{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.hexagonal-tiling","code":"thing-q3063631","url":"https://ver.cy/models/thing/q3063631/","name":"hexagonal tiling","alternateNames":[],"kind":"thing","status":"published","version":"0.1.0-research.1","language":"en","classifiers":{"family":"Thing Registry","category":"Cross-cutting context","entryKind":"thing","plane":"XCT","domain":["XCT.QLT"],"industry":[],"navPath":"","tags":[],"facets":{}},"whatItIs":"Hexagonal tiling is the edge-to-edge covering of the Euclidean plane by congruent regular hexagons, without gaps or overlapping interiors, with three hexagons meeting at every vertex.","purpose":"Enable an agent to recognise the regular hexagonal tiling of the Euclidean plane, assess its representations and finite patches, and choose transformations that preserve its geometry and incidence relationships.","scope":{"in":["The regular Euclidean tiling by congruent regular hexagons is included.","Cell, edge and vertex incidence and adjacency are included.","Geometric representations, coordinate conventions and symmetry are included.","Finite patches and their boundaries are included as representations of portions of the plane tiling.","Operations that preserve the regular plane tiling's defining properties are included."],"out":["Tilings by nonregular hexagons are excluded.","Tilings on surfaces other than the Euclidean plane are excluded.","Triangular tilings are excluded except for their dual relationship to the regular hexagonal tiling.","General polygonal tiling classification beyond the regular hexagonal case is excluded.","Physical honeycomb materials, manufacturing processes and mechanical properties are excluded.","Game rules, map semantics and simulation dynamics assigned to hexagonal cells are excluded.","Three-dimensional honeycombs and space-filling polyhedra are excluded."],"boundaries":[]},"distinguishingFeatures":["Each cell is a regular hexagon congruent to every other cell; six boundary edges alone do not establish this tiling.","The tiling covers the Euclidean plane without gaps or overlapping cell interiors; a disconnected collection of hexagons fails this test.","Cells have equal sides and 120-degree interior angles, with three cells meeting at each vertex.","The tiling has six edge-sharing neighbours per cell; the graph connecting cell centres is triangular and differs from the tiling's vertex-edge graph.","A finite drawing represents a patch of the plane tiling and must not silently inherit the boundary-free properties of the infinite plane."],"structure":{"bundles":[{"id":"hexagonal-identity","name":"Hexagonal identity","description":"Establishes what the term hexagonal tiling means for the represented instance.","layers":[{"id":"regularity-commitment","name":"Regularity commitment","description":"Separates the regular reference case from broader uses of hexagonal tiling.","findings":[{"id":"declared-hexagonal-meaning","name":"Declared hexagonal meaning","description":"Record whether hexagonal asserts regular congruent cells, merely six-sided polygons, or an abstract six-edge face structure.","questions":[{"text":"Does this instance require regular congruent hexagons or only six-sided cells?","id":"declared-hexagonal-meaning-q01","kind":"definition"},{"text":"Which source or construction establishes that interpretation?","id":"declared-hexagonal-meaning-q02","kind":"provenance"}]}]},{"id":"tiling-domain","name":"Tiling domain","description":"Defines where coverage is claimed and how a displayed patch relates to that claim.","findings":[{"id":"coverage-claim","name":"Coverage claim","description":"Record the domain and distinguish an infinite tiling rule from a finite patch or bounded-domain tiling.","questions":[{"text