a position to define what a depression means in our context. As a preliminary, we use the term basin for the set of terrain cells that share the same stream tree, effectively all belonging to the same catchment area and channeling water to a single root cell. If this root cell is an outflow, then the basin is classified as an outflow basin, otherwise, it constitutes a depression (Figure 2 (c)). ### 31. Tool result: search_text Exact matches 1. Source: Procedural Content Generation through Quality Diversity (#7GR3AQ), Ahmed Khalifa, Antonios Liapis, Daniele Gravina, Georgios N. Yannakakis, Julian Togelius, p. 3 Context: #DZ9GFC Procedural Content Generation through Quality Diversity #W6TBRK IV. CASES OF QUALITY DIVERSITY IN PCG #ZF4PWP A. Generation of 2D and 3D Objects Matching excerpt #964T2S: MAP-Elites was used to generate 2D images [37] and 3D objects [38]. Similar to DeLeNoX [16], this work combines quality-diversity search and machine learning. A Deep Neural Network (DNN) is first trained on classifying real-world images and then combined with MAP-Elites to generate 2D images or 3D objects. The classification output of the DNN distinguishes between 1000 different classes of images, which is used as diversity characterization of the input. Specifically, MAP-Elites uses the classification output of the DNN as a partition of the search space and tries to optimize every bin based on the confidence of the DNN. 2. Source: On implication and negation in partition logic (#C8FHDZ), , David Ellerman, p. 1 Context: #L8JZAG On implication and negation in partition logic #7WB5RZ 2. The logic of partitions Matching excerpt #HTEQ5J: The indit set of a partition is the equivalence relation defined by the partition, and the ditset of a partition is the complementary partition relation defined by the partition. 3. Source: On implication and negation in partition logic (#C8FHDZ), , David Ellerman, p. 3 Context: #L8JZAG On implication and negation in partition logic #FEMQRA 4. Three more equivalent ways to define implication for partitions #8P96YU 4.1. The adjunctive definition Matching excerpt #U4TAMP: The partition implication \sigma \Rightarrow \pi can be similarly characterized in partition logic substituting ditsets for subsets [2]. For a third partition \tau = \{D, D', \dots\} , the characterization is: 4. Source: On implication and negation in partition logic (#C8FHDZ), , David Ellerman, p. 6 Context: #L8JZAG On implication and negation in partition logic #LEZCCF 6. Valid formulas Matching excerpt #XWG2AT: There is a simple way to see that all partition tautologies are also subset tautologies, i.e., valid formulas of subset logic. Consider the partition algebra \Pi(2) on the two-element set 2 = \{0, 1\} . It has only two partitions, the discrete partition \mathbf{1} = \{\{0\}, \{1\}\} where 0 and 1 are distinguished, and the indiscrete partition \mathbf{0} = \{\{0, 1\}\} where they are not distinguished. The partition operations, such as join, meet, and implication, applied to those two partitions could be described in "truth tables" since \mathbf{0} and \mathbf{1} are the only partitions on 2. And those truth tables are the same as the Boolean subset operations on the two subsets of the one-element set. Hence we have an isomorphism between the partition algebra \Pi(2) on 2 and the power-set Boolean algebra \wp(1) = 2 for 1 as the one-element set. 3 Now consider any formula that is a valid formula in partition logic. Since it evaluates to \mathbf{1} for all partitions on any U where |U| \geq 2 , it does that for U = 2 , but then the isomorphism \Pi(2) \cong \wp(1) means that the same formula will be a truth table tautology in \wp(1) and thus it is a valid formula for subset logic in general. Hence all partition tautologies are subset tautologies. But the inclusion is strict. For instance, the law of excluded middle \sigma \vee \neg\sigma = \sigma \vee (\sigma \Rightarrow \mathbf{0}) is not a partition tautology since for any \sigma \neq \mathbf{0}, \mathbf{1} , \sigma \Rightarrow \mathbf{0} = \mathbf{0} , and \sigma \vee \neg\sigma = \sigma \vee \mathbf{0} = \sigma \neq \mathbf{1} . 5. Source: On implication and negation in partition logic (#C8FHDZ), , David Ellerman, p. 1 Context: #L8JZAG On implication and negation in partition logic #7WB5RZ 2. The logic of partitions Matching excerpt #B7W87A: A distinction of a partition is an ordered pair (u, u') of elements of U in distinct blocks of the partition. The set of distinctions (abbreviated "dits") of a partition is the ditset 6. Source: On implication and negation in partition logic (#C8FHDZ), , David Ellerman, p. 0 Context: #L8JZAG On implication and negation in partition logic #7WB5RZ 2. The logic of partitions Matching excerpt #ZBQE96: Our purpose here is briefly present the basics of partition logic that suffice to explore the role of negation and implication in that logic. A partition \pi = \{B, B', \dots\} on a set U is a set of non-empty subsets B, B', \dots (“blocks”) of U where the blocks are mutually exclusive (the intersection of distinct blocks is empty) and jointly exhaustive (the union of the blocks is U ). An equivalence relation is a binary relation E \subseteq U \times U that is reflexive, symmetric, and transitive. Every equivalence relation on a set U determines a partition on U where the equivalence classes are the mutually exclusive and jointly exhaustive blocks of the partition. Conversely, every partition on a set determines an equivalence relation on the set; two elements are equivalent if they are in the same block of the partition. The notions of a partition on a set and an equivalence relation on a set are thus interdefinable. Indeed, equivalence relations and partitions are often considered as the “same” as in the conventional practice (not used here) of defining the “lattice of partitions” as the lattice of equivalence relations [4]. 7. Source: On implication and negation in partition logic (#C8FHDZ), , David Ellerman, p. 6 Context: #L8JZAG On implication and negation in partition logic #LEZCCF 6. Valid formulas Matching excerpt #5AWNV7: The Boolean core \mathcal{B}_\pi of the upper segment [\pi, \mathbf{1}] for any partition \pi , provides a way to 'automatically' generate partition tautologies. Since \mathcal{B}_\pi is a Boolean algebra, any Boolean tautology comprised of \pi -negated partitions will also be a partition tautology. For instance, the law of excluded middle in \mathcal{B}_\pi has the form \frac{\pi}{\neg\pi} \vee \frac{\pi}{\neg\pi} \sigma which is the "weak law of excluded middle" in partition logic. It is a partition validity since it is a Boolean tautology that evaluates to \mathbf{1} no matter what partitions on U are substituted for \pi and \sigma . 8. Source: On implication and negation in partition logic (#C8FHDZ), , David Ellerman, p. 1 Context: #L8JZAG On implication and negation in partition logic #7WB5RZ 2. The logic of partitions Matching excerpt #PS79AA: Similarly an indistinction or indit of a partition is an ordered pair of elements in the same block of the partition so: 9. Source: On implication and negation in partition logic (#C8FHDZ), , David Ellerman, p. 7 Context: #L8JZAG On implication and negation in partition logic #LEZCCF 6. Valid formulas Matching excerpt #L4CR23: Then the single \pi -negation transform of any classical tautology will still be a tautology but now expressed in \mathcal{B}_\pi and thus it is also a partition tautology. Thus all partition tautologies are ordinary Boolean logic tautologies, and any ordinary subset tautology transforms into a partition tautology via the single \pi -negation transform. 10. Source: Interactive procedural street modeling (#V4TQYB), Eugene Zhang, Gregory Esch, Guoning Chen, Pascal Müller, Peter Wonka, p. 6 Context: #57PDWB Interactive Procedural Street Modeling #ZPVYV6 6 Street Graph Generation #AZZQZY 6.4 Local Street Graph Editing using Tensor Fields Matching excerpt #ZVWF4A: The original street network (outside the regions) and the user generated network (inside the regions) are connected by tracing boundary street segment forward until they hit the other network. Figure 15 illustrates this approach. Approximate matches 1. Source: Procedural Content Generation through Quality Diversity (#7GR3AQ), Ahmed Khalifa, Antonios Liapis, Daniele Gravina, Georgios N. Yannakakis, Julian Togelius, p. 1 Context: #DZ9GFC Procedural Content Generation through Quality Diversity #KHLS7N II. QUALITY DIVERSITY APPROACHES #974EW9 A. Divergence Components Score: 0.027 Related excerpt #DJQRBX: 2) Behavior Space Partitioning : Diversity can also be enforced by partitioning the behavior space using N different dimensions of behavior, where each dimension is discretized and stored as a grid or a map of cells. Each cell corresponds to a different area in the behavior space with different properties; this cell contains all individuals that have a certain behaviors. Partitioning could be uniform, as in MAP-Elites [12], or based on the distribution of the population, as in MAP-Elites with Sliding Boundaries [13]. A big difference between the use of a distance function and partitioning the space is that partitioning allows designers to control the granularity of the space. 2. Source: Wholeness as a Hierarchical Graph to Capture the Nature of Space (#BYG3BQ), Bin Jiang, p. 3 Context: #V2MHRV Wholeness as a Hierarchical Graph to Capture the Nature of Space #LWK7XQ 2. The 15 properties #ZEFTKL Thick boundaries Score: 0.027 Related excerpt #THP9B9: Centers are often differentiated by thick boundaries. For example, the different triangle sizes in the snowflake (Figure 2a) and the convex spaces of the urban layout have thick boundaries (Figure 2d). The five hierarchical levels of the axial map can be perceived as centers, represented by five different colors (Figure 2f), which apparently lack thick boundaries. In this regard, the different means used for the head/tail breaks process might be considered thin boundaries. 3. Source: Terrain Generation Using Procedural Models Based on Hydrology (#DMTA8Y), Adrien Peytavie, Bedřich Beneš, Jean-David Génevaux, Éric Galin, Éric Guérin, p. 5 Context: #RULAFW Terrain Generation Using Procedural Models Based on Hydrology #T6FSBF 5 River Classification #VE6H2H 5.2 Water-courses Labeling Score: 0.026 Related excerpt #VPDQ33: After the domain is segmented, we classify the water-course presented in every Voronoi cell V_i . Our method relies on the Rosgen classification [Rosgen 1994] that defines nine river categories depending on their slopes and trajectories. Each river class has a trajectory type (A+, A, B, C, D, DA, E, F, or G) and a digging profile of the riverbed (Fig. 9). The classification includes the geological composition of the riverbed (bedrock, rocks, stones, gravel, sand, silt, or clay) in this description. 4. Source: Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space (#MJKTBB), Bin Jiang, Chris de Rijke, p. 4 Context: #YXCQD2 Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space #KXUAWS 3. A recursive approach to computing the structural beauty of images #LB8R59 3.1 Head/tail breaks and two laws of living structure Score: 0.025 Related excerpt #NH49NJ: (Note: The dataset is divided around the average into the head for those larger than the average and the tail for those smaller than the average. The head is brought back to do the same division again and again recursively until a certain threshold is met. The tails and the last head constitute individual classes in the iterative order: [1/20, 1/21, \dots, 1/100] , [1/6, 1/7, \dots, 1/19] , [1/3, 1/4, 1/5] , and [1, 1/2] . 5. Source: Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space (#MJKTBB), Bin Jiang, Chris de Rijke, p. 7 Context: #YXCQD2 Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space #KXUAWS 3. A recursive approach to computing the structural beauty of images #KK923R 3.2 The recursive approach Score: 0.022 Related excerpt #YTCZ4Y: substructures, resulting in 768, 1,856, 1,206, 598, 112, and 26 substructures, respectively, at different levels of the recursion, and their centroids are shown as red dots (c1–c6). Overlapping all the six levels of the 4,566 ( = 768 + 1,856 + 1,206 + 598 + 112 + 26 ) centroids together represents the skeleton of the image (d). Among the 4,566 substructures, only 62 ( = 11 + 20 + 22 + 6 + 3 ) are decomposable substructures and their centroids (e), and the 63 centroids are visualized according to their degree of connectivity (both in- and out-links) by the dot sizes, with colors indicating their levels of recursion (f). 6. Source: Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space (#MJKTBB), Bin Jiang, Chris de Rijke, p. 4 Context: #YXCQD2 Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space #KXUAWS 3. A recursive approach to computing the structural beauty of images #LB8R59 3.1 Head/tail breaks and two laws of living structure Score: 0.02 Related excerpt #XUKG48: Head/tail breaks was initially developed as a classification scheme for a dataset with a heavy-tailed distribution. It is a de facto recursive function for deriving the inherent hierarchy of a dataset from the bottom up. That is, the dataset itself determines the inherent hierarchy of classes without any imposed criteria. Let us use the dataset containing 100 numbers [1, 1/2, 1/3, \dots, 1/100] (Figure 3) to illustrate how the dataset can be decomposed into the head for those greater than the average and the tail for those less than the average, and then proceed recursively or iteratively for the head, the head of the head, and so on. All tails and the last head constitute individual classes or hierarchical levels. 7. Source: Priority-Flood: An Optimal Depression-Filling and Watershed-Labeling Algorithm for Digital Elevation Models (#MTDKDE), Clarence Lehman, David Mulla, Richard Barnes, p. 12 Context: #XJ3LVD Priority-Flood: An Optimal Depression-Filling and Watershed-Labeling Algorithm for Digital Elevation Models #4K4XN3 7. Variants #LFYVTV 7.3. Watershed Labeling Score: 0.025 Related excerpt #L6E5XU: Following execution, watershed boundaries may be identified by locating adjacent cells with differing labels. To mark the border cells, consistently choose either the cell with the lower label, the cell with the higher label, or both. 8. Source: FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation (#NV2YRW), Aryamaan Jain, Bernhard Kerbl, Brandon Finley, Guillaume Cordonnier, James Gain, p. 5 Context: #6UY46T FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation #J5PECC 5. Depression routing #UJCLXC Algorithm 2: Propagation of basin identifiers Score: 0.02 Related excerpt #C9XNSJ: Basin identification. Our algorithm begins by segmenting out each stream tree and associating with its cells a unique basin identifier (hereafter referred to as a basin-id , or bid in the pseudo-code of Algorithm 2). To this end, we assign to each non-outflow local minima a unique identifier – leaving a common outflow identifier for all outflow basins – and use pointer jumping to copy this identifier in parallel to all upstream nodes of its stream tree (Algorithm 2). 9. Source: A Layered Particle-Based Fluid Model for Real-Time Rendering of Water (#RBS5K6), Daniel Scherzer, Florian Bagar, Michael Wimmer, p. 4 Context: #5NFAXU A Layered Particle-Based Fluid Model for Real-Time Rendering of Water #Z68DF9 5. Real-Time Foam #P6BSN6 5.2. Layer Creation Score: 0.023 Related excerpt #48RCD2: Now that particles have been classified as either particle or foam, we partition the fluid into layers , as shown in Figure 2. 10. Source: Geography as a Science of the Earth’s Surface Founded on the Third View of Space (#SKRF4C), Bin Jiang, p. 5 Context: #KSQ64X Geography as a Science of the Earth's Surface Founded on the Third View of Space #F4PFN9 3. Living versus nonliving structure: The “things” the two laws refer to Score: 0.023 Related excerpt #WQW77N: For the sake of simplicity, we use the 10 numbers [1, 1/2, 1/3, \dots, 1/10] to show how they are classified through the head/tail breaks (Figure 3, Jiang and Slocum 2020). The dataset is a whole, and its average is about 0.29, which partitions the whole into two subwholes: those greater than the average are called the head [1, 1/2, 1/3] , and those less than the average are called the tail [1/4, \dots, 1/10] . The average of the head subwhole is about 0.61, and it partitions the head subwhole into two subwholes again: those greater than the average are called the head [1] , and those less than the average are called the tail [1/2, 1/3] . Instead of expressing the dataset as a set of numbers, we state the 10 numbers as an iterative system consisting of three subwholes recursively defined: [1] , [1, 1/2, 1/3] , and [1, 1/2, 1/3, \dots, 1/10] . Instead of perceiving these numbers as a set of 10 numbers, we consider them as a coherent whole, consisting of three subwholes including the whole itself. Or alternatively, these numbers as a coherent structure consists of three substructures including the structure itself. The dataset [1, 1/2, 1/3, \dots, 1/10] , because of its inherent hierarchy of 3, is more living than the other dataset [1, 2, 3, \dots, 10] that is without any inherent hierarchy, or violates the notion of far more smalls than larges. ### 32. Tool result: search_text Exact matches 1. Source: Terrain Generation Using Procedural Models Based on Hydrology (#DMTA8Y), Adrien Peytavie, Bedřich Beneš, Jean-David Génevaux, Éric Galin, Éric Guérin, p. 5 Context: #RULAFW Terrain Generation Using Procedural Models Based on Hydrology #T6FSBF 5 River Classification #TRR58W 5.1 Segmentation and Elevation of Crests Matching excerpt #XA9BE4: Ridges. The computation of ridge elevation is important to guarantee a coherent flow. Each Voronoi cell has two types of edges: those that do not intersect the river graph and that define ridge lines, and those that carry a river entry e_k or outlet s . 2. Source: Priority-Flood: An Optimal Depression-Filling and Watershed-Labeling Algorithm for Digital Elevation Models (#MTDKDE), Clarence Lehman, David Mulla, Richard Barnes, p. 12 Context: #XJ3LVD Priority-Flood: An Optimal Depression-Filling and Watershed-Labeling Algorithm for Digital Elevation Models #4K4XN3 7. Variants #LFYVTV 7.3. Watershed Labeling Matching excerpt #NNC37P: Watershed labeling applies a common label—such as an integer number—to all cells which drain to a given outlet. The algorithm (Alg. 5) for this works in much the same way as the improved Priority-Flood (Alg. 2). The DEM is flooded inwards from its edges with the lowest cell always being processed first. Rather than filling cells in depressions, this algorithm merely prioritizes them to the level of their outlet. 3. Source: Priority-Flood: An Optimal Depression-Filling and Watershed-Labeling Algorithm for Digital Elevation Models (#MTDKDE), Clarence Lehman, David Mulla, Richard Barnes, p. 14 Context: #XJ3LVD Priority-Flood: An Optimal Depression-Filling and Watershed-Labeling Algorithm for Digital Elevation Models #9D8AS4 8. Coda Matching excerpt #477DMT: Algorithm 5 IMPROVED PRIORITY-FLOOD+WATERSHED LABELS: This variation of the IMPROVED PRIORITY-FLOOD follows the work of Beucher and Meyer [1992] and Beucher and Beucher [2011] . It applies a common label to all cells draining to an outlet. Line 21 should be interpreted as pushing a copy of the cell's coordinates into Pit with the copy's z -value set to c.z . If simultaneous watershed labeling and depression-filling is desired, change the original z value of n to c.z before making the copy. Upon entry , (1) DEM contains the elevations of every cell or the value NODATA for cells not part of the DEM. At exit , (1) Labels contains a label for every cell or the value NODATA for cells not part of the DEM. (2) All cells which drain to a common point at the edge of the DEM bear the same label. 4. Source: FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation (#NV2YRW), Aryamaan Jain, Bernhard Kerbl, Brandon Finley, Guillaume Cordonnier, James Gain, p. 6 Context: #6UY46T FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation #J5PECC 5. Depression routing #2HKXNP Algorithm 3: Finding saddles and outlets Matching excerpt #L94DMG: Input : Terrain cells \mathcal{T} with elevation z , set of depression basins \mathcal{D} Output : Per-depression saddle and outlet 1 // Compute border cells and store the border altitude in z' 2 foreach cell c \in \mathcal{T} in parallel do 3 if bid_c \neq bid_n of any 4-neighbour n of c ( n \in \mathcal{N}_c ) then 4 tag c as a border cell 5 znb \leftarrow \min(z_n \text{ for } n \in \mathcal{N}_c \text{ such that } bid_c \neq bid_n) 6 z'_c \leftarrow \max(z_c, znb) 7 end 8 end 9 // Saddles and outlets (with atomic lexicographic argmin) 10 foreach basin d \in \mathcal{D} in parallel do 11 saddle of d \leftarrow \text{argmin}((z'_c, bid_n) for each border cell c 12 and n \in \mathcal{N}_c such that bid_c = d and bid_n \neq d ) 13 outlet of d \leftarrow \text{argmin}(z_n for each neighbor n of the 14 saddle such that bid_n \neq d ) 15 end 16 // Remove cycles 17 foreach basin d \in \mathcal{D} in parallel do 18 d' \leftarrow basin of outlet of d 19 if bid of outlet of d' = bid of saddle of d then 20 if bid of outlet of d < bid of saddle of d then 21 Delete the saddle and outlet of d 22 end 23 end 24 end 5. Source: FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation (#NV2YRW), Aryamaan Jain, Bernhard Kerbl, Brandon Finley, Guillaume Cordonnier, James Gain, p. 5 Context: #6UY46T FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation #J5PECC 5. Depression routing #UJCLXC Algorithm 2: Propagation of basin identifiers Matching excerpt #U6XU5M: Although we could compute the minimum altitude using a variant of rake-compress in \mathcal{O}(\log n) , we found it more efficient to instead compute it in a single step with atomic operations. We also store the saddle , which is the source cell in the current basin with minimum weight, and the outlet , which is its lowest neighbouring cell across the basin border (see Figure 4). 6. Source: FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation (#NV2YRW), Aryamaan Jain, Bernhard Kerbl, Brandon Finley, Guillaume Cordonnier, James Gain, p. 5 Context: #6UY46T FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation #J5PECC 5. Depression routing #UJCLXC Algorithm 2: Propagation of basin identifiers Matching excerpt #3Z5QAL: For each border cell, this weight is set as the lowest elevation that will permit an overflow from it into an adjacent basin. Specifically, this is the maximum of the border cell's altitude and the lowest altitude among its neighboring cells. Then we choose the border cell with minimum weight per basin. 7. Source: FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation (#NV2YRW), Aryamaan Jain, Bernhard Kerbl, Brandon Finley, Guillaume Cordonnier, James Gain, p. 5 Context: #6UY46T FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation #J5PECC 5. Depression routing #UJCLXC Algorithm 2: Propagation of basin identifiers Matching excerpt #FHU2AM: Close-coupled loops can occur when two basins have saddle edges pointing at each other. We resolve this by simply deleting the edge with the higher saddle basin-id. We also apply a similar strategy to choose among saddles of identical elevation, which could lead to larger cycles. This is prevented by selecting saddles in lexicographic order (elevation concatenated with the basin-id of the outlet). This algorithm is summarized in Algorithm 3. 8. Source: Advected river textures (#WZMZGY), Dirk Arnold, Stephen Brooks, Tim Burrell, p. 4 Context: #JCB5RE Advected river textures #G2SPEY Previous Work #CSD47K Impulse Driven NS Matching excerpt #6PLNP5: Equipped with the flow hinting field and a primed fluid volume we begin adding impulses to each cell in the simulation as it runs. As we do so we need to ensure the simulation stability is not compromised, complex flows can still be visualized and the overall flow does not exhibit incorrect flow behavior such as a river reversing its direction. To accomplish this, the magnitude of the impulse added to a cell is made proportional to the cell’s velocity in relation to the desired flow volume. For example, if two cells had the same flow hint, with one having a velocity of 1.0 units of flow and the other 0.5, the faster moving cell would be affected twice as much as the slower moving cell during the application of the impulse. In this way, the impulse application scales linearly and the ratio between one cell and another does not change after the impulses are applied (given equal circumstances the difference in rate of flow between once cell and the next will have the same ratio after the impulse application). 9. Source: Advected river textures (#WZMZGY), Dirk Arnold, Stephen Brooks, Tim Burrell, p. 4 Context: #JCB5RE Advected river textures #G2SPEY Previous Work #CSD47K Impulse Driven NS Matching excerpt #AWH7WY: Our solution is related to a technique used in rigid-body physics engines: the application of impulses. We will apply small impulses to each cell in order to maintain flow. There are, however, issues with this approach that must be addressed. For example, in what direction should impulses be applied, particularly if the local flow is in opposition to the mean direction of the river? Another issue is what the initial state of the river should be and how are the first rounds of impulses to be applied? 10. Source: Advected river textures (#WZMZGY), Dirk Arnold, Stephen Brooks, Tim Burrell, p. 4 Context: #JCB5RE Advected river textures #G2SPEY Previous Work #PVQNK6 Bootstrapping the Hydrostatic Pressure Columns Matching excerpt #UJWCSS: The HSP computations are dependent on the velocity in the previous time step. Therefore in order to generate an HSP pressure map we simply substitute the initial phase of NS velocity output into the velocity component, u_0 , of the HSP velocity update equation. This bootstrapping procedure only provides the HSP simulation with the initial flow rate and direction of each hydrostatic pressure column cell. Once these values are input into the HSP solver, the resulting 3D flow information is entirely based on the 3D terrain underneath and inside the fluid volume. Approximate matches 1. Source: Terrain Generation Using Procedural Models Based on Hydrology (#DMTA8Y), Adrien Peytavie, Bedřich Beneš, Jean-David Génevaux, Éric Galin, Éric Guérin, p. 5 Context: #RULAFW Terrain Generation Using Procedural Models Based on Hydrology #T6FSBF 5 River Classification #TRR58W 5.1 Segmentation and Elevation of Crests Score: 0.026 Related excerpt #WL3SCW: Watersheds are associated with each water outlet s of a cell V_j and are defined as the set of upstream connected cells V_k \in V . 2. Source: Priority-Flood: An Optimal Depression-Filling and Watershed-Labeling Algorithm for Digital Elevation Models (#MTDKDE), Clarence Lehman, David Mulla, Richard Barnes, p. 12 Context: #XJ3LVD Priority-Flood: An Optimal Depression-Filling and Watershed-Labeling Algorithm for Digital Elevation Models #4K4XN3 7. Variants #LFYVTV 7.3. Watershed Labeling Score: 0.024 Related excerpt #NNC37P: Watershed labeling applies a common label—such as an integer number—to all cells which drain to a given outlet. The algorithm (Alg. 5) for this works in much the same way as the improved Priority-Flood (Alg. 2). The DEM is flooded inwards from its edges with the lowest cell always being processed first. Rather than filling cells in depressions, this algorithm merely prioritizes them to the level of their outlet. 3. Source: Priority-Flood: An Optimal Depression-Filling and Watershed-Labeling Algorithm for Digital Elevation Models (#MTDKDE), Clarence Lehman, David Mulla, Richard Barnes, p. 12 Context: #XJ3LVD Priority-Flood: An Optimal Depression-Filling and Watershed-Labeling Algorithm for Digital Elevation Models #4K4XN3 7. Variants #LFYVTV 7.3. Watershed Labeling Score: 0.023 Related excerpt #HBWRR5: Watershed outlets are identified as being unlabeled cells adjacent to a NoDATA cell. These cells are given a unique label which then floods inwards to cover all the cells in the watershed. 4. Source: FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation (#NV2YRW), Aryamaan Jain, Bernhard Kerbl, Brandon Finley, Guillaume Cordonnier, James Gain, p. 5 Context: #6UY46T FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation #J5PECC 5. Depression routing #UJCLXC Algorithm 2: Propagation of basin identifiers Score: 0.029 Related excerpt #V256NP: Input : Terrain cells \mathcal{T} , non-outflow local minima \mathcal{L} Output: Per-cell basin identifiers bid 5. Source: FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation (#NV2YRW), Aryamaan Jain, Bernhard Kerbl, Brandon Finley, Guillaume Cordonnier, James Gain, p. 5 Context: #6UY46T FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation #J5PECC 5. Depression routing #UJCLXC Algorithm 2: Propagation of basin identifiers Score: 0.029 Related excerpt #C9XNSJ: Basin identification. Our algorithm begins by segmenting out each stream tree and associating with its cells a unique basin identifier (hereafter referred to as a basin-id , or bid in the pseudo-code of Algorithm 2). To this end, we assign to each non-outflow local minima a unique identifier – leaving a common outflow identifier for all outflow basins – and use pointer jumping to copy this identifier in parallel to all upstream nodes of its stream tree (Algorithm 2). 6. Source: FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation (#NV2YRW), Aryamaan Jain, Bernhard Kerbl, Brandon Finley, Guillaume Cordonnier, James Gain, p. 6 Context: #6UY46T FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation #J5PECC 5. Depression routing #2HKXNP Algorithm 3: Finding saddles and outlets Score: 0.026 Related excerpt #L94DMG: Input : Terrain cells \mathcal{T} with elevation z , set of depression basins \mathcal{D} Output : Per-depression saddle and outlet 1 // Compute border cells and store the border altitude in z' 2 foreach cell c \in \mathcal{T} in parallel do 3 if bid_c \neq bid_n of any 4-neighbour n of c ( n \in \mathcal{N}_c ) then 4 tag c as a border cell 5 znb \leftarrow \min(z_n \text{ for } n \in \mathcal{N}_c \text{ such that } bid_c \neq bid_n) 6 z'_c \leftarrow \max(z_c, znb) 7 end 8 end 9 // Saddles and outlets (with atomic lexicographic argmin) 10 foreach basin d \in \mathcal{D} in parallel do 11 saddle of d \leftarrow \text{argmin}((z'_c, bid_n) for each border cell c 12 and n \in \mathcal{N}_c such that bid_c = d and bid_n \neq d ) 13 outlet of d \leftarrow \text{argmin}(z_n for each neighbor n of the 14 saddle such that bid_n \neq d ) 15 end 16 // Remove cycles 17 foreach basin d \in \mathcal{D} in parallel do 18 d' \leftarrow basin of outlet of d 19 if bid of outlet of d' = bid of saddle of d then 20 if bid of outlet of d < bid of saddle of d then 21 Delete the saddle and outlet of d 22 end 23 end 24 end 7. Source: FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation (#NV2YRW), Aryamaan Jain, Bernhard Kerbl, Brandon Finley, Guillaume Cordonnier, James Gain, p. 6 Context: #6UY46T FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation #J5PECC 5. Depression routing #V69QNF Algorithm 5: Depression Routing Score: 0.025 Related excerpt #CJMQ7G: Implementation. As with flow routing, we prevent read-after-write hazards while minimizing the extent of copying. For basin identification, we store the basin-id only for the local minima and not for all cells, and we use the updated recipients at the end of Algorithm 2 as a per-cell pointer to the downstream local minima, which we interrogate to obtain the basin-id. This change relaxes the need for any copy during basin identification, as read-after-write no longer prevents the correct convergence of the algorithm. Furthermore, we observe that this operation is required only for the first iteration. In subsequent iterations, we need only update the pointers to reflect the new connections out of the local minima, yielding an overall complexity of \mathcal{O}(\log(n) + \log^2(L)) . 8. Source: FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation (#NV2YRW), Aryamaan Jain, Bernhard Kerbl, Brandon Finley, Guillaume Cordonnier, James Gain, p. 5 Context: #6UY46T FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation #J5PECC 5. Depression routing #UJCLXC Algorithm 2: Propagation of basin identifiers Score: 0.024 Related excerpt #3Z5QAL: For each border cell, this weight is set as the lowest elevation that will permit an overflow from it into an adjacent basin. Specifically, this is the maximum of the border cell's altitude and the lowest altitude among its neighboring cells. Then we choose the border cell with minimum weight per basin. 9. Source: FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation (#NV2YRW), Aryamaan Jain, Bernhard Kerbl, Brandon Finley, Guillaume Cordonnier, James Gain, p. 3 Context: #6UY46T FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation #SD69QC 3. Overview #CPNMBK 3.2. Depression routing Score: 0.023 Related excerpt #A6G5PJ: We are now in a position to define what a depression means in our context. As a preliminary, we use the term basin for the set of terrain cells that share the same stream tree, effectively all belonging to the same catchment area and channeling water to a single root cell. If this root cell is an outflow, then the basin is classified as an outflow basin, otherwise, it constitutes a depression (Figure 2 (c)). 10. Source: FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation (#NV2YRW), Aryamaan Jain, Bernhard Kerbl, Brandon Finley, Guillaume Cordonnier, James Gain, p. 2 Context: #6UY46T FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation #SD69QC 3. Overview Score: 0.023 Related excerpt #S8B4Q9: There are three layers of terrain input to our algorithms. The first is the elevation grid. We refer to each element of the grid as a terrain cell c , with n being the total number of cells. We prefer to use the term “cell” as opposed to the more familiar “vertex” or “node” in this context because we want to make clear that this represents a rectangular area on the terrain surrounding the sample. The neighbors of c , denoted by N_c , are the 4 cells directly adjacent to c in the x and y directions (Figure 2 (b)). The second input layer is the amount of water (or any loose material) initially deposited on the terrain and subject to flow. This can be interpreted as rainfall or precipitation and is denoted as p_c . The final layer is a user-defined boolean mask b_c \mapsto \{0, 1\} indicating the presence (1) or absence (0) of outflow cells. These serve as boundary conditions and act to trap water flow. Typically, outflow cells would line the edges of the grid, but there are cases, such as groundwater sinks, estuaries, and sea shores where they would be placed within the domain. ### 33. Tool result: search_text Exact matches 1. Source: Structural Beauty: A Structure-Based Computational Approach to Quantifying the Beauty of an Image (#3XSLTA), Bin Jiang, Chris de Rijke, p. 6 Context: #ELWE7N Structural Beauty: A Structure-based Approach to Quantifying the Beauty of an Image #85YSXN 3. A computational approach to the goodness or structural beauty of an image #4BFT5R 3.2 The computational approach to structural beauty Matching excerpt #XPDDRQ: An image is viewed as an iterative system that consists of the figure, the figure of the figure, and the figure of the figure of the figure and so on at different levels of hierarchy. These figures are also called the subwholes of the image, which is the whole. The above process of deriving a living structure from the image itself can be recursively applied to these different subwholes. In other words, instead of the entire image, we take the figure as a subwhole to get its average pixel value, which is used to derive the figure of the figure, and the figure of the figure of the figure and so on. Subsequently, their living structures can be derived in the same way as the living structure of the image itself (Figure 3). The derived living structures have two parameters: (1) the number of substructures, and (2) the hierarchy of the substructures based on the head/tail breaks. Eventually, structural beauty is based on these two parameters. 2. Source: Structural Beauty: A Structure-Based Computational Approach to Quantifying the Beauty of an Image (#3XSLTA), Bin Jiang, Chris de Rijke, p. 12 Context: #ELWE7N Structural Beauty: A Structure-based Approach to Quantifying the Beauty of an Image #PQFTNG 6. Conclusion Matching excerpt #MHC3DH: Structural beauty, as defined and computed in this paper, presents a radical mindset change from subjective to objective beauty, thus significantly contributing to the effort on aesthetic measures and image understanding. We develop a computational approach to structural beauty or goodness of an image based on the living structure, a new way of image understanding. An image is commonly represented mechanically by many individual pixels, but human perception of the image is hardly pixel-based and is instead oriented towards a coherent whole (e.g., the figure of the figure of the figure and so on) or living structure. As a natural and organic representation, a living structure derived from an image constitutes the backbone or configuration of the image from a holistic perspective. It is governed by two fundamental laws: scaling law and Tobler’s law, which are respectively available across different levels and at each level of the hierarchy. There are far more small substructures than large ones, according to scaling law, whereas substructures are more or less similar in terms of Tobler’s law. These two laws of living structure underlie the computational approach to the goodness or structural beauty of an image. The living structure of an image is composed of many substructures with the inherent hierarchy of far more smalls than larges. The figure of the image can be further composed of many substructures with the inherent hierarchy of far more smalls than larges. Therefore, structural beauty or life (L), given as S (the number of substructures) times H (the number of hierarchical levels), is computed based on the rule that the more substructures, the more beautiful, and the higher hierarchy, the more beautiful. The measure of structural beauty or the computational approach in general is shown to be simple, effective, and efficient for ranking different images. 3. Source: Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space (#MJKTBB), Bin Jiang, Chris de Rijke, p. 1 Context: #YXCQD2 Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space #7WXQSD 1. Introduction Matching excerpt #ETVNUT: Any image is perceived by human eyes subconsciously as a coherent structure (or whole) with two large, contrasting substructures: figure and ground (Koffka 1936, Rubin 1921). The figure, which is also called the foreground, constitutes the focus of the visual field, while the ground is the background. The figure is a living structure, for it can be decomposed into many substructures with an inherent hierarchy of far more smalls than larges. The substructures are auto-generated segments (or sets of pixels) out of a gray-scale image by vectorizing the individual sets of pixels that are darker (or lighter) than the average pixel. There are far more small substructures than large ones across the hierarchy or the different levels of scale, yet the substructures on each of the hierarchy are more or less similar in size. It is essentially the recurring notion of far more small substructures than large ones that triggers a sense of livingness in the human mind and heart (Jiang 2019). This sense of livingness is called structural beauty (Jiang and De Rijke 2021) and it is shared among people, and even different peoples, regardless of their cultures, gender, and races. Thus, the livingness (L) or structural beauty is defined by the derived substructures, or more specifically the multiplication of their number (S) and their inherent hierarchy (H); that is, L = S \times H . Instead of working with the figure, we, in the present paper, work directly with the image itself and derive its substructures, and the substructures of the decomposable substructures recursively until all substructures are no longer decomposable. All the substructures at different iterations (or recursive levels) together constitute a coherent whole or a living structure: hence the notion of living images, the central theme of this paper. 4. Source: Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space (#MJKTBB), Bin Jiang, Chris de Rijke, p. 0 Context: #YXCQD2 Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space #K2N49X Abstract Matching excerpt #47VNT4: According to Gestalt theory, any image is perceived subconsciously as a coherent structure (or whole) with two contrast substructures: figure and ground. The figure consists of numerous auto-generated substructures with an inherent hierarchy of far more smalls than larges. Through these substructures, the structural beauty of an image (L), or equivalently the livingness of space, can be computed by the multiplication of the number of substructures (S) and their inherent hierarchy (H). This definition implies that the more substructures something has, the more living or more structurally beautiful it is, and the higher hierarchy of the substructures, the more living or more structurally beautiful. This is the non-recursive approach to the structural beauty of images or the livingness of space. In this paper we develop a recursive approach, which derives all substructures of an image (instead of its figure) and continues the deriving process for those decomposable substructures until none of them are decomposable. All of the substructures derived at different iterations (or recursive levels) together constitute a living structure; hence the notion of living images. We have applied the recursive approach to a set of images that have been previously studied in the literature and found that (1) the number of substructures of an image is far lower (3 percent on average) than the number of pixels and the centroids of the substructures can effectively capture the skeleton or saliency of the image; (2) all the images have the recursive levels more than three, indicating that they are indeed living images; (3) no more than 2 percent of the substructures are decomposable, implying that a vast amount of the substructures are not decomposable; (4) structural beauty can be well measured by the recursively defined substructures, as well as their decomposable subsets. Despite a slightly higher computational cost, the recursive approach is proved to be more robust than the non-recursive approach. The recursive approach and the non-recursive approach both provide a powerful means to study the livingness or vitality of space in cities and communities. 5. Source: Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space (#MJKTBB), Bin Jiang, Chris de Rijke, p. 13 Context: #YXCQD2 Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space #AMFSDX 6. Conclusion Matching excerpt #K7EAKA: Any space possesses a certain degree of livingness or structural beauty, although the degree varies from one to another depending on its internal geometry of substructures. Reflected in an image, a substructure is a set of pixels whose pixel values are greater (or less) than the average pixel value. Thus, an image can be viewed as a set of recursively defined substructures, rather than a set of pixels or a set of human recognizable objects as conventionally conceived. The major difference between substructures and objects lies in the fact that substructures are defined by pixels themselves from the bottom up, while objects are defined by human eyes. In this paper, we developed a recursive approach to the structural beauty of images by considering all recursively defined substructures. Through the case studies, we have verified that the recursive approach is better or more robust than the non-recursive approach, although both approaches are based on the same principle: the more substructures the more beautiful, the higher hierarchy of the substructures the more beautiful. We have also verified that the decomposable substructures alone can be used to differentiate two images in terms of their structural beauty, by multiplying the number of decomposable substructures and their levels of recursion. This implies that the more decomposable substructures the more beautiful, and the more levels of recursion the more beautiful. 6. Source: Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space (#MJKTBB), Bin Jiang, Chris de Rijke, p. 1 Context: #YXCQD2 Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space #7WXQSD 1. Introduction Matching excerpt #JGHLF6: In this paper we consider an image – or space in general – to be a living structure that is composed of recursively defined substructures. This is a holistic view of perceiving an image or space as a coherent whole, so it differs fundamentally from conventional thinking (e.g., Umbaugh 2017, Davies 2017). Conventional image understanding tends to identify a few features or objects that can be named by words or recognizable by human eyes – so-called computer vision. For example, a human face image consists of numerous substructures with far more smalls than large ones, but our natural language can only name certain features, such as the eyes, the nose, the mouth, the ears, and the hair. In other words, a vast majority of substructures cannot be named by words. In general terms, a gray-scale image can be decomposed, around the average pixel value m_1 , into dark pixels (darker than m_1 ) which may be called the figure (say, 52 percent) and light pixels (lighter than m_1 ), which may be called the ground (for example, 48 percent). Although the figure (or the dark pixels) is perceived as a whole, it consists of numerous substructures with an inherent hierarchy of far more smalls than larges. Interestingly, the figure can be further decomposed in a recursive manner, around the average pixel value ( m_i ) of the figure, into dark and light pixels, leading to numerous substructures with far more smalls than larges. This decomposition process is referred to here as recursion . 7. Source: Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space (#MJKTBB), Bin Jiang, Chris de Rijke, p. 10 Context: #YXCQD2 Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space #563UCF 4. Case studies #AKW33C 4.3 Decomposable substructures and their centroids Matching excerpt #PWZ7T4: If the centroids of the substructures of an image constitute the skeleton of the image, as shown and discussed above, those decomposable substructures and their centroids appear to be the most salient spots of the image. Figure 7 (L1–L8) demonstrates the decomposable substructures (black grounds) and their centroids (red dots). We can note that the decomposable substructures can be seen as sketches that well represent the corresponding images themselves. The red dots look scattered, yet they are the centroids of the decomposable substructures. In fact, the centroids are not scattered at all, and instead they form interconnected wholes or graphs (G1–G8 of Figure 7). The number of nodes (or substructures) and the number of colors (or hierarchy) of graphs can help differentiate the structural beauty of images. The more substructures, the more beautiful the image, and the higher the hierarchy, the more beautiful the image, as indicated by column V in Table 3. 8. Source: Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space (#MJKTBB), Bin Jiang, Chris de Rijke, p. 6 Context: #YXCQD2 Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space #KXUAWS 3. A recursive approach to computing the structural beauty of images #KK923R 3.2 The recursive approach Matching excerpt #JWLNAT: where S and H denote the number and the hierarchy of substructures, respectively. This definition implies that the more substructures an image has, the more beautiful it is, and the higher the hierarchy of the substructures of the image, the more beautiful it is. 9. Source: Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space (#MJKTBB), Bin Jiang, Chris de Rijke, p. 13 Context: #YXCQD2 Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space #266PRV 5. The livingness of space: Related work, application, and implication Matching excerpt #PETABM: Different people may perceive this image differently, depending on which objects or features they are familiar with or interested in. The fishpond image contains many objects or features, which are termed as centers (Alexander 2002–2005) or substructures. Identifying these substructures is probably the first step of image understanding, but the livingness of image lies on how these individual substructures constitute a coherent whole. The following passage (Alexander 2002–2005) is a typical description of the livingness of the image. 10. Source: Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space (#MJKTBB), Bin Jiang, Chris de Rijke, p. 12 Context: #YXCQD2 Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space #266PRV 5. The livingness of space: Related work, application, and implication Matching excerpt #PRFDAG: This study or the livingness of space in general adds a deep implication on image understanding and computer vision. For example, the morphological skeleton (Umbaugh 2017) and saliency of images (Kadir and Brady 2001, Borji and Itti 2013) can be effectively captured by the centroids of substructures. However, research questions remain regarding how the living structure perspective is comparable to previous approaches to morphological skeleton and saliency maps. While conventional image understanding concentrates on things (objects or features) that human eyes can recognize or natural language can describe with “words” (Deng et al. 2009), the recursive approach to structural beauty captures all substructures that may or may not correspond to human recognizable or language describable things by “words”. The auto-generated substructures, and the recursively generated substructures in particular, capture the wholeness of the image. In this connection, the substructures represent a new way of image understanding. While the human recognized objects or features tend to be fragmented as “words”, the substructures represent a coherent whole or living structure. To make this point clear, let's examine in detail another living image the fishpond image (Figure 9, Table 6) with which structural beauty is calculated from both the figure and the ground. Approximate matches 1. Source: Wholeness as a Hierarchical Graph to Capture the Nature of Space (#BYG3BQ), Bin Jiang, p. 3 Context: #V2MHRV Wholeness as a Hierarchical Graph to Capture the Nature of Space #LWK7XQ 2. The 15 properties #ZEFTKL Thick boundaries Score: 0.025 Related excerpt #THP9B9: Centers are often differentiated by thick boundaries. For example, the different triangle sizes in the snowflake (Figure 2a) and the convex spaces of the urban layout have thick boundaries (Figure 2d). The five hierarchical levels of the axial map can be perceived as centers, represented by five different colors (Figure 2f), which apparently lack thick boundaries. In this regard, the different means used for the head/tail breaks process might be considered thin boundaries. 2. Source: Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space (#MJKTBB), Bin Jiang, Chris de Rijke, p. 7 Context: #YXCQD2 Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space #KXUAWS 3. A recursive approach to computing the structural beauty of images #KK923R 3.2 The recursive approach Score: 0.029 Related excerpt #YTCZ4Y: substructures, resulting in 768, 1,856, 1,206, 598, 112, and 26 substructures, respectively, at different levels of the recursion, and their centroids are shown as red dots (c1–c6). Overlapping all the six levels of the 4,566 ( = 768 + 1,856 + 1,206 + 598 + 112 + 26 ) centroids together represents the skeleton of the image (d). Among the 4,566 substructures, only 62 ( = 11 + 20 + 22 + 6 + 3 ) are decomposable substructures and their centroids (e), and the 63 centroids are visualized according to their degree of connectivity (both in- and out-links) by the dot sizes, with colors indicating their levels of recursion (f). 3. Source: Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space (#MJKTBB), Bin Jiang, Chris de Rijke, p. 5 Context: #YXCQD2 Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space #KXUAWS 3. A recursive approach to computing the structural beauty of images #KK923R 3.2 The recursive approach Score: 0.029 Related excerpt #2XBW2N: The recursive approach starts with the first iteration or the first level of recursion (as mentioned above) and use the first iteration substructures to clip the original image, and then continue what was done in the iteration again and again, until all decomposable substructures are decomposed. As shown in Figure 4, the first iteration leads to 768 substructures (Panel b) and their centroids (Panel c1), 11 of which are decomposable, leading to 1,856 substructures (whose centroids in Panel c2), 20 of which are decomposable, leading to 1,206 substructures (whose centroids in Panel c3), 22 of which are decomposable, leading to 598 substructures (whose centroids in Panel c4), six of which are decomposable, leading to 112 substructures (whose centroids in Panel c5), three of which are decomposable, leading to 26 substructures (whose centroids in Panel c6). Table 2 provides detailed statistics for the recursive process. All the centroids of the recursively derived substructures effectively capture the skeleton or saliency of the image (Panel d). The centroids of the 63 = 1 + 11 + 20 + 22 + 6 + 3 decomposable substructures are shown in Panel e. 4. Source: Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space (#MJKTBB), Bin Jiang, Chris de Rijke, p. 11 Context: #YXCQD2 Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space #563UCF 4. Case studies #AKW33C 4.3 Decomposable substructures and their centroids Score: 0.027 Related excerpt #U8GBXK: (Note: Centroids of the decomposable substructures of an image constitute an interconnected whole or graph. Each graph represents the skeleton of the corresponding image, and its structural beauty can be judged according to the rule: the more nodes, the more beautiful the image, and the higher hierarchy (indicated by colors), the more beautiful the image. There is little doubt for every pair of graphs that the left graph is more living or more beautiful than the right. In addition, the graphs can be perceived as the organized complexity, a key concept in Jacobs (1961), inspired by the work by Weaver (1948).) 5. Source: Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space (#MJKTBB), Bin Jiang, Chris de Rijke, p. 10 Context: #YXCQD2 Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space #563UCF 4. Case studies #AKW33C 4.3 Decomposable substructures and their centroids Score: 0.026 Related excerpt #PWZ7T4: If the centroids of the substructures of an image constitute the skeleton of the image, as shown and discussed above, those decomposable substructures and their centroids appear to be the most salient spots of the image. Figure 7 (L1–L8) demonstrates the decomposable substructures (black grounds) and their centroids (red dots). We can note that the decomposable substructures can be seen as sketches that well represent the corresponding images themselves. The red dots look scattered, yet they are the centroids of the decomposable substructures. In fact, the centroids are not scattered at all, and instead they form interconnected wholes or graphs (G1–G8 of Figure 7). The number of nodes (or substructures) and the number of colors (or hierarchy) of graphs can help differentiate the structural beauty of images. The more substructures, the more beautiful the image, and the higher the hierarchy, the more beautiful the image, as indicated by column V in Table 3. 6. Source: Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space (#MJKTBB), Bin Jiang, Chris de Rijke, p. 5 Context: #YXCQD2 Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space #KXUAWS 3. A recursive approach to computing the structural beauty of images #KK923R 3.2 The recursive approach Score: 0.026 Related excerpt #FXLSNG: Applying the head/tail breaks to a gray-scale image will help get the substructures of the image. More specifically, we first determine the figure or the ground of the image depending on the percentage of the dark or light pixels. The high percentage is usually referred to as the ground, while the low percentage is referred to as the figure (Robin 1921). The first dichotomy process helps differentiate the figure from the image and helps derive the first round of substructures. To illustrate, let us take the weather-beaten face image as an example (Figure 4). It consists of 390 by 672 (262,080) pixels, each of which has a gray scale between 26 and 254. The average pixel value of those 262K pixels is 163, which partitions all the pixels into two groups: 120,036 dark pixels (46 percent of the total) as the figure, and 142,044 light pixels (54 percent) – the so-called dichotomy. The dark pixels are set to black, while the light pixels are set to white to create a binary image (binarization), which is further vectorized into a living structure consisting of numerous substructures. To this point, the illustration fulfills the first iteration, as shown in Panel a of Figure 4. As a reminder, the first iteration or the first level of recursion that starts from the image is NOT identical to the no-recursive approach (Jiang and De Rijke 2021), which is based on the figure of the image for deriving all non-recursively defined substructures. 7. Source: Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space (#MJKTBB), Bin Jiang, Chris de Rijke, p. 7 Context: #YXCQD2 Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space #KXUAWS 3. A recursive approach to computing the structural beauty of images #KK923R 3.2 The recursive approach Score: 0.025 Related excerpt #NLBDDP: A short note on the concept of substructures follows. Conventional image understanding and computer vision tends to identify individual objects that human beings can recognize in a human face image, such as the eyes, nose, and mouth, but hundreds and thousands of objects we cannot find proper words to name. Instead, the concept of substructures is defined naturally or organically by all pixels. In other words, all the pixels collectively determine an average pixel value that is used to delineate individual substructures, a kind of wisdom of crowds thinking (Surowiecki 2004). More importantly, unlike objects that are fragmented pieces, the substructures constitute a coherent whole. It is the concept of whole or its substructures that make the approach unique and different from the conventional image understanding. 8. Source: Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space (#MJKTBB), Bin Jiang, Chris de Rijke, p. 7 Context: #YXCQD2 Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space #563UCF 4. Case studies #7V2LAA 4.2 Centroids of the substructures capture the skeleton of the images Score: 0.025 Related excerpt #Z8VVE6: The centroids of the substructures capture very well the skeleton or saliency of the images. This is shown in Figure 5 as living structures in red, whose numbers are shown in column “substructures” in Table 3. They account for an average of 3 percent of the pixels of an image. We conjecture that these centroids are what is captured by a painter while he/she is drawing a sketch of the image. To demonstrate how the centroids capture the skeleton or saliency, Figure 6 shows three enlarged pairs from Figure 5. The centroids of the substructures are representative of the corresponding images. This is probably the reason that structural beauty calculated from these substructures (or their centroids) can effectively differentiate the two images. 9. Source: Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space (#MJKTBB), Bin Jiang, Chris de Rijke, p. 1 Context: #YXCQD2 Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space #7WXQSD 1. Introduction Score: 0.024 Related excerpt #JGHLF6: In this paper we consider an image – or space in general – to be a living structure that is composed of recursively defined substructures. This is a holistic view of perceiving an image or space as a coherent whole, so it differs fundamentally from conventional thinking (e.g., Umbaugh 2017, Davies 2017). Conventional image understanding tends to identify a few features or objects that can be named by words or recognizable by human eyes – so-called computer vision. For example, a human face image consists of numerous substructures with far more smalls than large ones, but our natural language can only name certain features, such as the eyes, the nose, the mouth, the ears, and the hair. In other words, a vast majority of substructures cannot be named by words. In general terms, a gray-scale image can be decomposed, around the average pixel value m_1 , into dark pixels (darker than m_1 ) which may be called the figure (say, 52 percent) and light pixels (lighter than m_1 ), which may be called the ground (for example, 48 percent). Although the figure (or the dark pixels) is perceived as a whole, it consists of numerous substructures with an inherent hierarchy of far more smalls than larges. Interestingly, the figure can be further decomposed in a recursive manner, around the average pixel value ( m_i ) of the figure, into dark and light pixels, leading to numerous substructures with far more smalls than larges. This decomposition process is referred to here as recursion . 10. Source: Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space (#MJKTBB), Bin Jiang, Chris de Rijke, p. 4 Context: #YXCQD2 Living Images: A Recursive Approach to Computing the Structural Beauty of Images or the Livingness of Space #KXUAWS 3. A recursive approach to computing the structural beauty of images Score: 0.023 Related excerpt #2TLGAU: In this section, we introduce a recursive approach to the structural beauty of images by extending the substructures to recursive ones. Before that, we will also illustrate the head/tail breaks (Jiang 2013) as a recursive function for deriving the underlying living structure or the inherent hierarchy of a dataset. The dataset contains 100 numbers as a working example to mimic the rank–size distribution: the largest size is about twice as large as the second largest, approximately three times as large as the third largest, and so on (Zipf 1949). This working example also helps to illustrate the two laws of living structure. ### 34. Tool result: search_text Exact matches 1. Source: Interactive procedural street modeling (#V4TQYB), Eugene Zhang, Gregory Esch, Guoning Chen, Pascal Müller, Peter Wonka, p. 5 Context: #57PDWB Interactive Procedural Street Modeling #ZPVYV6 6 Street Graph Generation #JW8A9D 6.2 Minor Street Graph Generation from Tensor Fields Matching excerpt #PUZXXV: Once the major street graph G_M has been constructed, it can be used to generate the minor street graph G_m . The process of generating G_m is similar to that of G_M with the following key difference. The edges in G_M and the boundaries of topographical features divide the domain into regions, inside each of which the user creates a continuous tensor field (see Figure 3 (6)). The tensor field can be discontinuous across region boundaries, i.e., major roads. This implies the tensor field used for minor road tracing is not necessarily the same as we use for major road tracing above. Figure 11 shows the minor road network generated based on the major road network. Note that minor roads do not necessarily follow the same directions as major roads. We point out that the idea of flow tiles proposed by Chenney [2004] for modeling a vector field can also be adopted to achieve the wealth of minor road patterns. 2. Source: Interactive procedural street modeling (#V4TQYB), Eugene Zhang, Gregory Esch, Guoning Chen, Pascal Müller, Peter Wonka, p. 2 Context: #57PDWB Interactive Procedural Street Modeling #P35TFV 3 Pipeline Overview Matching excerpt #QCYCTQ: (2) that give rise to a major street network (3). Then the user refines the initial major road layout by placing a new tensor field design element inducing a radial structure in the tensor field (4) as well as the street graph (5). Using our segmentation algorithm, the user performs additional local tensor field modifications (6) and generate a minor road network (7). The user uses a rotation noise field to create irregular structures near the top (8) and produces the final result (9). The visualization of tensor fields shown in this paper is based on [van Wijk 2002; Zhang et al. 2007]. 3. Source: Interactive procedural street modeling (#V4TQYB), Eugene Zhang, Gregory Esch, Guoning Chen, Pascal Müller, Peter Wonka, p. 0 Context: #57PDWB Interactive Procedural Street Modeling #H9D7PX 1 Introduction Matching excerpt #A2NHD8: An important aspect of street patterns is the existence of two dominant directions due to the need for efficient use of space. Interestingly, tensor fields give rise to two sets of hyperstreamlines (defined in Section 4): one follows the major eigenvector field, and the other the minor eigenvector field. These observations have inspired our approach in which interactive tensor field design techniques are used to guide the road network generation. This concept is illustrated in Figures 1 and 3. The user can interactively edit a street network by either modifying the underlying tensor field or by changing the graph representing the street network. This allows for efficient modeling because we can combine global and local modeling operations, constraints, and procedural methods. 4. Source: Interactive procedural street modeling (#V4TQYB), Eugene Zhang, Gregory Esch, Guoning Chen, Pascal Müller, Peter Wonka, p. 6 Context: #57PDWB Interactive Procedural Street Modeling #ZPVYV6 6 Street Graph Generation #AZZQZY 6.4 Local Street Graph Editing using Tensor Fields Matching excerpt #36WURJ: Our system allows the user to specify regions inside which the existing street network is erased and replaced with one that is created from a locally defined tensor field. Such an approach lends the power of tensor field design to graph editing. In our system, the user can explicitly specifies a region to modify or uses the brush interface that we discussed in Section 5 to obtain a region. 5. Source: Interactive procedural street modeling (#V4TQYB), Eugene Zhang, Gregory Esch, Guoning Chen, Pascal Müller, Peter Wonka, p. 1 Context: #57PDWB Interactive Procedural Street Modeling #P35TFV 3 Pipeline Overview Matching excerpt #BMDPVE: Stage Two is the street graph generation step. Streets are computed as hyperstreamlines (Section 4) of the tensor field. In Section 6 we explain how to generate the street network, edit it, and modify existing street networks using a combination of graph-based and tensor field editing. Street networks are modeled using a hierarchy: major roads and minor roads . Major roads are typically major business roads and local highways, and minor roads are usually residential and back roads. A street network is stored as a graph G = (V, E) where V is a set of nodes and E is a set of edges. Nodes with three or more incident edges are crossings . Road attributes, such as road width, road type, pavement markings, and the type of lanes, are stored at nodes and edges. 6. Source: Interactive procedural street modeling (#V4TQYB), Eugene Zhang, Gregory Esch, Guoning Chen, Pascal Müller, Peter Wonka, p. 2 Context: #57PDWB Interactive Procedural Street Modeling #QNKCB5 5 Tensor Field Generation #FCEMBJ 5.1 Generation of Basis Fields Matching excerpt #6X3MNM: The tensor field is generated based on user constraints (desirable patterns) and topography information (water and park boundaries, terrain height, etc). Near the city center, the user may wish to create a typical North-South and East-West pattern. In contrast, near the coastline, it is often natural to design the road network to follow the coastline. To provide sufficient flexibility in addressing these different and often competing needs, we seek a tensor field design framework that allows both global and local control. 7. Source: Interactive procedural street modeling (#V4TQYB), Eugene Zhang, Gregory Esch, Guoning Chen, Pascal Müller, Peter Wonka, p. 2 Context: #57PDWB Interactive Procedural Street Modeling #QNKCB5 5 Tensor Field Generation Matching excerpt #RTLSM6: In this section, we describe how to generate a tensor field in the domain using our system. The approach is to edit tensor fields by specifying constraints such as regular and radial patterns, brush strokes, topography information, and rotation fields. While we borrow some vector and tensor field design techniques such as the use of basis fields and field smoothing from previous work [Zhang et al. 2006; Zhang et al. 2007; Chen et al. 2007], we contribute the application of the idea to street network modeling and introduce a novel brush interface that facilitates the specification of user constraints, the use of rotation fields to relax the orthogonality in a tensor field network, the combination of noise and tensor field design, hierarchical segmentation and editing, automatic incorporation of water and height maps in the generation of a tensor field, and the introduction of discontinuities. 8. Source: Interactive procedural street modeling (#V4TQYB), Eugene Zhang, Gregory Esch, Guoning Chen, Pascal Müller, Peter Wonka, p. 4 Context: #57PDWB Interactive Procedural Street Modeling #ZPVYV6 6 Street Graph Generation Matching excerpt #ZT498R: In this section, we describe how to generate a street network from a tensor field. We also describe how our system allows an existing street network to be modified directly as a graph or through local tensor field design. 9. Source: Interactive procedural street modeling (#V4TQYB), Eugene Zhang, Gregory Esch, Guoning Chen, Pascal Müller, Peter Wonka, p. 2 Context: #57PDWB Interactive Procedural Street Modeling #ZJ293A 4 Tensor Field Background Matching excerpt #4T4FXQ: Another important and relevant concept is the hyperstreamline , which describes curves that are tangent to an eigenvector field everywhere along their paths. A hyperstreamline is either major or minor depending on the type of the underlying eigenvector field. Note that the major and minor eigenvectors of a tensor field are not related to major and minor roads in a street network. For example, the tensor field corresponding to the major street network has its own major and minor hyperstreamlines. Hyperstreamlines have been used previously to visualize tensor fields [Wilson and Brannon 10. Source: Interactive procedural street modeling (#V4TQYB), Eugene Zhang, Gregory Esch, Guoning Chen, Pascal Müller, Peter Wonka, p. 5 Context: #57PDWB Interactive Procedural Street Modeling #ZPVYV6 6 Street Graph Generation #4SLJV9 6.1 Major Street Graph Generation from Tensor Fields Matching excerpt #J3F7DX: Transitions in Density: At city borders the road density decreases. Transitions in density are a phenomenon of the street graph and not the underlying tensor field. In our system, we use road density maps (or population density maps) to control d_{sep} in the road tracing algorithm described above. Figure 10 provides an illustrative example demonstrating how our system imitates the transition of density. Approximate matches 1. Source: Interactive procedural street modeling (#V4TQYB), Eugene Zhang, Gregory Esch, Guoning Chen, Pascal Müller, Peter Wonka, p. 2 Context: #57PDWB Interactive Procedural Street Modeling #QNKCB5 5 Tensor Field Generation #FCEMBJ 5.1 Generation of Basis Fields Score: 0.029 Related excerpt #6X3MNM: The tensor field is generated based on user constraints (desirable patterns) and topography information (water and park boundaries, terrain height, etc). Near the city center, the user may wish to create a typical North-South and East-West pattern. In contrast, near the coastline, it is often natural to design the road network to follow the coastline. To provide sufficient flexibility in addressing these different and often competing needs, we seek a tensor field design framework that allows both global and local control. 2. Source: Interactive procedural street modeling (#V4TQYB), Eugene Zhang, Gregory Esch, Guoning Chen, Pascal Müller, Peter Wonka, p. 6 Context: #57PDWB Interactive Procedural Street Modeling #ZPVYV6 6 Street Graph Generation #Y8ER56 6.3 Street Graph Editing Score: 0.028 Related excerpt #EGCNHM: where road networks share some similarities with fracture patterns. One example are major roads in rural Missouri (see Figure 14 left). In this case local topography dominates the road layout. We have some possibility to match these patterns with a tensor field and added noise. Figure 14 (right) shows a map generated using rotation on the graph (i.e. rotating the street segments). The rotation field is generated using Perlin noise discussed in Section 5.3. 3. Source: Interactive procedural street modeling (#V4TQYB), Eugene Zhang, Gregory Esch, Guoning Chen, Pascal Müller, Peter Wonka, p. 2 Context: #57PDWB Interactive Procedural Street Modeling #QNKCB5 5 Tensor Field Generation #FCEMBJ 5.1 Generation of Basis Fields Score: 0.028 Related excerpt #885URQ: Grid: An important building block for most cities is the grid pattern. Parcels are generated by two orthogonal sets of parallel roads. A grid pattern can be defined by a regular element indicating the direction of the major eigenvector field. See Figure 4 for a tensor field guiding streets in a regular grid pattern. Given the direction (u_x, u_y) defined at a point \mathbf{p}_0 we can compute l = \sqrt{u_x^2 + u_y^2} and \theta = \arctan(\frac{u_y}{u_x}) and define the following basis field (the constant direction field) [Zhang et al. 2007]: 4. Source: Interactive procedural street modeling (#V4TQYB), Eugene Zhang, Gregory Esch, Guoning Chen, Pascal Müller, Peter Wonka, p. 4 Context: #57PDWB Interactive Procedural Street Modeling #ZPVYV6 6 Street Graph Generation Score: 0.026 Related excerpt #ZT498R: In this section, we describe how to generate a street network from a tensor field. We also describe how our system allows an existing street network to be modified directly as a graph or through local tensor field design. 5. Source: Interactive procedural street modeling (#V4TQYB), Eugene Zhang, Gregory Esch, Guoning Chen, Pascal Müller, Peter Wonka, p. 2 Context: #57PDWB Interactive Procedural Street Modeling #P35TFV 3 Pipeline Overview Score: 0.025 Related excerpt #QCYCTQ: (2) that give rise to a major street network (3). Then the user refines the initial major road layout by placing a new tensor field design element inducing a radial structure in the tensor field (4) as well as the street graph (5). Using our segmentation algorithm, the user performs additional local tensor field modifications (6) and generate a minor road network (7). The user uses a rotation noise field to create irregular structures near the top (8) and produces the final result (9). The visualization of tensor fields shown in this paper is based on [van Wijk 2002; Zhang et al. 2007]. 6. Source: Interactive procedural street modeling (#V4TQYB), Eugene Zhang, Gregory Esch, Guoning Chen, Pascal Müller, Peter Wonka, p. 0 Context: #57PDWB Interactive Procedural Street Modeling #H9D7PX 1 Introduction Score: 0.025 Related excerpt #A2NHD8: An important aspect of street patterns is the existence of two dominant directions due to the need for efficient use of space. Interestingly, tensor fields give rise to two sets of hyperstreamlines (defined in Section 4): one follows the major eigenvector field, and the other the minor eigenvector field. These observations have inspired our approach in which interactive tensor field design techniques are used to guide the road network generation. This concept is illustrated in Figures 1 and 3. The user can interactively edit a street network by either modifying the underlying tensor field or by changing the graph representing the street network. This allows for efficient modeling because we can combine global and local modeling operations, constraints, and procedural methods. 7. Source: Interactive procedural street modeling (#V4TQYB), Eugene Zhang, Gregory Esch, Guoning Chen, Pascal Müller, Peter Wonka, p. 6 Context: #57PDWB Interactive Procedural Street Modeling #ZPVYV6 6 Street Graph Generation #AZZQZY 6.4 Local Street Graph Editing using Tensor Fields Score: 0.025 Related excerpt #36WURJ: Our system allows the user to specify regions inside which the existing street network is erased and replaced with one that is created from a locally defined tensor field. Such an approach lends the power of tensor field design to graph editing. In our system, the user can explicitly specifies a region to modify or uses the brush interface that we discussed in Section 5 to obtain a region. 8. Source: Interactive procedural street modeling (#V4TQYB), Eugene Zhang, Gregory Esch, Guoning Chen, Pascal Müller, Peter Wonka, p. 7 Context: #57PDWB Interactive Procedural Street Modeling #YE5T4M 8 Discussion Score: 0.024 Related excerpt #URGYL2: Strengths: The inherent strengths of tensor fields include the possibility to model street patterns, which usually contain two preferred directions that are often mutually perpendicular. Furthermore, tensor field design allows the user to quickly generate an initial street layout which can be further modified at either the tensor field level or the graph level. This flexibility is unmatched by editing tools that only operate on the graph level, especially when creating the typical street patterns such as the regular East-West and North-South patterns. 9. Source: Interactive procedural street modeling (#V4TQYB), Eugene Zhang, Gregory Esch, Guoning Chen, Pascal Müller, Peter Wonka, p. 3 Context: #57PDWB Interactive Procedural Street Modeling #QNKCB5 5 Tensor Field Generation #FCEMBJ 5.1 Generation of Basis Fields Score: 0.024 Related excerpt #7R43W5: Heightfield: The natural elevation is an important constraint for most road construction. We observe that roads are often built by taking into account the gradient of the height field. To derive a tensor field from a heightfield H(x, y) , we compute the gradient \nabla H = (\partial H / \partial x, \partial H / \partial y) . We then use the tensor field T(x, y) = R \begin{pmatrix} \cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{pmatrix} whose minor eigenvector field matches the gradient of the heightfield everywhere, i.e. \theta = \arctan(\frac{\partial H / \partial y}{\partial H / \partial x}) + \frac{\pi}{2} and R = \sqrt{(\partial H / \partial x)^2 + (\partial H / \partial y)^2} . 10. Source: Interactive procedural street modeling (#V4TQYB), Eugene Zhang, Gregory Esch, Guoning Chen, Pascal Müller, Peter Wonka, p. 1 Context: #57PDWB Interactive Procedural Street Modeling #P35TFV 3 Pipeline Overview Score: 0.024 Related excerpt #BMDPVE: Stage Two is the street graph generation step. Streets are computed as hyperstreamlines (Section 4) of the tensor field. In Section 6 we explain how to generate the street network, edit it, and modify existing street networks using a combination of graph-based and tensor field editing. Street networks are modeled using a hierarchy: major roads and minor roads . Major roads are typically major business roads and local highways, and minor roads are usually residential and back roads. A street network is stored as a graph G = (V, E) where V is a set of nodes and E is a set of edges. Nodes with three or more incident edges are crossings . Road attributes, such as road width, road type, pavement markings, and the type of lanes, are stored at nodes and edges. ### 35. Tool result: search_text Exact matches 1. Source: Procedural Riverscapes (#AK7NGE), A. Peytavie, B. Benes, E. Galin, E. Guérin, J. Gain, T. Dupont, Y. Cortial, p. 2 Context: #JJE8HN Procedural Riverscapes #Z6DVCN 4. River Network Graph Matching excerpt #JGM58Q: We begin by computing the drainage area A_{ij} for every cell C_{ij} of the input terrain \mathcal{T} (using the method of Freeman [Fre91]). The discrete river network \mathcal{D} is then simply the set of cells that have a drainage area greater than a user-controlled threshold value (see 2. Source: Terrain Generation Using Procedural Models Based on Hydrology (#DMTA8Y), Adrien Peytavie, Bedřich Beneš, Jean-David Génevaux, Éric Galin, Éric Guérin, p. 0 Context: #RULAFW Terrain Generation Using Procedural Models Based on Hydrology #SFQZPA 1 Introduction Matching excerpt #CL3N2Q: We propose a novel procedural approach, using river networks, for terrain modeling. The user optionally defines the river mouths and sketches the most important rivers on the terrain, and our approach generates the complete river network with the corresponding terrain, as shown in Fig. 1. The user can also control the river network and terrain generation with a set of intuitive parameters. Our method can represent large terrain models with complex river networks and geomorphologically consistent patterns that conform with observations from landscape and river science and yet provide a high level of controllability. The actual river geometry is generated by converting the drainage network data into a subset of river types that are taken from a well-known classification in hydrology [Rosen 1994]. The terrain is stored in a novel hierarchical continuous data representation that is inspired by constructive solid geometry (CSG). The terrain features are stored in the tree leaves, and the internal nodes define operations (blending, subtraction) on them. Contrary to most of the previous work, our terrain is represented by an analytic continuous function and not as a raster-based height field. Yet, our terrain is composed of many primitives and not a single abstract function. This allows us to generate large-scale terrains with an unlimited and locally varying level of detail. 3. Source: Terrain Generation Using Procedural Models Based on Hydrology (#DMTA8Y), Adrien Peytavie, Bedřich Beneš, Jean-David Génevaux, Éric Galin, Éric Guérin, p. 1 Context: #RULAFW Terrain Generation Using Procedural Models Based on Hydrology #CZMG8P 2 Related Work Matching excerpt #QKZVHE: Various techniques exist that attempt to incorporate rivers into the procedural terrain generation. Probably the first one is the paper by Kelley et al. [1988], who proposed a procedural method to generate watersheds. Their approach resembles ours because the river network is generated first and the terrain second. However, our algorithm creates large terrains represented by a continuous procedural model from a hydrographically and geomorphologically consistent river drainage network. It can also be used to generate terrains from partial input sketches. Prusinkiewicz et al. [1993] combined context-sensitive L-systems with the midpoint displacement method in an approach that imprints the rivers into fractal terrains. Later Belhadj and Audibert [2005] presented a modified stochastic subdivision algorithm that constrains ridges and river curves generated by fractional Brownian motion. Teoh [2009] presented an algorithm for terrain generation that also starts by producing the river network. However, our approach is based on models from hydrology, provides better control over the terrain generation process, and generates implicit terrain decomposition into continuous patches. Similarly, Derzapf et al. [2011] generated river networks on a planetary scale. 4. Source: Terrain Generation Using Procedural Models Based on Hydrology (#DMTA8Y), Adrien Peytavie, Bedřich Beneš, Jean-David Génevaux, Éric Galin, Éric Guérin, p. 3 Context: #RULAFW Terrain Generation Using Procedural Models Based on Hydrology #WWSSDJ 4 River Network Generation #DBN56J 4.2 River Network Generation #WXUE6Z 4.2.1 Node Selection Matching excerpt #MEQE9Y: The node that will be expanded is selected from the list of candidate nodes by using a heuristic that takes into account the node elevation and its priority index. Combining those two criteria allows a simultaneous creation of several hierarchical drainage networks competing for space. Moreover, modifying the relative importance of the node elevation and its priority index provides the user with a control over the network shape. If the priority index is preferred, the algorithm favors networks created from river mouths independent of their relative elevation. In contrast, when the node with the lowest elevation is selected, the algorithm first generates the drainage network in the lowlands (Fig. 5). 5. Source: Terrain Generation Using Procedural Models Based on Hydrology (#DMTA8Y), Adrien Peytavie, Bedřich Beneš, Jean-David Génevaux, Éric Galin, Éric Guérin, p. 3 Context: #RULAFW Terrain Generation Using Procedural Models Based on Hydrology #WWSSDJ 4 River Network Generation #DBN56J 4.2 River Network Generation #WXUE6Z 4.2.1 Node Selection Matching excerpt #Q6XWAJ: The parameter \zeta \in [0; +\infty[ controls the length of the drainage network by limiting the elevation range between two nodes \mathcal{X}_\zeta (Fig. 5). For small values of \zeta \approx 0 , it prioritizes the nodes with the lowest elevation and causes rivers to have more ramifications. When \zeta increases, the candidate node with the highest priority index is chosen, causing the formation of larger river-drainage networks. 6. Source: Terrain Generation Using Procedural Models Based on Hydrology (#DMTA8Y), Adrien Peytavie, Bedřich Beneš, Jean-David Génevaux, Éric Galin, Éric Guérin, p. 3 Context: #RULAFW Terrain Generation Using Procedural Models Based on Hydrology #WWSSDJ 4 River Network Generation #DBN56J 4.2 River Network Generation #VTGN6K 4.2.2 Node Expansion Matching excerpt #BMNJ7A: River Slope map. During the expansion step, the elevation of each new node should be higher than its ancestors to guarantee a consistent water flow. This elevation is computed according to a local river slope-magnitude value that is provided either by the user (Fig. 1 and 17) or generated procedurally (Fig. 18). Either way, the river slope-magnitude (a scalar value) defines only the height variation and provides no information on the direction of the expansion. Mapped on the whole terrain, this river slope map provides an intuitive way to describe how the drainage network will expand. 7. Source: Terrain Generation Using Procedural Models Based on Hydrology (#DMTA8Y), Adrien Peytavie, Bedřich Beneš, Jean-David Génevaux, Éric Galin, Éric Guérin, p. 1 Context: #RULAFW Terrain Generation Using Procedural Models Based on Hydrology #3G9YBV 3 Algorithm Overview Matching excerpt #EL3FAM: From this input, the system first generates the drainage river network. The network is created inside the domain formed by the contour and is represented as a geometric graph. The graph is generated by a progressive growth from the seeds placed on the domain contour and the input rivers already sketched by the user. The expansion algorithm is inspired by Horton-Strahler's ordering [Horton 1945], which quantifies the complexity of a tree structure. 8. Source: Physically-based analytical erosion for fast terrain generation (#DWXKYQ), Boris Gailleton, Guillaume Cordonnier, Petros Tzathas, Philippe Steer, p. 7 Context: #JDHNVB Physically-based analytical erosion for fast terrain generation #4VW4H2 5. Combining river network and elevations #USAEE8 5.3. Optimization-based altitude correction Matching excerpt #U5GU9Q: Our multigrid algorithm efficiently updates the river network to remove the discontinuities but is not applicable in all situations. For instance, a user might want to preserve the original drainage or use a procedurally generated river network [GGG + 13, GBG + 19]. For these cases, we assume that the river network is given and fixed, and we find the minimal correction to the analytical solution that removes the discontinuities. Figure 4 shows the discontinuities seen after applying the analytical solutions directly to the initial river network (left), and how the discontinuities were removed by our optimization method (right). 9. Source: Physically-based analytical erosion for fast terrain generation (#DWXKYQ), Boris Gailleton, Guillaume Cordonnier, Petros Tzathas, Philippe Steer, p. 11 Context: #JDHNVB Physically-based analytical erosion for fast terrain generation #PZFF56 7. Results #V6AZDH 7.3. Applicability of our method Matching excerpt #RKFYWV: Canyons , especially the well-known examples such as the Grand Canyon , do not exhibit the large flat-bottomed areas between the cliffs observed in Figure 12. The difference comes from the boundary conditions: our results in Figure 12 assumed a flat boundary condition, while a canyon is typically tributary to another river downstream. To implement this, we assume a constant a = a(0) to extend D_{x,t} negatively beyond the bound and we assume a constant slope to deduce z_0(D_{x,t}) . We illustrate the impact of this new boundary condition in Figure 13 where we set a single boundary node to which we assign this small slope boundary condition . We additionally set a source point on the other side of the terrain where we impose a strong drainage area A to enforce the formation of a main river connecting these two points. Erosion at time t = 700ky shows a fully formed canyon with multiple tributary rivers branching out. Note that the new boundary condition resulted in the desired V-shaped walls surrounding the main river. 10. Source: Physically-based analytical erosion for fast terrain generation (#DWXKYQ), Boris Gailleton, Guillaume Cordonnier, Petros Tzathas, Philippe Steer, p. 3 Context: #JDHNVB Physically-based analytical erosion for fast terrain generation #QW6CT7 3. Background and overview #Q878EF 3.2. Challenges and algorithm Matching excerpt #S3XW5N: We follow [Ste21] and order the computation along the river network. This network consists of a set of trees that covers the terrain and represents the progressive merging of high-altitude small streams down to the larger rivers. We obtain the 2D analytical solution by solving the 1D problem on each upstream path embedded in the tree structures. Similarly to the previous implicit time stepping schemes for the Stream Power Law [BW13, CBC*16], we separate the computation into two parts: first we accumulate the drainage area by following the river directions from the high elevations (ridges) of the terrain to the bounds, then we evaluate the analytical solution upstream from the bounds to the ridges. Approximate matches 1. Source: Procedural Riverscapes (#AK7NGE), A. Peytavie, B. Benes, E. Galin, E. Guérin, J. Gain, T. Dupont, Y. Cortial, p. 5 Context: #JJE8HN Procedural Riverscapes #9ZYJP2 6. Animated Procedural River Model Score: 0.026 Related excerpt #8W6ZMV: Our river modeling approach is centered on a hierarchical blend-flow tree (illustrated in Figure 15) that merges animated procedural riverflow primitives using blending operators. We take inspiration from the Feature Hierarchy of Genevaux et al. [GGP + 15] and generalize it to support animated content. The riverflow primitives are leaf nodes in the tree and encapsulate temporally self-similar patterns commonly observed in rivers. Each primitive is responsible for animating a stretch of cohesive water surface, such as waves, whirlpools, or wakes, over a compact support. They have parameterized inputs for user control, a low memory footprint, and are capable of generating precise and varied animated content (see the accompanying video). Blend operators are internal nodes of the tree that act to combine and aggregate overlapping leaf nodes and sub-trees. 2. Source: Procedural Riverscapes (#AK7NGE), A. Peytavie, B. Benes, E. Galin, E. Guérin, J. Gain, T. Dupont, Y. Cortial, p. 3 Context: #JJE8HN Procedural Riverscapes #Z6DVCN 4. River Network Graph Score: 0.025 Related excerpt #WNSBS7: P_{ij} = A_{ij}^{1/2} \cdot S_{ij} , and averaged. The final derivation is of the Horton-Strahler number [Hor45], a numerical measure of branching complexity, with higher numbered river segments having more feeder tributaries. With this the river network graph is fully labeled and can be passed directly to the River Network Amplification process described in Section 5. 3. Source: Procedural Riverscapes (#AK7NGE), A. Peytavie, B. Benes, E. Galin, E. Guérin, J. Gain, T. Dupont, Y. Cortial, p. 0 Context: #JJE8HN Procedural Riverscapes #RMH5XA 1. Introduction Score: 0.023 Related excerpt #9ESE2P: In more detail, we take as input a digital elevation model, evaluate its hydrological characteristics, and specify the course of a detailed, possibly branching river network (see Figure 1). Detailed riverbed cross-sections are then derived using Rosgen classification and flow characteristics and the resulting geometry can be inscribed into the terrain heightfield. Then an attendant blend-flow tree is generated automatically. For instance, cascade primitives are placed after step-wise drops in elevation, while basins will be populated with calm water primitives. Our key observation is that visually a river surface is in a steady flow state, and displays only small periodic, and random perturbations. For example, the unperturbed wake behind a submerged rock varies subtly in form, but not in position. Movement is also predominantly in 2\frac{1}{2}D , with the occurrence of locally significant patterns such as vortices, ripples, whirlpools, and small cascades. Rather than implementing a full fluid simulation with the attendant scaling issues, we blend animated procedural primitives to capture these cyclical patterns. 4. Source: Terrain Generation Using Procedural Models Based on Hydrology (#DMTA8Y), Adrien Peytavie, Bedřich Beneš, Jean-David Génevaux, Éric Galin, Éric Guérin, p. 1 Context: #RULAFW Terrain Generation Using Procedural Models Based on Hydrology #3G9YBV 3 Algorithm Overview Score: 0.028 Related excerpt #EL3FAM: From this input, the system first generates the drainage river network. The network is created inside the domain formed by the contour and is represented as a geometric graph. The graph is generated by a progressive growth from the seeds placed on the domain contour and the input rivers already sketched by the user. The expansion algorithm is inspired by Horton-Strahler's ordering [Horton 1945], which quantifies the complexity of a tree structure. 5. Source: Terrain Generation Using Procedural Models Based on Hydrology (#DMTA8Y), Adrien Peytavie, Bedřich Beneš, Jean-David Génevaux, Éric Galin, Éric Guérin, p. 1 Context: #RULAFW Terrain Generation Using Procedural Models Based on Hydrology #3G9YBV 3 Algorithm Overview Score: 0.025 Related excerpt #TJ6DHX: The output of the river network generator is a set of 3D polylines with increasing elevation from the outlet to the spring. The rivers and their parts are then classified into distinct procedural primitives. We use building blocks such as junctions, springs, deltas, and river trajectories, for the final river rendering. This categorization is inspired by the Rosgen classification [Rosgen 1994], which is used in hydrology and geomorphology. 6. Source: Terrain Generation Using Procedural Models Based on Hydrology (#DMTA8Y), Adrien Peytavie, Bedřich Beneš, Jean-David Génevaux, Éric Galin, Éric Guérin, p. 2 Context: #RULAFW Terrain Generation Using Procedural Models Based on Hydrology #WWSSDJ 4 River Network Generation #YQ7VH4 4.1 Initial Candidate Nodes Score: 0.023 Related excerpt #WJF6JY: The first step consists of creating the set of initial candidate nodes that will be expanded later. The candidate nodes are located at the river mouths on the contour \Gamma . Alternatively, if the user specified input sketches representing some parts of the rivers, the initial nodes are placed on their extremities and on regularly jittered sample locations along their paths as shown in Fig. 3. Each node has been assigned a priority index that defines its importance. Both the position and the priority index define the overall appearance of the resulting river network hierarchy. 7. Source: Terrain Generation Using Procedural Models Based on Hydrology (#DMTA8Y), Adrien Peytavie, Bedřich Beneš, Jean-David Génevaux, Éric Galin, Éric Guérin, p. 2 Context: #RULAFW Terrain Generation Using Procedural Models Based on Hydrology #3G9YBV 3 Algorithm Overview Score: 0.022 Related excerpt #KAMGFF: Once the river network is defined, the algorithm extracts the graph topology and geometry that is used for the terrain generation in the next step. We decompose the terrain into a set of patches by computing the Voronoi cells corresponding to the nodes of the river graph. The algorithm then generates the hierarchical watershed structure by traversing the geometric graph and gathering information of the Voronoi cells. This step enables us to compute the area of the watersheds and subwatersheds and to evaluate the flow of the water-courses at every node in the graph. 8. Source: Terrain Generation Using Procedural Models Based on Hydrology (#DMTA8Y), Adrien Peytavie, Bedřich Beneš, Jean-David Génevaux, Éric Galin, Éric Guérin, p. 2 Context: #RULAFW Terrain Generation Using Procedural Models Based on Hydrology #WWSSDJ 4 River Network Generation #DBN56J 4.2 River Network Generation Score: 0.022 Related excerpt #L2BUXU: The river network is generated by incrementally growing and elevating the geometric graphs \mathcal{G} using a probabilistic approach. The graph network has a set of candidate nodes \mathcal{X} that is expanded using a selection algorithm and several rules. We iteratively perform the following three steps: 9. Source: Physically-based analytical erosion for fast terrain generation (#DWXKYQ), Boris Gailleton, Guillaume Cordonnier, Petros Tzathas, Philippe Steer, p. 5 Context: #JDHNVB Physically-based analytical erosion for fast terrain generation #UE2TM4 4. Analytical solutions of the stream power law #5CR379 4.3. Extension to the 2D terrain domain Score: 0.022 Related excerpt #Z3ZLJU: This ordering results in a set of river trees whose roots correspond to the bounds of the terrain and whose leaves correspond to the ridges. The inset figure below shows the river network for a 32 \times 32 terrain, different colors corresponding to different trees. 10. Source: Scalable real‐time animation of rivers (#XDESU9), Eric Bruneton, Fabrice Neyret, Nicolas Holzschuch, Qizhi Yu, p. 3 Context: #B9LYTR Scalable real-time animation of rivers #S3P9T2 4. Computing flow velocities #89NKTG 4.1. Finding the volumetric flow rate from the hydro geometry Score: 0.029 Related excerpt #QXKFZQ: The hydrographic network is a directed graph, expressing the connections between channels, and flowing from the source to the ocean. In the case of streams joining into rivers, themselves joining into larger rivers, the graph is a tree. At the delta of a river, the graph is an inverted tree. If there are no islands, we have an acyclic graph; each obstacle or island results in a cycle in the graph. Islands and obstacles can either be treated at this level, or be treated as a special case in the interpolation method. ### 36. Assistant tool calls #### Tool call: `search_text` ```json { "document_id": "B6P8L4", "limit": 8, "query": "destinations attraction destination points trail system areas regions" } ``` #### Tool call: `search_text` ```json { "document_id": "UYLTYJ", "limit": 8, "query": "destinations attraction destination points trail system areas regions" } ``` #### Tool call: `search_text` ```json { "document_id": "PXG56P", "limit": 8, "query": "partition regions centers boundaries whole differentiation" } ``` #### Tool call: `search_text` ```json { "document_id": "XW22YY", "limit": 8, "query": "land regions zones boundaries neighborhoods parcels blocks" } ``` #### Tool call: `search_text` ```json { "document_id": "RBS5K6", "limit": 8, "query": "partition fluid layers particles classified" } ``` #### Tool call: `search_text` ```json { "document_id": "CQBDX4", "limit": 8, "query": "clusters classes categories partition latent space levels tiles" } ``` ### 37. Tool result: search_text Exact matches 1. Source: Active walker model for the formation of human and animal trail systems (#B6P8L4), Dirk Helbing, Frank Schweitzer, Joachim Keltsch, Péter Molnár, p. 20 Context: #E9C9Y2 Active Walker Model for the Formation of Human and Animal Trail Systems #T6PSLK V. SUMMARY AND OUTLOOK Matching excerpt #YZ2YVF: The structure of the resulting trail system can considerably vary with the species. This depends decisively on the main effect which counteracts the trail attraction. Whereas our model ants find their destinations (the food sources) by chance, pedestrians can directly orient towards their destinations, so that fluctuations are no necessary model component in this case. Thus, for certain ant species a dendritic trail system is found, the detailed form of which depends on random events, i.e. the concrete history of its evolution. Pedestrians, however, produce a minimal detour system, i.e. an optimal compromise between a direct way system and a minimal way system. 2. Source: Active walker model for the formation of human and animal trail systems (#B6P8L4), Dirk Helbing, Frank Schweitzer, Joachim Keltsch, Péter Molnár, p. 13 Context: #E9C9Y2 Active Walker Model for the Formation of Human and Animal Trail Systems #3WQXTG IV. HUMAN TRAIL FORMATION Matching excerpt #JAAMW3: At the beginning, pedestrians take the direct ways to their respective destinations. However, after some time pedestrians begin to use already existing trails, since this is more comfortable than to clear new ways. The frequency of usage decides which trails are reinforced and which ones vanish in the course of time. If the attractiveness of the forming trails is large, the final trail system is a minimal way system (which is the shortest way system that connects all entry points and destinations). However, because of the pedestrians' dislike of taking detours the evolution of the trail system normally stops before this state is reached. In other words, a so-called minimal detour system develops if the model parameters are chosen realistically (cf. Fig. 5). The resulting trails can considerably differ from the direct ways which the pedestrians would use if these were equally comfortable. 3. Source: Active walker model for the formation of human and animal trail systems (#B6P8L4), Dirk Helbing, Frank Schweitzer, Joachim Keltsch, Péter Molnár, p. 15 Context: #E9C9Y2 Active Walker Model for the Formation of Human and Animal Trail Systems #3WQXTG IV. HUMAN TRAIL FORMATION #CAN22D B. Macroscopic formulation of trail formation Matching excerpt #RRJ2BJ: From the above ‘microscopic’ model of trail formation we will now derive the related ‘macroscopic’ equations. For this purpose we need to distinguish different subpopulations a of individuals \alpha . By a(\tau) we denote the time-dependent set of individuals \alpha who have started from the same entry point \mathbf{p}_a with the same destination \mathbf{d}_a . Therefore, the different sets a correspond to the possible (directed) combinations between existing entry points and destinations. 4. Source: Active walker model for the formation of human and animal trail systems (#B6P8L4), Dirk Helbing, Frank Schweitzer, Joachim Keltsch, Péter Molnár, p. 20 Context: #E9C9Y2 Active Walker Model for the Formation of Human and Animal Trail Systems #T6PSLK V. SUMMARY AND OUTLOOK Matching excerpt #QNTA7X: As a consequence, we could derive a macroscopic model for the trail formation by pedestrians, but not for ants. It implied a self-consistent field method for a very efficient calculation of the finally evolving trail system. This is determined by the location of the entry points and destinations (e.g. houses, shops, or parking lots) and the rates of choosing the possible connections between them. Apart from this it depends on two parameters only, which was demonstrated by scaling to dimensionless equations. These are related to the trail attractiveness and the average velocity of motion. 5. Source: Active walker model for the formation of human and animal trail systems (#B6P8L4), Dirk Helbing, Frank Schweitzer, Joachim Keltsch, Péter Molnár, p. 22 Context: #E9C9Y2 Active Walker Model for the Formation of Human and Animal Trail Systems #T6PSLK V. SUMMARY AND OUTLOOK #CN7EEL B. Implications for urban planners: Optimization of way systems Matching excerpt #AR9S9D: Computer simulations of our pedestrian trail formation model will be a valuable tool for designing convenient way systems (cf. Fig. 7). For planning purposes the model parameters \lambda and \kappa must be specified in a realistic way. Then, one needs to simulate the expected flows of pedestrians that enter the considered system at certain entry points with the intention to reach certain destinations. Already existing ways can be taken into account by the function G'_0(\mathbf{x}) . According to our model, a trail system will evolve which minimizes overall detours and thereby provides an optimal compromise between a direct and a minimal way system. It is expected that the corresponding ways meet the pedestrian requirements best: They will most likely be accepted and actually used, since they take into account the route choice habits of pedestrians. For the simulation of realistic situations, the results can serve as planning guidelines for architects, landscape gardeners, and urban planners. 6. Source: Active walker model for the formation of human and animal trail systems (#B6P8L4), Dirk Helbing, Frank Schweitzer, Joachim Keltsch, Péter Molnár, p. 2 Context: #E9C9Y2 Active Walker Model for the Formation of Human and Animal Trail Systems #TTL9MC I. INTRODUCTION Matching excerpt #KXXUJY: As our experience tells us, trails are adapted to the requirements of their users. In the course of time, frequently used trails become more developed, making them more attractive, whereas rarely used trails vanish again. Trails with large detours become optimized by creating shortcuts. New destinations or entry points are connected to an existing trail system. These dynamical processes occur basically without any common planning or direct communication among the users. Instead, the adaptation process can be understood as a self-organization phenomenon, resulting from the non-linear feedback between the users and the trails [35]. 7. Source: Active walker model for the formation of human and animal trail systems (#B6P8L4), Dirk Helbing, Frank Schweitzer, Joachim Keltsch, Péter Molnár, p. 6 Context: #E9C9Y2 Active Walker Model for the Formation of Human and Animal Trail Systems #YEVN37 II. ACTIVE WALKER MODEL OF TRAIL FORMATION Matching excerpt #3YJ2PE: Therefore, the mechanism of trail formation is based on some kind of agglomeration process , which is delocalized due to the directedness of the walkers' motion. Starting with a plain, spatially homogeneous ground, the walkers will move arbitrarily. However, by continuously leaving markings, they produce trails which have an attractive effect on nearby walkers. Thus, the agents begin to use already existing trails after some time. By this, a kind of selection process between trails occurs (cf. [43]): Frequently used trails are reinforced, which makes them even more attractive, whereas rarely used trails may vanish again. The trails begin to bundle, especially where different trails meet or intersect. Therefore, even walkers with different entry points and destinations use and maintain common parts of the trail system. 8. Source: Active walker model for the formation of human and animal trail systems (#B6P8L4), Dirk Helbing, Frank Schweitzer, Joachim Keltsch, Péter Molnár, p. 1 Context: #E9C9Y2 Active Walker Model for the Formation of Human and Animal Trail Systems #4PR2LM Abstract Matching excerpt #LFS7S4: Whereas pedestrians leave footprints on the ground, ants produce chemical markings for their orientation. Nevertheless, it is more important that pedestrians steer towards a certain destination, while ants usually find their food sources by chance, i.e. they reach their destination in a stochastic way. As a consequence, the typical structure of the evolving trail systems depends on the respective species. Some ant species produce a dendritic trail system, whereas pedestrians generate a minimal detour system. Approximate matches 1. Source: Active walker model for the formation of human and animal trail systems (#B6P8L4), Dirk Helbing, Frank Schweitzer, Joachim Keltsch, Péter Molnár, p. 2 Context: #E9C9Y2 Active Walker Model for the Formation of Human and Animal Trail Systems #TTL9MC I. INTRODUCTION Score: 0.03 Related excerpt #KXXUJY: As our experience tells us, trails are adapted to the requirements of their users. In the course of time, frequently used trails become more developed, making them more attractive, whereas rarely used trails vanish again. Trails with large detours become optimized by creating shortcuts. New destinations or entry points are connected to an existing trail system. These dynamical processes occur basically without any common planning or direct communication among the users. Instead, the adaptation process can be understood as a self-organization phenomenon, resulting from the non-linear feedback between the users and the trails [35]. 2. Source: Active walker model for the formation of human and animal trail systems (#B6P8L4), Dirk Helbing, Frank Schweitzer, Joachim Keltsch, Péter Molnár, p. 13 Context: #E9C9Y2 Active Walker Model for the Formation of Human and Animal Trail Systems #3WQXTG IV. HUMAN TRAIL FORMATION Score: 0.029 Related excerpt #JAAMW3: At the beginning, pedestrians take the direct ways to their respective destinations. However, after some time pedestrians begin to use already existing trails, since this is more comfortable than to clear new ways. The frequency of usage decides which trails are reinforced and which ones vanish in the course of time. If the attractiveness of the forming trails is large, the final trail system is a minimal way system (which is the shortest way system that connects all entry points and destinations). However, because of the pedestrians' dislike of taking detours the evolution of the trail system normally stops before this state is reached. In other words, a so-called minimal detour system develops if the model parameters are chosen realistically (cf. Fig. 5). The resulting trails can considerably differ from the direct ways which the pedestrians would use if these were equally comfortable. 3. Source: Active walker model for the formation of human and animal trail systems (#B6P8L4), Dirk Helbing, Frank Schweitzer, Joachim Keltsch, Péter Molnár, p. 20 Context: #E9C9Y2 Active Walker Model for the Formation of Human and Animal Trail Systems #T6PSLK V. SUMMARY AND OUTLOOK Score: 0.028 Related excerpt #YZ2YVF: The structure of the resulting trail system can considerably vary with the species. This depends decisively on the main effect which counteracts the trail attraction. Whereas our model ants find their destinations (the food sources) by chance, pedestrians can directly orient towards their destinations, so that fluctuations are no necessary model component in this case. Thus, for certain ant species a dendritic trail system is found, the detailed form of which depends on random events, i.e. the concrete history of its evolution. Pedestrians, however, produce a minimal detour system, i.e. an optimal compromise between a direct way system and a minimal way system. 4. Source: Active walker model for the formation of human and animal trail systems (#B6P8L4), Dirk Helbing, Frank Schweitzer, Joachim Keltsch, Péter Molnár, p. 6 Context: #E9C9Y2 Active Walker Model for the Formation of Human and Animal Trail Systems #YEVN37 II. ACTIVE WALKER MODEL OF TRAIL FORMATION Score: 0.027 Related excerpt #3YJ2PE: Therefore, the mechanism of trail formation is based on some kind of agglomeration process , which is delocalized due to the directedness of the walkers' motion. Starting with a plain, spatially homogeneous ground, the walkers will move arbitrarily. However, by continuously leaving markings, they produce trails which have an attractive effect on nearby walkers. Thus, the agents begin to use already existing trails after some time. By this, a kind of selection process between trails occurs (cf. [43]): Frequently used trails are reinforced, which makes them even more attractive, whereas rarely used trails may vanish again. The trails begin to bundle, especially where different trails meet or intersect. Therefore, even walkers with different entry points and destinations use and maintain common parts of the trail system. 5. Source: Active walker model for the formation of human and animal trail systems (#B6P8L4), Dirk Helbing, Frank Schweitzer, Joachim Keltsch, Péter Molnár, p. 9 Context: #E9C9Y2 Active Walker Model for the Formation of Human and Animal Trail Systems #P3AR99 III. TRUNK TRAIL FORMATION BY ANTS Score: 0.026 Related excerpt #96FW7W: The perception of already existing trails will have an attractive effect \mathbf{f}_{tr}(\mathbf{r}_\alpha, \mathbf{v}_\alpha, t) to the active walkers. This has been defined by the gradients of the trail potentials, 6. Source: Active walker model for the formation of human and animal trail systems (#B6P8L4), Dirk Helbing, Frank Schweitzer, Joachim Keltsch, Péter Molnár, p. 11 Context: #E9C9Y2 Active Walker Model for the Formation of Human and Animal Trail Systems #3WQXTG IV. HUMAN TRAIL FORMATION Score: 0.026 Related excerpt #JN4H6B: Trail formation by pedestrians has been investigated only very recently [60]. It can be interpreted as a complex interplay between pedestrian motion, human orientation, and environmental changes: On the one hand, pedestrians tend to take the shortest way to their destination. On the other hand, they avoid to walk on bumpy ground, since this is uncomfortable. Therefore, they prefer to use existing trails, but they build a new shortcut, if the relative detour would be too large. In the latter case they generate a new trail, since footprints clear some vegetation. Examples of the resulting trail systems can be found in green areas, like public parks (cf. Fig. 3). 7. Source: Active walker model for the formation of human and animal trail systems (#B6P8L4), Dirk Helbing, Frank Schweitzer, Joachim Keltsch, Péter Molnár, p. 13 Context: #E9C9Y2 Active Walker Model for the Formation of Human and Animal Trail Systems #3WQXTG IV. HUMAN TRAIL FORMATION Score: 0.024 Related excerpt #C4GYR3: Finally, we need to specify the trail potential V_{tr} for pedestrians. Obviously a trail must be recognized by the walkers and near enough in order to be used. Whereas the ground potential G(\mathbf{r}, t) describes the existence of a trail segment at position \mathbf{r} , the trail potential V_{tr}(\mathbf{r}_\alpha, t) reflects the attractiveness of a trail from the actual position \mathbf{r}_\alpha(t) of the walker. Since this will decrease with the distance \|\mathbf{r} - \mathbf{r}_\alpha\| , we have applied the relation 8. Source: Active walker model for the formation of human and animal trail systems (#B6P8L4), Dirk Helbing, Frank Schweitzer, Joachim Keltsch, Péter Molnár, p. 1 Context: #E9C9Y2 Active Walker Model for the Formation of Human and Animal Trail Systems #4PR2LM Abstract Score: 0.024 Related excerpt #PCL35Q: The trail formation model can be used as a tool for the optimization of pedestrian facilities: It allows urban planners to design convenient way systems which actually meet the route choice habits of pedestrians. ### 38. Tool result: search_text Exact matches 1. Source: Modelling the Evolution of Human Trail Systems (#UYLTYJ), Dirk Helbing, Joachim Keltsch, Péter Molnár, p. 3 Context: #BSSYS2 Modelling the Evolution of Human Trail Systems Matching excerpt #PG7BAQ: A direct way system (which provides the shortest connections, but covers a lot of space) only develops if all ways are almost equally comfortable. If the advantage \kappa of using existing trails is large, the final trail system is a minimal way system (which is the shortest way system that connects all entry points and destinations). For realistic values of \kappa , the evolution of the trail system stops before this state is reached (Figure 2). Thus, \kappa is related to the average relative detour of the walkers. We conjecture that the resulting way system is the shortest one which is compatible with a certain accepted relative detour. In this sense, it yields an optimal compromise between convenience and shortness. 2. Source: Modelling the Evolution of Human Trail Systems (#UYLTYJ), Dirk Helbing, Joachim Keltsch, Péter Molnár, p. 3 Context: #BSSYS2 Modelling the Evolution of Human Trail Systems Matching excerpt #7WG2F4: Therefore, we suggest to use the above model as a tool for urban planners and landscape gardeners, who have the dilemma to build most comfortable way systems at minimal construction costs. For planning purposes one needs to know the entry points and destinations within the considered area and the rates of usage of their connections. If necessary, these can be estimated by trip chaining models [22], which are also needed in cases of complex lines of access and sight. The effects of the physical terrain and already existing ways can be taken into account by the function G_0(\vec{r}) . By varying the model parameter \kappa , the overall length of the resulting trail system can be influenced (Figures 2 and 3). In the same way, one can check its structural stability. Presently, we are evaluating typical parameter values of \lambda and \kappa by comparison of simulation results with real pedestrian flows which are reconstructed from video films by image processing. These values shall be used for designing convenient way systems in residential areas, parks, and recreation areas by means of computer simulations (Figure 3). We expect that such way systems will actually be accepted, since they take into account the route choice habits of pedestrians. 3. Source: Modelling the Evolution of Human Trail Systems (#UYLTYJ), Dirk Helbing, Joachim Keltsch, Péter Molnár, p. 2 Context: #BSSYS2 Modelling the Evolution of Human Trail Systems Matching excerpt #FCSQLN: On a plain, homogeneous ground, the walking direction \vec{e}_\alpha of pedestrian \alpha is determined by the direction of the next destination \vec{d}_\alpha , i.e. \vec{e}_\alpha(\vec{r}_\alpha) = (\vec{d}_\alpha - \vec{r}_\alpha) / \|\vec{d}_\alpha - \vec{r}_\alpha\| . Without a destination, a pedestrian is expected to move into the direction of the largest increase of ground attraction, which is given by the (normalized) gradient \vec{\nabla}_{\vec{r}_\alpha} V_{\text{tr}}(\vec{r}_\alpha, t) of the trail potential. However, since the choice of the walking direction \vec{e}_\alpha is influenced by the destination and existing trails at the same time, the orientation relation 4. Source: Modelling the Evolution of Human Trail Systems (#UYLTYJ), Dirk Helbing, Joachim Keltsch, Péter Molnár, p. 3 Context: #BSSYS2 Modelling the Evolution of Human Trail Systems Matching excerpt #ST8FZE: At the beginning, pedestrians take the direct ways to their respective destinations. However, after some time they begin to use already existing trails, since this is more comfortable than to clear new ways. By this, a kind of selection process [20,21,16] between trails sets in: Frequently used trails are more attractive than others. For this reason they are chosen very often, and the resulting reinforcement makes them even more attractive. However, the weathering effect destroys rarely used trails and limits the maximum length of the way system which can be supported by a certain rate of trail usage. As a consequence, the trails begin to bundle, especially where different trails meet or intersect. This explains, why pedestrians with different destinations use and produce common parts of the trail system (Figures 2 and 3). 5. Source: Modelling the Evolution of Human Trail Systems (#UYLTYJ), Dirk Helbing, Joachim Keltsch, Péter Molnár, p. 2 Context: #BSSYS2 Modelling the Evolution of Human Trail Systems Matching excerpt #PMPCXQ: Our simulations base on a discretization of the considered area in small quadratic elements of equal size, which converts the integral (2) into a sum. Temporal and spatial derivatives are approximated by difference quotients. The presented examples begin with plain, homogeneous ground. All pedestrians have their own destinations and entry points, from which they start at a randomly chosen point in time. In Figure 2 (Figure 3) pedestrians move between all possible pairs of three (four) fixed places. While in Figure 4 the entry points and destinations are distributed over the small ends of the ground. 6. Source: Modelling the Evolution of Human Trail Systems (#UYLTYJ), Dirk Helbing, Joachim Keltsch, Péter Molnár, p. 1 Context: #BSSYS2 Modelling the Evolution of Human Trail Systems Matching excerpt #F6A63D: Previous studies have shown that various observed self-organization phenomena in pedestrian crowds can be simulated very realistically. This includes the emergence of lanes of uniform walking direction and oscillatory changes of the passing direction at bottlenecks [7,10]. Another interesting collective effect of pedestrian motion, which we have investigated very recently, is the formation of trail systems in green areas. In many cases, the pedestrians' desire to take the shortest way and the specific properties of the terrain are insufficient for an explanation of the trail characteristics. It is essential to include the effect of human orientation. To simulate the typical features of trail systems, we have extended the aforementioned model of pedestrian motion to an active walker model by introducing equations for environmental changes and their impact on the chosen walking direction. 7. Source: Modelling the Evolution of Human Trail Systems (#UYLTYJ), Dirk Helbing, Joachim Keltsch, Péter Molnár, p. 2 Context: #BSSYS2 Modelling the Evolution of Human Trail Systems Matching excerpt #4DF7BV: A comparison of simulation results with photographs shows that the above described model is in good agreement with empirical observations. In particular, the evolution of the unexpected ‘island’ in the middle of the trail system in Figure 1 can be correctly described (Figure 2). The goodness of fit of the model is quite surprising, since it contains only two independent parameters \kappa = IT/\sigma^2 and \lambda = V^0 T/\sigma , where V^0 denotes the average of the desired velocities v_\alpha^0 . This can be shown by scaling the model to dimensionless equations. The parameter \lambda was kept constant. 8. Source: Modelling the Evolution of Human Trail Systems (#UYLTYJ), Dirk Helbing, Joachim Keltsch, Péter Molnár, p. 2 Context: #BSSYS2 Modelling the Evolution of Human Trail Systems Matching excerpt #FXKR3R: The trail potential V_{\text{tr}}(\vec{r}_\alpha, t) reflects the attractiveness of walking at place \vec{r}_\alpha . It describes indirect long-range interactions via environmental changes, which are essential for the characteristics of the evolving patterns [18]. Approximate matches 1. Source: Modelling the Evolution of Human Trail Systems (#UYLTYJ), Dirk Helbing, Joachim Keltsch, Péter Molnár, p. 1 Context: #BSSYS2 Modelling the Evolution of Human Trail Systems Score: 0.03 Related excerpt #XDF23G: The attractiveness of a trail segment at place \vec{r} from the perspective of place \vec{r}_\alpha decreases with its distance \|\vec{r} - \vec{r}_\alpha(t)\| and depends on the visibility \sigma(\vec{r}_\alpha) . Considering this by a factor \exp(-\|\vec{r} - \vec{r}_\alpha\|/\sigma(\vec{r}_\alpha)) and taking the spatial average by integration of the weighted ground structure over the green area, we obtain 2. Source: Modelling the Evolution of Human Trail Systems (#UYLTYJ), Dirk Helbing, Joachim Keltsch, Péter Molnár, p. 2 Context: #BSSYS2 Modelling the Evolution of Human Trail Systems Score: 0.028 Related excerpt #FXKR3R: The trail potential V_{\text{tr}}(\vec{r}_\alpha, t) reflects the attractiveness of walking at place \vec{r}_\alpha . It describes indirect long-range interactions via environmental changes, which are essential for the characteristics of the evolving patterns [18]. 3. Source: Modelling the Evolution of Human Trail Systems (#UYLTYJ), Dirk Helbing, Joachim Keltsch, Péter Molnár, p. 3 Context: #BSSYS2 Modelling the Evolution of Human Trail Systems Score: 0.028 Related excerpt #ST8FZE: At the beginning, pedestrians take the direct ways to their respective destinations. However, after some time they begin to use already existing trails, since this is more comfortable than to clear new ways. By this, a kind of selection process [20,21,16] between trails sets in: Frequently used trails are more attractive than others. For this reason they are chosen very often, and the resulting reinforcement makes them even more attractive. However, the weathering effect destroys rarely used trails and limits the maximum length of the way system which can be supported by a certain rate of trail usage. As a consequence, the trails begin to bundle, especially where different trails meet or intersect. This explains, why pedestrians with different destinations use and produce common parts of the trail system (Figures 2 and 3). 4. Source: Modelling the Evolution of Human Trail Systems (#UYLTYJ), Dirk Helbing, Joachim Keltsch, Péter Molnár, p. 3 Context: #BSSYS2 Modelling the Evolution of Human Trail Systems Score: 0.027 Related excerpt #PG7BAQ: A direct way system (which provides the shortest connections, but covers a lot of space) only develops if all ways are almost equally comfortable. If the advantage \kappa of using existing trails is large, the final trail system is a minimal way system (which is the shortest way system that connects all entry points and destinations). For realistic values of \kappa , the evolution of the trail system stops before this state is reached (Figure 2). Thus, \kappa is related to the average relative detour of the walkers. We conjecture that the resulting way system is the shortest one which is compatible with a certain accepted relative detour. In this sense, it yields an optimal compromise between convenience and shortness. 5. Source: Modelling the Evolution of Human Trail Systems (#UYLTYJ), Dirk Helbing, Joachim Keltsch, Péter Molnár, p. 3 Context: #BSSYS2 Modelling the Evolution of Human Trail Systems Score: 0.026 Related excerpt #7WG2F4: Therefore, we suggest to use the above model as a tool for urban planners and landscape gardeners, who have the dilemma to build most comfortable way systems at minimal construction costs. For planning purposes one needs to know the entry points and destinations within the considered area and the rates of usage of their connections. If necessary, these can be estimated by trip chaining models [22], which are also needed in cases of complex lines of access and sight. The effects of the physical terrain and already existing ways can be taken into account by the function G_0(\vec{r}) . By varying the model parameter \kappa , the overall length of the resulting trail system can be influenced (Figures 2 and 3). In the same way, one can check its structural stability. Presently, we are evaluating typical parameter values of \lambda and \kappa by comparison of simulation results with real pedestrian flows which are reconstructed from video films by image processing. These values shall be used for designing convenient way systems in residential areas, parks, and recreation areas by means of computer simulations (Figure 3). We expect that such way systems will actually be accepted, since they take into account the route choice habits of pedestrians. 6. Source: Modelling the Evolution of Human Trail Systems (#UYLTYJ), Dirk Helbing, Joachim Keltsch, Péter Molnár, p. 0 Context: #BSSYS2 Modelling the Evolution of Human Trail Systems Score: 0.025 Related excerpt #CR8C3T: Many human social phenomena, such as cooperation [1–3], the growth of settlements [4], traffic dynamics [5–7] and pedestrian movement [7–10], appear to be accessible to mathematical descriptions that invoke self-organization [11,12]. Here we develop a model of pedestrian motion to explore the evolution of trails in urban green spaces such as parks. Our aim is to address such questions as what the topological structures of these trail systems are [13], and whether optimal path systems can be predicted for urban planning. We use an ‘active walker’ model [14–19] that takes into account pedestrian motion and orientation and the concomitant feedbacks with the surrounding environment. Such models have previously been applied to the study of complex structure formation in physical [14–16], chemical [17] and biological [18,19] systems. We find that our model is able to reproduce many of the observed large-scale spatial features of trail systems. 7. Source: Modelling the Evolution of Human Trail Systems (#UYLTYJ), Dirk Helbing, Joachim Keltsch, Péter Molnár, p. 0 Context: #BSSYS2 Modelling the Evolution of Human Trail Systems Score: 0.013 Related excerpt #M2D4A8: The Center of Theoretical Studies of Physical Systems, 223 James P. Brawley Drive, Atlanta, Georgia 30314, USA 8. Source: Modelling the Evolution of Human Trail Systems (#UYLTYJ), Dirk Helbing, Joachim Keltsch, Péter Molnár, p. 2 Context: #BSSYS2 Modelling the Evolution of Human Trail Systems Score: 0.012 Related excerpt #FCSQLN: On a plain, homogeneous ground, the walking direction \vec{e}_\alpha of pedestrian \alpha is determined by the direction of the next destination \vec{d}_\alpha , i.e. \vec{e}_\alpha(\vec{r}_\alpha) = (\vec{d}_\alpha - \vec{r}_\alpha) / \|\vec{d}_\alpha - \vec{r}_\alpha\| . Without a destination, a pedestrian is expected to move into the direction of the largest increase of ground attraction, which is given by the (normalized) gradient \vec{\nabla}_{\vec{r}_\alpha} V_{\text{tr}}(\vec{r}_\alpha, t) of the trail potential. However, since the choice of the walking direction \vec{e}_\alpha is influenced by the destination and existing trails at the same time, the orientation relation ### 39. Tool result: search_text Exact matches 1. Source: Harmony-Seeking Computations: A Science of Non-Classical Dynamics Based on the Progressive Evolution of the Larger Whole (#PXG56P), Christopher Alexander, p. 3 Context: #ULGVY6 II Fifteen Properties Matching excerpt #UYGJ7T: intensified when the ground, against which it is contrasted, is clarified and itself becomes made of centers: all this differentiation arises from the degree or sharpness of contrast that is attained. Note, though, that too-sharp a degree of contrast is offset by NOT-SEPARATENESS, below. 2. Source: Harmony-Seeking Computations: A Science of Non-Classical Dynamics Based on the Progressive Evolution of the Larger Whole (#PXG56P), Christopher Alexander, p. 32 Context: #QUCRWC VI STRUCTURE-PRESERVING TRANSFORMATIONS: HELPING A LARGER WHOLE TO FORM #FSRDPE The SP-Postulate: Always Helping A Larger Whole To Form Matching excerpt #5NK6M3: STRONG CENTERS LEVELS OF SCALE BOUNDARIES ALTERNATING REPETITION POSITIVE SPACE GOOD SHAPE LOCAL SYMMETRIES DEEP INTERLOCK AND AMBIGUITY CONTRAST GRADIENTS ROUGHNESS ECHOES THE VOID SIMPLICITY AND INNER CALM NOT SEPARATENESS 3. Source: Harmony-Seeking Computations: A Science of Non-Classical Dynamics Based on the Progressive Evolution of the Larger Whole (#PXG56P), Christopher Alexander, p. 0 Context: #ASGYJG Preface Matching excerpt #PRJ2E8: There is a structure, visible in any given part of the world, which we may call the wholeness . The wholeness is an abstract mathematical structure, existing in space. It captures what we may loosely consider as the global character of a given configuration, in itself and in relation to the world around it. The wholeness is a structure which exists at many levels of scale, and covers the interrelationships of the configurations at different scales. The primary entities of which the structure is built are centers, centers which become activated in the space as a result of the configuration as a whole. Centers have different levels of strength or coherence. The coherence of a configuration is caused by relationships among other centers. In particular, there are fifteen types of relationships among centers which increase or intensify the strength of any given center. These fifteen properties are listed below, and define the way that configurations within a configuration help each other. Within this scheme, unfolding of new configurations is a natural process, and can be understood and followed. We thus have a basis for making computations about unfolding. These are somewhat similar to the bifurcations that have been observed and analyzed in complex non-linear systems, but they are much richer and more complex than the theory of bifurcations can at present contemplate. Unfolding occurs as a result of structure-preserving (SP-) transformations. These SP-transformations are combinations and sequences of 15 possible spatial transformations based on the fifteen properties that determine how coherent centers may be built from one another. An advanced computational theory of these SP transformations does not yet exist, but it is my aim, in this paper, to show you how unfolding is built from these transformations, and how the outline of a new (computable) theory of unfolding can be established. 4. Source: Harmony-Seeking Computations: A Science of Non-Classical Dynamics Based on the Progressive Evolution of the Larger Whole (#PXG56P), Christopher Alexander, p. 58 Context: #8MPFPH X ECOLOGY OF THE ORDINARY #A8HF33 The Uniqueness Of Each Region in the Generated Structure Matching excerpt #KBY4TY: Still more exciting though, is that the same quality of uniqueness and subtle differentiation, continues to smaller and smaller scales. If we examine the drawings, we shall see that the detailed configuration of terraces, entrances, paths, lawns, stairs and archways, produces unique results in each part of the larger whole, and in each part of the individual gardens. This is not because of a shallow desire to make each thing different for its own sake (sometimes the driving force behind the more commercial post modern developments). It occurs because the effect of harmony seeking computations, on only slightly different starting conditions, is to generate entirely new and different configurations, but all members of a fairly simple family. 5. Source: Harmony-Seeking Computations: A Science of Non-Classical Dynamics Based on the Progressive Evolution of the Larger Whole (#PXG56P), Christopher Alexander, p. 32 Context: #QUCRWC VI STRUCTURE-PRESERVING TRANSFORMATIONS: HELPING A LARGER WHOLE TO FORM #FSRDPE The SP-Postulate: Always Helping A Larger Whole To Form Matching excerpt #5BC6M2: At the same time, there is a larger whole, often an order of magnitude bigger than \mathcal{L} . The transformation which is structure-preserving, preserves the structure of the \mathcal{W} , and to do so modifies the \mathcal{L} , and modifies it in relation to the whole context around it. Thus the output from this step is a modification geometrically within \mathcal{L} , but it is a function of both \mathcal{L} and \mathcal{W} . In addition, there is a sense in which \mathcal{L} is being fitted to \mathcal{W} , it is being made to fit \mathcal{W} , to be congruent with \mathcal{W} , adapted to \mathcal{W} , harmonious with \mathcal{W} . Further, in order to modify \mathcal{L} in this way, new centers are going to be created within and around \mathcal{L} . We may refer to these new centers as \mathcal{N}_i , and since there may be several of them, we may think of them as \mathcal{N}_1, \mathcal{N}_2, \mathcal{N}_3 , etc. 6. Source: Harmony-Seeking Computations: A Science of Non-Classical Dynamics Based on the Progressive Evolution of the Larger Whole (#PXG56P), Christopher Alexander, p. 44 Context: #8DNY7R VIII STRUCTURE-PRESERVING TRANSFORMATIONS AND SYMMETRY BREAKING #5UEEL4 The Possibility that Structure-Preserving Transformations are Deep Generalizations of Symmetry-Breaking. #WYT8BH Example 17. Dewdrops On A Spider's Thread Matching excerpt #LNKPPL: In my view, the symmetry-breaking idea is not yet, by itself, sufficiently profound to be useful as a general theory explaining real-world complex configurations, or to account for the harmony-seeking phenomenon I am describing in this paper. As I have said earlier, it has been demonstrated that “the” wholeness consists, in part, of the entire system of overlapping local symmetries at a wide variety of scales in a configuration. 37 We therefore need to have a view where somehow the underlying structure of all these symmetries, working together, is preserved. And further, the centers that are present in a given wholeness are not all LOCAL SYMMETRIES. Other centers are formed by GRADIENTS, ECHOES, BOUNDARIES, DEEP INTERLOCK, POSITIVE SPACE, NOT SEPARATENESS, and so on. These other properties and that entire structure, too , have to be preserved when a harmony-seeking computation starts with a currently existing structure, and finds its way to a stronger structure that is latent in this overall configuration, and can be brought out by a few transformations. 7. Source: Harmony-Seeking Computations: A Science of Non-Classical Dynamics Based on the Progressive Evolution of the Larger Whole (#PXG56P), Christopher Alexander, p. 12 Context: #Q94AYK IV HARMONY-SEEKING COMPUTATIONS #XE4TCT What Are The Underlying Qualities Common to Different Examples of Harmony-Seeking Steps in Different Systems? #QJLGS4 Example 4: Hayricks in a Field Matching excerpt #4Q8VNF: The people who built and placed these ricks, were, consciously or unconsciously, performing a harmony-seeking computation. There are ECHOES of shape and size between land and hayricks, ECHOES of a certain kind of curve, LOCAL SYMMETRIES in the ricks themselves, and the placing of the ricks emphasizes naturally occurring STRONG CENTERS that are generated by shelves and flattened places, bounded so that the hayricks nestle in the land, are subdued and congruent, and inside the structure which exists. The hayricks are kindly to the land, but they are placed with enormous care. They follow the wholeness. And if I were to move them slightly, to different positions, the placement and the whole then created, the ensemble, would be less profound and less harmonious. 8. Source: Harmony-Seeking Computations: A Science of Non-Classical Dynamics Based on the Progressive Evolution of the Larger Whole (#PXG56P), Christopher Alexander, p. 62 Context: #A6ZZPA XI CONCLUSIONS #MRNGTP Structure-Preserving Transformations #PBRJGY The Paradigm Being Followed In Each SP-Transformation Matching excerpt #FGHMHR: The intuitive act is nevertheless a computation, and we may be able to pin down what kind of computation it is. Then, if we can succeed in making a blind computation, even perhaps one day performed by a computer working in a new way, achieve similar holistic results, that will be because the thing we recognize intuitively and emotionally as whole and coherent, is, mathematically, a particular recursively generated structure of symmetries and centers : and it is this underlying structure which allows the human mind, and natural processes, both, to follow this path and to seek wholeness in the way they do. Most important, we may become conscious about this process, and consciously use this kind of computation to improve the coherence and harmony of our physical world. Approximate matches 1. Source: Harmony-Seeking Computations: A Science of Non-Classical Dynamics Based on the Progressive Evolution of the Larger Whole (#PXG56P), Christopher Alexander, p. 32 Context: #QUCRWC VI STRUCTURE-PRESERVING TRANSFORMATIONS: HELPING A LARGER WHOLE TO FORM #FSRDPE The SP-Postulate: Always Helping A Larger Whole To Form Score: 0.03 Related excerpt #S3MN33: In each case there is a whole, \mathcal{W} , and within the whole a latent center which is being modified, transformed, shaped, or reshaped, by a certain step. This latent center is the focus of the transformation, and the latent center sets the boundary of the geometrical and physical transformations that are then actively being undertaken. Let's call this focal latent center \mathcal{L} . 2. Source: Harmony-Seeking Computations: A Science of Non-Classical Dynamics Based on the Progressive Evolution of the Larger Whole (#PXG56P), Christopher Alexander, p. 60 Context: #A6ZZPA XI CONCLUSIONS #MRNGTP Structure-Preserving Transformations #HWWGSL Models Of The Wholeness In A Given Configuration? Score: 0.029 Related excerpt #CF88F5: Postulate A1. In any configuration we see certain salient wholes, or centers. Each of these wholes is an identified, spatially contiguous subset of the configuration, which corresponds to something we see, or experience, as an “entity.” 3. Source: Harmony-Seeking Computations: A Science of Non-Classical Dynamics Based on the Progressive Evolution of the Larger Whole (#PXG56P), Christopher Alexander, p. 3 Context: #ULGVY6 II Fifteen Properties Score: 0.029 Related excerpt #UYGJ7T: intensified when the ground, against which it is contrasted, is clarified and itself becomes made of centers: all this differentiation arises from the degree or sharpness of contrast that is attained. Note, though, that too-sharp a degree of contrast is offset by NOT-SEPARATENESS, below. 4. Source: Harmony-Seeking Computations: A Science of Non-Classical Dynamics Based on the Progressive Evolution of the Larger Whole (#PXG56P), Christopher Alexander, p. 35 Context: #567Q7Z VII HARMONY-SEEKING RATHER THAN MERELY “EMERGENCE” #S4HQZJ A Whole-Based, Harmony-Seeking Process Which Works By Continually Strengthening Latent Centers. #WE6WWG In Detail, What Exactly Does It Mean For A System To Help The Larger System It Is Embedded In? Score: 0.026 Related excerpt #D53VF2: Let us return to the example of St Mark's Square. At each cycle the process identifies a latent center in the larger configuration. This latent center is an area or potential center which is weak, and which – if strengthened -- would improve the coherence of the whole. The area immediately around that latent center is healed or made more whole by the injection of the repaired latent center. 5. Source: Harmony-Seeking Computations: A Science of Non-Classical Dynamics Based on the Progressive Evolution of the Larger Whole (#PXG56P), Christopher Alexander, p. 9 Context: #Q94AYK IV HARMONY-SEEKING COMPUTATIONS #W2TBVW The Essence Of Harmony-Seeking Computation #9FBT9U Example 3: Growth Of The City Of Amsterdam Score: 0.023 Related excerpt #XEK4DE: In the third state of the city, late eighteenth century (below), the concentric structure of canals and streets has been intensified by adding further layers and filling in a much larger area in a way that supports and continues the structure of the second state. In the transformations applied here there has been particular emphasis on BOUNDARIES (in the canals, walkways and polders), on POSITIVE SPACE, LOCAL SYMMETRIES, GOOD SHAPE, DEEP INTERLOCK in the surrounding fortifications, and on ALTERNATING REPETITION throughout. 6. Source: Harmony-Seeking Computations: A Science of Non-Classical Dynamics Based on the Progressive Evolution of the Larger Whole (#PXG56P), Christopher Alexander, p. 55 Context: #8MPFPH X ECOLOGY OF THE ORDINARY #TFM623 Example 20. The Ecology Of The Ordinary Score: 0.023 Related excerpt #4B4QNJ: Cycle 1. The process starts with a triangular brown-field site in Somerville, 5.5 acres in area, between a railroad line, a bike path, and an existing neighborhood → The computation then identifies latent centers in the site: the bike path, along the south side, Warwick avenue along the west, and Lowell Street in the south east corner. 7. Source: Harmony-Seeking Computations: A Science of Non-Classical Dynamics Based on the Progressive Evolution of the Larger Whole (#PXG56P), Christopher Alexander, p. 36 Context: #567Q7Z VII HARMONY-SEEKING RATHER THAN MERELY “EMERGENCE” #S4HQZJ A Whole-Based, Harmony-Seeking Process Which Works By Continually Strengthening Latent Centers. #WE6WWG In Detail, What Exactly Does It Mean For A System To Help The Larger System It Is Embedded In? Score: 0.023 Related excerpt #V7GAR8: Abstractly we may express this concept through the following diagram. The red area is a latent center, a weak center which has the capacity to heal the blue area beyond it: and it is able to do so by creation of the smaller white centers -- they are built to create coherence in the red center. However, the key point is that the healing will not take place, unless the red center then also helps to heal the larger blue center beyond it. 8. Source: Harmony-Seeking Computations: A Science of Non-Classical Dynamics Based on the Progressive Evolution of the Larger Whole (#PXG56P), Christopher Alexander, p. 58 Context: #8MPFPH X ECOLOGY OF THE ORDINARY #A8HF33 The Uniqueness Of Each Region in the Generated Structure Score: 0.022 Related excerpt #NL2UKH: For example, at the largest level of scale, each of the fifteen or so courtyard gardens has a slightly different shape and configuration. That occurs because the application of the principal transformations (POSITIVE SPACE, LOCAL SYMMETRIES, ECHOES and GRADIENTS, ALTERNATING REPETITION and CONTRAST), generates a different configuration for each one according to its starting point – that is, according to its context. ### 40. Tool result: search_text Exact matches 1. Source: Generative Codes: The Path to Building Welcoming, Beautiful, Sustainable Neighborhoods (#XW22YY), Brian Hanson, Christopher Alexander, Maggie Moore Alexander, Michael Mehaffy, Randall Schmidt, p. 1 Context: #CU9CAT Generative Codes #H8LQMH A Simple Question: What Is It That We Really Need From The Neighborhood Where We Live? Matching excerpt #3XM62W: When successful, it binds land and people together, into a social-spatial fabric or tapestry. When we list the items at the beginning of this section, it is that fabric or tapestry, of which we are dreaming. We will never get that kind of neighborhood, unless we consciously set out to make that fabric. The fabric must be generated by the processes we use. And in the processes we support, that try to build houses and public space and neighborhoods, it is this tapestry and fabric that must be generated. Without it, nothing valuable can ensue. With it, the neighborhood has a very strong chance of life. 2. Source: Generative Codes: The Path to Building Welcoming, Beautiful, Sustainable Neighborhoods (#XW22YY), Brian Hanson, Christopher Alexander, Maggie Moore Alexander, Michael Mehaffy, Randall Schmidt, p. 3 Context: #CU9CAT Generative Codes #5RWPGZ Historical Background Matching excerpt #4ATPNF: In the modern era, the first conscious, and deliberately thought out efforts to guide and control neighborhoods, were already types of code. These were the zoning ordinances, introduced in Chicago in the last decade of the 19 th century. New York City adopted the first zoning regulations to apply city-wide in 1916 as a reaction to construction of The Equitable Building (which still stands at 120 Broadway). 4 The advent of building codes, in their modern form, also started in the 19 th century. Yet what is generally accepted as the first building code was in the Code of Hammurabi dating to 1700 BC. And legal covenants, attached by deed trust to a particular piece of land, are also used in many countries, and may also be viewed as a type of code. 3. Source: Generative Codes: The Path to Building Welcoming, Beautiful, Sustainable Neighborhoods (#XW22YY), Brian Hanson, Christopher Alexander, Maggie Moore Alexander, Michael Mehaffy, Randall Schmidt, p. 19 Context: #CU9CAT Generative Codes #D2PCQS A Decisive and Lasting Change Matching excerpt #DCTJJF: The reorganization of development, creation of new legal controls and guidelines which fundamentally alter the way a developer enters into the growth process of a community, must be the bottom line of a successful policy for building and rebuilding neighborhoods. Generative codes, together with the radical shifts in power and control, and changes in responsibility of planning officers, inhabitants, and builders alike, are we believe -- in one form or another -- the only possible foundation for the way successful neighborhoods can be created. This must become the foundation of a national policy on neighborhoods. 4. Source: Generative Codes: The Path to Building Welcoming, Beautiful, Sustainable Neighborhoods (#XW22YY), Brian Hanson, Christopher Alexander, Maggie Moore Alexander, Michael Mehaffy, Randall Schmidt, p. 1 Context: #CU9CAT Generative Codes #H8LQMH A Simple Question: What Is It That We Really Need From The Neighborhood Where We Live? Matching excerpt #MDNUVZ: The reason is not hard to find. Making a neighborhood which has these qualities, is a human process. It is generated by a long chain of human events, involving respect for people, respect for one another, respect for land and place, and respect for age-old ways of making things: the origin of every genuine human structure. Above all it comes from the land, and it comes from the people. 5. Source: Generative Codes: The Path to Building Welcoming, Beautiful, Sustainable Neighborhoods (#XW22YY), Brian Hanson, Christopher Alexander, Maggie Moore Alexander, Michael Mehaffy, Randall Schmidt, p. 17 Context: #CU9CAT Generative Codes #7PFDSQ Placing Practical Emphasis on Respect For Individuals, Respect For Land, and Respect for Continuity. Matching excerpt #VTPWGS: In either case, the developer takes the risk, the bank lends money against this risk, and the insurance against risk which both developers and banks experience, is provided by a highly rigid and mechanical process. Unfortunately, that makes the recipients of the housing -- the public who walk and use the land within these areas – pay an immense price for this insurance: namely, that they have an environment which is inhuman, sterile, and impersonal, thus disconnecting people from society and from land. 6. Source: Generative Codes: The Path to Building Welcoming, Beautiful, Sustainable Neighborhoods (#XW22YY), Brian Hanson, Christopher Alexander, Maggie Moore Alexander, Michael Mehaffy, Randall Schmidt, p. 5 Context: #CU9CAT Generative Codes #5RWPGZ Historical Background Matching excerpt #FAAY88: We believe that certain recent experiments have now begun to demonstrate – at least on a preliminary basis -- that newly built neighborhoods which are created in this better and more generative way, fundamentally alter the way that people living and working there feel about the place, and about themselves. 7. Source: Generative Codes: The Path to Building Welcoming, Beautiful, Sustainable Neighborhoods (#XW22YY), Brian Hanson, Christopher Alexander, Maggie Moore Alexander, Michael Mehaffy, Randall Schmidt, p. 0 Context: #CU9CAT Generative Codes #H8LQMH A Simple Question: What Is It That We Really Need From The Neighborhood Where We Live? Matching excerpt #VG4545: And, of course, we also hope for these qualities in a newly built neighborhood, or in a refurbished neighborhood. This is the dream, one might say, of every developer. A developer with a conscience, who dreams of building neighborhoods, hopes and wishes to build for people, something that has these qualities. 8. Source: Generative Codes: The Path to Building Welcoming, Beautiful, Sustainable Neighborhoods (#XW22YY), Brian Hanson, Christopher Alexander, Maggie Moore Alexander, Michael Mehaffy, Randall Schmidt, p. 18 Context: #CU9CAT Generative Codes #D2PCQS A Decisive and Lasting Change Matching excerpt #EF25YV: In the last fifty years, it has almost always been assumed that the way to get construction of neighborhoods to meet the growing world population, is through the “good offices” of a developer: a person, or an institution, who is willing to take the financial risk, undertake the huge effort of management, and who, in short, will get things done. Approximate matches 1. Source: Generative Codes: The Path to Building Welcoming, Beautiful, Sustainable Neighborhoods (#XW22YY), Brian Hanson, Christopher Alexander, Maggie Moore Alexander, Michael Mehaffy, Randall Schmidt, p. 20 Context: #CU9CAT Generative Codes #LUPYTU Notes Score: 0.03 Related excerpt #QZZTUB: 17 A neighborhood for seventy families who laid out and built their own houses in the town of Santa Rose de Cabal, in the mountains of Colombia. 2. Source: Generative Codes: The Path to Building Welcoming, Beautiful, Sustainable Neighborhoods (#XW22YY), Brian Hanson, Christopher Alexander, Maggie Moore Alexander, Michael Mehaffy, Randall Schmidt, p. 3 Context: #CU9CAT Generative Codes #5RWPGZ Historical Background Score: 0.028 Related excerpt #WZUEX3: Gradually, during the last two or three decades of the 20 th century, the shortcomings of this prevailing, “old” system of urban codes became clear and inspired a number of major changes. The big innovation came in the late 70’s and early 80’s from Alexander’s pattern languages, and from Andrés Duany’s subsequent effort to introduce form-based codes similar to patterns as tools for guiding development in cities and neighborhoods. At first this movement, (by now world-wide), focused on the inadequacy and poverty of the older code elements themselves, and replaced them with NU rules that deal with sidewalks, front yards, windows, building facades, street widths, parking, mixed use, and so forth. Many of these rules represent a distinct functional improvement over previously existing ideas of development, suburban tracts, and urban housing. But like the elements of existing building codes and zoning ordinances, the elements of these “New Urbanist” codes are rules, enforced by law, which require that a certain number of geometric conditions are met within a neighborhood. 3. Source: Generative Codes: The Path to Building Welcoming, Beautiful, Sustainable Neighborhoods (#XW22YY), Brian Hanson, Christopher Alexander, Maggie Moore Alexander, Michael Mehaffy, Randall Schmidt, p. 0 Context: #CU9CAT Generative Codes #H8LQMH A Simple Question: What Is It That We Really Need From The Neighborhood Where We Live? Score: 0.027 Related excerpt #82WS6G: A sense of privacy -- we are left alone when we want to be alone. Friendly people who know you, and whom you greet and occasionally talk to. Safety -- safety from violence, from theft. Physical safety from traffic and noise. Safety for children. Safety at night. A beautiful place -- something which lifts your heart when you walk around or look out of the window. Intimate and personal. Physical safety from traffic and noise. Safety for children. Trees and gardens. A place to sit in public that is really a wonderful place. Streets and public places where everyone feels at home, instead of where nobody feels at home. Uniqueness of the neighborhood, so we know it when we are home and when we get home. Water, perhaps.. 4. Source: Generative Codes: The Path to Building Welcoming, Beautiful, Sustainable Neighborhoods (#XW22YY), Brian Hanson, Christopher Alexander, Maggie Moore Alexander, Michael Mehaffy, Randall Schmidt, p. 0 Context: #CU9CAT Generative Codes #H8LQMH A Simple Question: What Is It That We Really Need From The Neighborhood Where We Live? Score: 0.027 Related excerpt #4ULVN8: Most of us share a general, intuitive understanding of the qualities we would like to have in the neighborhood around us. It is not very complicated. 5. Source: Generative Codes: The Path to Building Welcoming, Beautiful, Sustainable Neighborhoods (#XW22YY), Brian Hanson, Christopher Alexander, Maggie Moore Alexander, Michael Mehaffy, Randall Schmidt, p. 1 Context: #CU9CAT Generative Codes #H8LQMH A Simple Question: What Is It That We Really Need From The Neighborhood Where We Live? Score: 0.016 Related excerpt #3XM62W: When successful, it binds land and people together, into a social-spatial fabric or tapestry. When we list the items at the beginning of this section, it is that fabric or tapestry, of which we are dreaming. We will never get that kind of neighborhood, unless we consciously set out to make that fabric. The fabric must be generated by the processes we use. And in the processes we support, that try to build houses and public space and neighborhoods, it is this tapestry and fabric that must be generated. Without it, nothing valuable can ensue. With it, the neighborhood has a very strong chance of life. 6. Source: Generative Codes: The Path to Building Welcoming, Beautiful, Sustainable Neighborhoods (#XW22YY), Brian Hanson, Christopher Alexander, Maggie Moore Alexander, Michael Mehaffy, Randall Schmidt, p. 11 Context: #CU9CAT Generative Codes #MU7M8B The Process of Procurement Score: 0.014 Related excerpt #HWD8DD: Developers who provide some capital for the construction Engineers who deal with soils, and/or with existing conditions requiring remedial action. Project manager who oversees the process in its entirety Community advocates who play a role in helping make sure that individual persons are represented and involved all the way along. Banks or lenders who have a financial stake in the new neighborhood Construction contractors and subcontractors Craftsmen Children of the families Ecologists taking care of the existing fauna and flora Local business support services 7. Source: Generative Codes: The Path to Building Welcoming, Beautiful, Sustainable Neighborhoods (#XW22YY), Brian Hanson, Christopher Alexander, Maggie Moore Alexander, Michael Mehaffy, Randall Schmidt, p. 19 Context: #CU9CAT Generative Codes #D2PCQS A Decisive and Lasting Change Score: 0.014 Related excerpt #DCTJJF: The reorganization of development, creation of new legal controls and guidelines which fundamentally alter the way a developer enters into the growth process of a community, must be the bottom line of a successful policy for building and rebuilding neighborhoods. Generative codes, together with the radical shifts in power and control, and changes in responsibility of planning officers, inhabitants, and builders alike, are we believe -- in one form or another -- the only possible foundation for the way successful neighborhoods can be created. This must become the foundation of a national policy on neighborhoods. 8. Source: Generative Codes: The Path to Building Welcoming, Beautiful, Sustainable Neighborhoods (#XW22YY), Brian Hanson, Christopher Alexander, Maggie Moore Alexander, Michael Mehaffy, Randall Schmidt, p. 0 Context: #CU9CAT Generative Codes #H8LQMH A Simple Question: What Is It That We Really Need From The Neighborhood Where We Live? Score: 0.014 Related excerpt #VG4545: And, of course, we also hope for these qualities in a newly built neighborhood, or in a refurbished neighborhood. This is the dream, one might say, of every developer. A developer with a conscience, who dreams of building neighborhoods, hopes and wishes to build for people, something that has these qualities. ### 41. Tool result: search_text Exact matches 1. Source: A Layered Particle-Based Fluid Model for Real-Time Rendering of Water (#RBS5K6), Daniel Scherzer, Florian Bagar, Michael Wimmer, p. 4 Context: #5NFAXU A Layered Particle-Based Fluid Model for Real-Time Rendering of Water #Z68DF9 5. Real-Time Foam #P6BSN6 5.2. Layer Creation Matching excerpt #48RCD2: Now that particles have been classified as either particle or foam, we partition the fluid into layers , as shown in Figure 2. 2. Source: A Layered Particle-Based Fluid Model for Real-Time Rendering of Water (#RBS5K6), Daniel Scherzer, Florian Bagar, Michael Wimmer, p. 3 Context: #5NFAXU A Layered Particle-Based Fluid Model for Real-Time Rendering of Water #Z68DF9 5. Real-Time Foam Matching excerpt #BB9PDK: In order to capture these two main effects in a real-time setting, we separate foam particles from water particles and arrange the resulting foam and water particles in separate layers and render them using volumetric back-to-front compositing. Although our layered representation does not account for discontinuity in the fluid volume which occurs if there are several layers of water and foam, the two most common cases mentioned above are covered by this model. 3. Source: A Layered Particle-Based Fluid Model for Real-Time Rendering of Water (#RBS5K6), Daniel Scherzer, Florian Bagar, Michael Wimmer, p. 2 Context: #5NFAXU A Layered Particle-Based Fluid Model for Real-Time Rendering of Water #34V9L7 4. Adaptive Curvature Flow Matching excerpt #8PQTK5: The first step in rendering a fluid using particles is to create the fluid surface. This is done by splatting the particles 4. Source: A Layered Particle-Based Fluid Model for Real-Time Rendering of Water (#RBS5K6), Daniel Scherzer, Florian Bagar, Michael Wimmer, p. 5 Context: #5NFAXU A Layered Particle-Based Fluid Model for Real-Time Rendering of Water #JEPXQL 7. Conclusions and Future Work Matching excerpt #YJNSYU: We presented a new method for rendering particle-based fluids with foam in real time. The first contribution is an adaptive curvature flow smoothing method that avoids over- or under-smoothing as present in previous methods. Our second contribution is a fast physically guided foam rendering algorithm based on Weber number thresholding and a layered compositing algorithm. Our approach provides more realistic fluid rendering at comparable cost to previous methods, and is simple to implement and integrate into existing engines. In future work, we plan to use the volumetric information available in the layers to generate soft shadows. We will also investigate whether situations that require more than 3 layers are likely to appear. 5. Source: A Layered Particle-Based Fluid Model for Real-Time Rendering of Water (#RBS5K6), Daniel Scherzer, Florian Bagar, Michael Wimmer, p. 4 Context: #5NFAXU A Layered Particle-Based Fluid Model for Real-Time Rendering of Water #Z68DF9 5. Real-Time Foam #P6BSN6 5.2. Layer Creation Matching excerpt #LSFXLY: By using two water layers, one in front and one behind the foam layer, we can simulate foam inside water, as happens at the end of a waterfall (see for instance Figure 8, middle or Figure 5). We first determine the front water surface and the front foam surface by splatting water and foam particles into separate depth buffers (the splatting step was described in Section 4). Curvature flow is only applied to the front water surface. 6. Source: A Layered Particle-Based Fluid Model for Real-Time Rendering of Water (#RBS5K6), Daniel Scherzer, Florian Bagar, Michael Wimmer, p. 1 Context: #5NFAXU A Layered Particle-Based Fluid Model for Real-Time Rendering of Water #JZMDNB 2. Previous Work Matching excerpt #JWYUZ6: Simulation of liquids like water can be classified into Eulerian- and Lagrangian-approaches. The former build on a fixed grid in space, using finite element techniques [Sta99] to solve the Navier Stokes Equations. However, these approaches are not intuitive for flows because they limit the simulation to the space where the grid is defined. Lagrangian-approaches, like Smoothed Particle Hydrodynamics (SPH), introduced for computer graphics by [DG96], simulate a fluid by moving discrete volume elements, and are therefore not restricted concerning the simulation space. 7. Source: A Layered Particle-Based Fluid Model for Real-Time Rendering of Water (#RBS5K6), Daniel Scherzer, Florian Bagar, Michael Wimmer, p. 2 Context: #5NFAXU A Layered Particle-Based Fluid Model for Real-Time Rendering of Water #VR4JRQ 3. Overview Matching excerpt #VLHP44: The original algorithm calculates the water depth by splatting the particles, then smooths the depth buffer using curvature flow filtering, then calculates water thickness by accumulating particle depths in a separate thickness buffer, and finally composites the results. Our algorithm extends this by adapting the curvature flow filter for the viewer distance, and by adding a foam layer that can lie between two water layers. Our algorithm then performs the following steps once per frame after the scene has been rendered into a texture (see Figure 3): 8. Source: A Layered Particle-Based Fluid Model for Real-Time Rendering of Water (#RBS5K6), Daniel Scherzer, Florian Bagar, Michael Wimmer, p. 3 Context: #5NFAXU A Layered Particle-Based Fluid Model for Real-Time Rendering of Water #34V9L7 4. Adaptive Curvature Flow Matching excerpt #6HBEL2: However, close inspection of the screen-space curvature formulation reveals that the reference coordinate system for calculating the curvature is the window coordinate system. This means that, given equal iteration sizes, a particle that appears larger on screen (because it is closer due to perspective) will have a significantly larger radius in this coordinate system and therefore significantly lower curvature than a particle that is farther away. The resulting artifact is that smoothing will have a lower effect on closer particles, which therefore retain the unwanted spherical appearance, whereas particles far from the viewer will be overly smoothed, so that the fluid surface loses its defining characteristics such as highlights. This can be observed in Figure 4, left. Approximate matches 1. Source: A Layered Particle-Based Fluid Model for Real-Time Rendering of Water (#RBS5K6), Daniel Scherzer, Florian Bagar, Michael Wimmer, p. 4 Context: #5NFAXU A Layered Particle-Based Fluid Model for Real-Time Rendering of Water #Z68DF9 5. Real-Time Foam #P6BSN6 5.2. Layer Creation Score: 0.03 Related excerpt #48RCD2: Now that particles have been classified as either particle or foam, we partition the fluid into layers , as shown in Figure 2. 2. Source: A Layered Particle-Based Fluid Model for Real-Time Rendering of Water (#RBS5K6), Daniel Scherzer, Florian Bagar, Michael Wimmer, p. 1 Context: #5NFAXU A Layered Particle-Based Fluid Model for Real-Time Rendering of Water #JZMDNB 2. Previous Work Score: 0.029 Related excerpt #9WHWU9: al. [CPPK07] extend SPH by considering the dissolved gas within the fluid. Similar to our work they use a layered representation where the different parts of the fluid volume are separately rendered and composited into the final image. 3. Source: A Layered Particle-Based Fluid Model for Real-Time Rendering of Water (#RBS5K6), Daniel Scherzer, Florian Bagar, Michael Wimmer, p. 2 Context: #5NFAXU A Layered Particle-Based Fluid Model for Real-Time Rendering of Water #34V9L7 4. Adaptive Curvature Flow Score: 0.027 Related excerpt #8PQTK5: The first step in rendering a fluid using particles is to create the fluid surface. This is done by splatting the particles 4. Source: A Layered Particle-Based Fluid Model for Real-Time Rendering of Water (#RBS5K6), Daniel Scherzer, Florian Bagar, Michael Wimmer, p. 1 Context: #5NFAXU A Layered Particle-Based Fluid Model for Real-Time Rendering of Water #JZMDNB 2. Previous Work Score: 0.027 Related excerpt #JWYUZ6: Simulation of liquids like water can be classified into Eulerian- and Lagrangian-approaches. The former build on a fixed grid in space, using finite element techniques [Sta99] to solve the Navier Stokes Equations. However, these approaches are not intuitive for flows because they limit the simulation to the space where the grid is defined. Lagrangian-approaches, like Smoothed Particle Hydrodynamics (SPH), introduced for computer graphics by [DG96], simulate a fluid by moving discrete volume elements, and are therefore not restricted concerning the simulation space. 5. Source: A Layered Particle-Based Fluid Model for Real-Time Rendering of Water (#RBS5K6), Daniel Scherzer, Florian Bagar, Michael Wimmer, p. 3 Context: #5NFAXU A Layered Particle-Based Fluid Model for Real-Time Rendering of Water #Z68DF9 5. Real-Time Foam Score: 0.026 Related excerpt #BB9PDK: In order to capture these two main effects in a real-time setting, we separate foam particles from water particles and arrange the resulting foam and water particles in separate layers and render them using volumetric back-to-front compositing. Although our layered representation does not account for discontinuity in the fluid volume which occurs if there are several layers of water and foam, the two most common cases mentioned above are covered by this model. 6. Source: A Layered Particle-Based Fluid Model for Real-Time Rendering of Water (#RBS5K6), Daniel Scherzer, Florian Bagar, Michael Wimmer, p. 4 Context: #5NFAXU A Layered Particle-Based Fluid Model for Real-Time Rendering of Water #Z68DF9 5. Real-Time Foam #P6BSN6 5.2. Layer Creation Score: 0.025 Related excerpt #72YLNM: Since water and foam are volumetric phenomena, the amount of water respectively foam between two layer surfaces needs to be determined in order to allow correct compositing and attenuation. Similar to [vdLS09], the thickness of a layer is determined by additively splatting every particle belonging to the volume into a buffer. In contrast to the depth surface calculation, the splat kernel gives the thickness of the particle at each particle sampling point. Accumulating particle thicknesses is a reasonable approximation because particles from the physics simulation can be assumed to be largely non-overlapping. 7. Source: A Layered Particle-Based Fluid Model for Real-Time Rendering of Water (#RBS5K6), Daniel Scherzer, Florian Bagar, Michael Wimmer, p. 3 Context: #5NFAXU A Layered Particle-Based Fluid Model for Real-Time Rendering of Water #Z68DF9 5. Real-Time Foam #T4PBCD 5.1. Foam Formation Score: 0.024 Related excerpt #DNLBHX: First, we classify particles as water or foam. Following [MMS09], we base the classification on the Weber number [Sir99], which is a dimensionless physical quantity that describes the relative influence of the inertia of a fluid to its surface tension. The Weber number is defined as the ratio of the kinetic energy to the surface energy: 8. Source: A Layered Particle-Based Fluid Model for Real-Time Rendering of Water (#RBS5K6), Daniel Scherzer, Florian Bagar, Michael Wimmer, p. 4 Context: #5NFAXU A Layered Particle-Based Fluid Model for Real-Time Rendering of Water #Z68DF9 5. Real-Time Foam #P6BSN6 5.2. Layer Creation Score: 0.023 Related excerpt #LSFXLY: By using two water layers, one in front and one behind the foam layer, we can simulate foam inside water, as happens at the end of a waterfall (see for instance Figure 8, middle or Figure 5). We first determine the front water surface and the front foam surface by splatting water and foam particles into separate depth buffers (the splatting step was described in Section 4). Curvature flow is only applied to the front water surface. ### 42. Tool result: search_text Exact matches 1. Source: Procedural Content Generation via Machine Learning (PCGML) (#CQBDX4), Aaron Isaksen, Adam Summerville, Amy K. Hoover, Andy Nealen, Christoffer Holmgård, Julian Togelius, Matthew Guzdial, Sam Snodgrass, p. 7 Context: #6EE2TX Procedural Content Generation via Machine Learning (PCGML) #6BC6KQ III. METHODS OF PCGML #QNMYX8 B. Grids #UKUFBH 3) Matrix Factorization Matching excerpt #3GNRWF: While approaches to generating levels typically focus on platformers (e.g. Super Mario Bros. ), Summerville et al. [32] generate levels for The Legend of Zelda [60] series. Their approach relies on segmenting data hierarchically by first generating the high-level topological structure of a dungeon (discussed in Section III-C) and then the rooms represented as a grid of tiles via Principal Component Analysis (PCA). The PCA algorithm finds a compressed representation of the original 2-D room arrays by taking the eigenvectors of the original high dimensional space and retaining only the most informative (inspired by EigenFaces[61]). These compressed representations are represented as weight vectors that can be interpolated between to generate new room content as seen in Figure 9. 2. Source: Procedural Content Generation via Machine Learning (PCGML) (#CQBDX4), Aaron Isaksen, Adam Summerville, Amy K. Hoover, Andy Nealen, Christoffer Holmgård, Julian Togelius, Matthew Guzdial, Sam Snodgrass, p. 6 Context: #6EE2TX Procedural Content Generation via Machine Learning (PCGML) #6BC6KQ III. METHODS OF PCGML #QNMYX8 B. Grids #LTJS3M 2) Back Propagation Matching excerpt #X2N698: In [22] Jain et al. show how autoencoders [54] may be trained to reproduce levels from the original Super Mario Bros. game. The autoencoders are trained on series of vertical level windows and compress the typical features of Mario levels into a more general representation. They experimented with the width of the level windows and found that four tiles seems to work best for Mario levels. They proceeded to use these networks to discriminate generated levels from original levels, and to generate new levels as transformation from noise. They also demonstrated how a trained autoencoder may be used to repair unplayable levels, changing tile-level features to make levels playable, by inputting a broken/unplayable level to the autoencoder and receiving a repaired one as output (see Figure 7). 3. Source: Procedural Content Generation via Machine Learning (PCGML) (#CQBDX4), Aaron Isaksen, Adam Summerville, Amy K. Hoover, Andy Nealen, Christoffer Holmgård, Julian Togelius, Matthew Guzdial, Sam Snodgrass, p. 6 Context: #6EE2TX Procedural Content Generation via Machine Learning (PCGML) #6BC6KQ III. METHODS OF PCGML #QNMYX8 B. Grids #AQ7YV6 1) Frequency Counting Matching excerpt #FZFYY9: They train an MdMC by building a probability table according to the frequency of the tiles in training data, given the network structure of the MdMC, the set of training levels, and the set of tile types. A new level is then sampled one tile at a time by probabilistically choosing the next tile based upon the types of the previous tiles and the learned probability table. 4. Source: Procedural Content Generation via Machine Learning (PCGML) (#CQBDX4), Aaron Isaksen, Adam Summerville, Amy K. Hoover, Andy Nealen, Christoffer Holmgård, Julian Togelius, Matthew Guzdial, Sam Snodgrass, p. 9 Context: #6EE2TX Procedural Content Generation via Machine Learning (PCGML) #6BC6KQ III. METHODS OF PCGML #PQ6DK6 D. Discussion of Approaches Matching excerpt #D6MLXP: It is hard to compare and contrast the different approaches’ performance as there is no agreed upon test for generator “goodness” (i.e. Playability of levels? Capturing of original levels’ style? etc.) and the different approaches do not list the time it takes to train or generate a level. However, a few general statements can be made. The LSTM approaches [21], [42] will nearly certainly take more time to train and generate than the MdMC approaches. Both generate one tile at a time, but the LSTM approach inherently requires more computation to generate. Furthermore, the MdMC can be trained in a single pass over the levels, but it is highly unlikely that training of the LSTM can be stopped after a single training epoch. We note that the Matrix Factorization approaches have potentially the worst memory usage, as all levels must be held in memory at once. In practice, given the relatively small datasets, this is unlikely a concern, but could pose an issue for situations that call for much larger or many more levels. The back-propagation neural network approaches assume a fixed architecture and as such are the only training size independent approach. In practice, the generator size for the other approaches (MdMC, latent style-graph, evolved neural network, matrix factorized) are likely to be smaller than the back-propagated neural network, but there are no guarantees. A similar concern for the latent style-graph approach is that generation time is dependent on the complexity of the training data (more dense, less regular levels will have more latent nodes and subsequently more relative position edges) unlike the other approaches which perform the same generation act at each step of generation. 5. Source: Procedural Content Generation via Machine Learning (PCGML) (#CQBDX4), Aaron Isaksen, Adam Summerville, Amy K. Hoover, Andy Nealen, Christoffer Holmgård, Julian Togelius, Matthew Guzdial, Sam Snodgrass, p. 6 Context: #6EE2TX Procedural Content Generation via Machine Learning (PCGML) #6BC6KQ III. METHODS OF PCGML #QNMYX8 B. Grids #AQ7YV6 1) Frequency Counting Matching excerpt #UD73WD: An extension to the previously discussed one dimensional Markov chains are Multi-dimensional Markov Chains (MdMCs) [46], wherein the state represents a surrounding neighborhood and not just a single linear dimension. Snodgrass and Ontan [47] present an approach to level generation using MdMCs. An MdMC differs from a standard Markov chain in that it allows for dependencies in multiple directions and from multiple states, whereas a standard Markov chain only allows for dependence on the previous state alone. In their work, Snodgrass and Ontan represent video game levels as 2-D arrays of tiles representing features in the levels. For example, in Super Mario Bros. they use tiles representing the ground, enemies, and ?-blocks, etc. These tile types are used as the states in the MdMC. That is, the type of the next tile is dependent upon the types of surrounding tiles, given the network structure of the MdMC (i.e. the states that the current state’s value depends on). 6. Source: Procedural Content Generation via Machine Learning (PCGML) (#CQBDX4), Aaron Isaksen, Adam Summerville, Amy K. Hoover, Andy Nealen, Christoffer Holmgård, Julian Togelius, Matthew Guzdial, Sam Snodgrass, p. 6 Context: #6EE2TX Procedural Content Generation via Machine Learning (PCGML) #6BC6KQ III. METHODS OF PCGML #QNMYX8 B. Grids #AQ7YV6 1) Frequency Counting Matching excerpt #SKB6TL: A recent approach by Gumin [51] is loosely inspired by quantum mechanics and uses a “super-position” of tiles to generate images and levels from a representative example tile set. This approach is a variant of MRF, except instead of solely sampling, samples are chosen via “collapsing of the wave function” (i.e. probabilistically choosing a tile and propagating constraints that that choice enforces). This in turn can propagate other changes and either deterministically chooses tiles that no longer have any other possible choices or reduces the possible set of other tiles. The probabilities and configurations are determined by finding each N \times N window in the input, and the number of times that window occurs. This approach was initially explored for bitmap generation, but has since been expanded for use with 3-D tile sets as well as for level generation [52], [53]. The source code and examples of the bitmap project can be found online [51]. 7. Source: Procedural Content Generation via Machine Learning (PCGML) (#CQBDX4), Aaron Isaksen, Adam Summerville, Amy K. Hoover, Andy Nealen, Christoffer Holmgård, Julian Togelius, Matthew Guzdial, Sam Snodgrass, p. 7 Context: #6EE2TX Procedural Content Generation via Machine Learning (PCGML) #6BC6KQ III. METHODS OF PCGML #QNMYX8 B. Grids #UKUFBH 3) Matrix Factorization Matching excerpt #2VRNS3: Some approaches to level generation find latent level features in high-dimensional data through matrix factorization, which infers features by compressing existing data into a series of smaller matrices. While often generators create levels with a limited expressive range [57], Shaker and Abou-Zleikha [58] create more expressive Super Mario Bros. levels by first generating thousands with five known, non-ML-based generators (i.e. Notch, Parameterized, Grammatical Evolution, Launchpad, and Hopper). These levels are then compressed into vectors indicating the content type at each column and transformed into T matrices for each type of content: Platforms, Hills, Gaps, Items, and Enemies. Through a multiplicative update algorithm [59] for non-negative matrix factorization (NNMF), these levels are factored into T approximate “part matrices” which represent level patterns and T coefficient matrices corresponding to weights for 8. Source: Procedural Content Generation via Machine Learning (PCGML) (#CQBDX4), Aaron Isaksen, Adam Summerville, Amy K. Hoover, Andy Nealen, Christoffer Holmgård, Julian Togelius, Matthew Guzdial, Sam Snodgrass, p. 12 Context: #6EE2TX Procedural Content Generation via Machine Learning (PCGML) #VAKEPG IV. OPEN PROBLEMS AND OUTLOOK #D82E9U G. Exposing and Exploring the Generative Space Matching excerpt #9S3FMA: The nature of Generative Adversarial Networks has allowed for the discovery of latent factors that hold specific semantics (e.g., subtracting the latent space image of blank-faced person from a smiling one is the smile vector) [102]. Interpolating and extrapolating along these latent dimensions allows a user to generate content while freely tuning the parameters they see fit. We envision that this type of approach could lead to similar results in PCG (e.g., subtract Mario 1-1 from Mario 1-2 for the underground dimension, subtract Mario 1-1 from Mario 8-3 for the difficulty dimension). Furthermore, if multiple games are used as training input, it is theoretically possible that one could interpolate between two games (e.g., find the half-way point between Mario and Sonic) or find other more esoteric combinations ( Contra – Mario + Zelda = ???). Approximate matches 1. Source: Procedural Content Generation via Machine Learning (PCGML) (#CQBDX4), Aaron Isaksen, Adam Summerville, Amy K. Hoover, Andy Nealen, Christoffer Holmgård, Julian Togelius, Matthew Guzdial, Sam Snodgrass, p. 7 Context: #6EE2TX Procedural Content Generation via Machine Learning (PCGML) #6BC6KQ III. METHODS OF PCGML #QNMYX8 B. Grids #UKUFBH 3) Matrix Factorization Score: 0.028 Related excerpt #2VRNS3: Some approaches to level generation find latent level features in high-dimensional data through matrix factorization, which infers features by compressing existing data into a series of smaller matrices. While often generators create levels with a limited expressive range [57], Shaker and Abou-Zleikha [58] create more expressive Super Mario Bros. levels by first generating thousands with five known, non-ML-based generators (i.e. Notch, Parameterized, Grammatical Evolution, Launchpad, and Hopper). These levels are then compressed into vectors indicating the content type at each column and transformed into T matrices for each type of content: Platforms, Hills, Gaps, Items, and Enemies. Through a multiplicative update algorithm [59] for non-negative matrix factorization (NNMF), these levels are factored into T approximate “part matrices” which represent level patterns and T coefficient matrices corresponding to weights for 2. Source: Procedural Content Generation via Machine Learning (PCGML) (#CQBDX4), Aaron Isaksen, Adam Summerville, Amy K. Hoover, Andy Nealen, Christoffer Holmgård, Julian Togelius, Matthew Guzdial, Sam Snodgrass, p. 6 Context: #6EE2TX Procedural Content Generation via Machine Learning (PCGML) #6BC6KQ III. METHODS OF PCGML #QNMYX8 B. Grids #AQ7YV6 1) Frequency Counting Score: 0.028 Related excerpt #SKB6TL: A recent approach by Gumin [51] is loosely inspired by quantum mechanics and uses a “super-position” of tiles to generate images and levels from a representative example tile set. This approach is a variant of MRF, except instead of solely sampling, samples are chosen via “collapsing of the wave function” (i.e. probabilistically choosing a tile and propagating constraints that that choice enforces). This in turn can propagate other changes and either deterministically chooses tiles that no longer have any other possible choices or reduces the possible set of other tiles. The probabilities and configurations are determined by finding each N \times N window in the input, and the number of times that window occurs. This approach was initially explored for bitmap generation, but has since been expanded for use with 3-D tile sets as well as for level generation [52], [53]. The source code and examples of the bitmap project can be found online [51]. 3. Source: Procedural Content Generation via Machine Learning (PCGML) (#CQBDX4), Aaron Isaksen, Adam Summerville, Amy K. Hoover, Andy Nealen, Christoffer Holmgård, Julian Togelius, Matthew Guzdial, Sam Snodgrass, p. 8 Context: #6EE2TX Procedural Content Generation via Machine Learning (PCGML) #6BC6KQ III. METHODS OF PCGML #FFTSJC C. Graphs #7EXCGN 1) Expectation Maximization Score: 0.016 Related excerpt #BDS4HS: Guzdial and Riedl’s model structure was adapted from [64], a graph structure meant to encode styles of shapes and their probabilistic relationships. The shapes in this case refer to collections of identical sprites tiled over space in different configurations. For further details please see [62], but it can be understood as a learned shape grammar, identifying individual shapes and probabilistic rules on how to combine them. First the chunks of level geometry were clustered to derive styles of level chunks, then the shapes within that chunk were clustered again to derive styles of shapes, and lastly the styles of shapes were clustered to determine how they could be combined to form novel chunks of level. After this point generating a new level requires generating novel level chunks in sequences derived from the gameplay videos. An example screen of generated content using this approach is shown in Figure 10. 4. Source: Procedural Content Generation via Machine Learning (PCGML) (#CQBDX4), Aaron Isaksen, Adam Summerville, Amy K. Hoover, Andy Nealen, Christoffer Holmgård, Julian Togelius, Matthew Guzdial, Sam Snodgrass, p. 5 Context: #6EE2TX Procedural Content Generation via Machine Learning (PCGML) #6BC6KQ III. METHODS OF PCGML #UPLPFN 3) Back Propagation Score: 0.016 Related excerpt #7PQ2V5: 6 of standard of RNNs. Summerville and Matteas used a tile representation representing levels as a linear string with 3 different representations used for experimentation. They also included simulated player path information in the input data, forcing the generator to generate exemplar paths in addition to the level geometry. Finally, they included information about how deep into the string the level geometry was, causing the generator to learn both level progression and when a level should end. 5. Source: Procedural Content Generation via Machine Learning (PCGML) (#CQBDX4), Aaron Isaksen, Adam Summerville, Amy K. Hoover, Andy Nealen, Christoffer Holmgård, Julian Togelius, Matthew Guzdial, Sam Snodgrass, p. 9 Context: #6EE2TX Procedural Content Generation via Machine Learning (PCGML) #6BC6KQ III. METHODS OF PCGML #PQ6DK6 D. Discussion of Approaches Score: 0.015 Related excerpt #5Z6GWM: Broadly, we see that platformer level generation is the most common target of generation and is represented by all types of training and all types of representation. The graph representation is the most general in form [62], with no notion of level shape or size – only the relative positions of entities. The sequence and grid based approaches all explicitly encode some aspect of the level shape, either by fixing the height at generation time [34], [31], [36], [21], [42] or at run time [47], [50], [26], [49], [48] — while the matrix factorization approach requires fixing both height and width [58]. Similarly, the convolutional grid approach [29] for RTS map generation requires a fixed map size. 6. Source: Procedural Content Generation via Machine Learning (PCGML) (#CQBDX4), Aaron Isaksen, Adam Summerville, Amy K. Hoover, Andy Nealen, Christoffer Holmgård, Julian Togelius, Matthew Guzdial, Sam Snodgrass, p. 7 Context: #6EE2TX Procedural Content Generation via Machine Learning (PCGML) #6BC6KQ III. METHODS OF PCGML #QNMYX8 B. Grids #UKUFBH 3) Matrix Factorization Score: 0.015 Related excerpt #3GNRWF: While approaches to generating levels typically focus on platformers (e.g. Super Mario Bros. ), Summerville et al. [32] generate levels for The Legend of Zelda [60] series. Their approach relies on segmenting data hierarchically by first generating the high-level topological structure of a dungeon (discussed in Section III-C) and then the rooms represented as a grid of tiles via Principal Component Analysis (PCA). The PCA algorithm finds a compressed representation of the original 2-D room arrays by taking the eigenvectors of the original high dimensional space and retaining only the most informative (inspired by EigenFaces[61]). These compressed representations are represented as weight vectors that can be interpolated between to generate new room content as seen in Figure 9. 7. Source: Procedural Content Generation via Machine Learning (PCGML) (#CQBDX4), Aaron Isaksen, Adam Summerville, Amy K. Hoover, Andy Nealen, Christoffer Holmgård, Julian Togelius, Matthew Guzdial, Sam Snodgrass, p. 6 Context: #6EE2TX Procedural Content Generation via Machine Learning (PCGML) #6BC6KQ III. METHODS OF PCGML #QNMYX8 B. Grids #AQ7YV6 1) Frequency Counting Score: 0.015 Related excerpt #FZFYY9: They train an MdMC by building a probability table according to the frequency of the tiles in training data, given the network structure of the MdMC, the set of training levels, and the set of tile types. A new level is then sampled one tile at a time by probabilistically choosing the next tile based upon the types of the previous tiles and the learned probability table. 8. Source: Procedural Content Generation via Machine Learning (PCGML) (#CQBDX4), Aaron Isaksen, Adam Summerville, Amy K. Hoover, Andy Nealen, Christoffer Holmgård, Julian Togelius, Matthew Guzdial, Sam Snodgrass, p. 5 Context: #6EE2TX Procedural Content Generation via Machine Learning (PCGML) #6BC6KQ III. METHODS OF PCGML #QNMYX8 B. Grids Score: 0.014 Related excerpt #PBHQ6N: Most game levels (particularly of the pre-3D era) can be thought of as two-dimensional grids. Sometimes these representations are lossy (e.g. a non-tile entity is forcibly aligned to the grid, even if it could actually be at a non-tile position), but are generally a natural representation for many different kinds of levels (e.g. platformers, dungeons, real time strategy maps, etc.). ### 43. Assistant tool calls #### Tool call: `write_note` ```json { "block_ids": [ "ZBQE96", "BLF82B", "NNC37P", "477DMT", "A6G5PJ", "C9XNSJ", "L94DMG", "KAMGFF", "WL3SCW", "PUZXXV", "FXLSNG", "WQW77N", "DJQRBX", "964T2S", "48RCD2", "RRJ2BJ", "LNKPPL" ], "text": "Ellerman’s partition logic suggests a useful cross-corpus lens, but only where blocks are mutually exclusive and jointly exhaustive (#ZBQE96). The clearest exact case is hydrological catchments: terrain cells are equivalent when they drain to the same outlet, and watershed labeling assigns one common label to every such equivalence class (#NNC37P, #477DMT). FastFlow similarly defines a basin as cells sharing a stream tree/root (#A6G5PJ) and propagates basin identifiers upstream (#C9XNSJ); saddle crossings then connect or merge basin blocks (#L94DMG), suggesting dynamic coarsening of a catchment partition. Hydrological terrain generation also constructs Voronoi cells and hierarchical watersheds/subwatersheds (#KAMGFF, #WL3SCW), giving nested partitions at multiple scales. Other exact or near-exact corpus examples include planar regions cut by major streets/topographic boundaries (#PUZXXV); recursive figure/ground and connected-pixel segmentation (#FXLSNG); head/tail classes (#WQW77N); MAP-Elites bins that partition behavior/search space (#DJQRBX, #964T2S); fluid particles classified into rendering layers (#48RCD2); and walker populations grouped by entry–destination pair (#RRJ2BJ). Alexanderian centers should not be treated as a partition without qualification because their local symmetries and centers overlap across scales (#LNKPPL). Conceptually, a deterministic map from each terrain cell to its terminal outlet realizes Ellerman’s function-to-partition idea (#BLF82B): catchments are the inverse-image fibers of the outlet map.", "title": "Partition lens across the corpus" } ``` ### 44. Tool result: write_note NOTE SAVED #TUCFMG IRI: https://m.sheaf.less.rest/TUCFMG ### 45. Assistant I added this micro-abstract to #C8FHDZ: > Develops implication as a refinement-sensitive operation on set partitions, showing how relative negation yields local Boolean cores within the non-distributive algebra of partitions. ## The catchment-basin example is a literal partition This is more than a loose analogy. Let $U$ be the terrain cells and let $$ o:U\rightarrow O $$ send each cell to the terminal outlet reached by following its drainage path. The inverse images $$ o^{-1}(q),\qquad q\in O $$ are mutually exclusive and jointly exhaustive catchment basins. Thus two cells are equivalent precisely when they drain to the same outlet. This directly instantiates Ellerman’s observation that the inverse-image fibers of a function form a partition (#BLF82B). The corpus states this almost verbatim: - Priority-Flood assigns a common label to every cell draining to a given outlet (#NNC37P), producing one label for every terrain cell (#477DMT). - FastFlow defines a basin as the cells sharing one stream tree and terminal root (#A6G5PJ), then propagates a basin identifier from each minimum to its upstream cells (#C9XNSJ). This partition also has useful dynamics: - **Watersheds and subwatersheds** form nested partitions at different scales (#KAMGFF, #WL3SCW). The subwatershed partition refines the watershed partition. - **Drainage divides** are the spatial manifestation of distinctions: adjacent cells on opposite sides belong to different blocks. - **Saddles and overflow connections** can join previously separate basins (#L94DMG). In partition language, this is a coarsening operation: distinctions disappear as basin blocks merge. If $\sigma$ is the coarse catchment partition and $\pi$ the finer subcatchment partition, then every block of $\pi$ lies inside a block of $\sigma$, so: $$ \sigma\preceq\pi \quad\Longleftrightarrow\quad \sigma\Rightarrow\pi=\mathbf 1. $$ That gives partition implication a concrete landscape interpretation: it tests, block by block, whether one hydrological regionalization respects another. ## Other partition-like notions in the corpus ### Strong, fairly literal cases 1. **Voronoi terrain cells** Hydrological terrain generation decomposes the domain into Voronoi cells associated with river-graph nodes (#KAMGFF). Subject to the usual boundary convention, these are a spatial partition. 2. **Street-defined urban regions** Major streets and topographic boundaries divide a domain into regions, each of which can receive its own local tensor field and minor-road pattern (#PUZXXV). This is a partition of urban space induced by infrastructure. 3. **Figure/ground and image segmentation** Jiang and de Rijke divide pixels into dark and light classes around an average value, then derive connected substructures from the binary image (#FXLSNG). Each segmentation stage gives a partition of the relevant pixel set; recursion creates a hierarchy of local refinements. 4. **Head/tail breaks** A dataset is divided into values above and below its mean, after which the head is divided again (#WQW77N). This is naturally a sequence of nested partitions rather than one flat classification. 5. **Quality-diversity search spaces** MAP-Elites discretizes behavioral dimensions into cells, each containing individuals with the corresponding behavior (#DJQRBX). One example explicitly describes classifier outputs as a partition of the search space (#964T2S). 6. **Fluid-particle classes and layers** A water renderer first classifies particles as water or foam and then partitions the fluid into rendering layers (#48RCD2). Here the underlying set is particles rather than geographical space. 7. **Pedestrian populations** The active-walker model divides individuals into subpopulations sharing the same entry point and destination (#RRJ2BJ). These origin–destination classes partition the walker population, even though their physical trails may later overlap and bundle. ### Important non-example or qualified case Alexanderian **centers** and “subwholes” should not automatically be called partition blocks. Centers can overlap, interlock, and operate at several scales (#LNKPPL), whereas a strict partition requires disjoint blocks. They may be better represented as an overlapping cover, hypergraph, or hierarchy. A partition appears only after choosing a rule that assigns every point to exactly one dominant center. So the most promising bridge is: > **A landscape can carry multiple partitions—by outlet, stream tree, Voronoi generator, street-bounded region, land-use class, or destination—and design or simulation changes can be studied as refinement, coarsening, or comparison between those partitions.** I saved this cross-corpus connection as research note #TUCFMG.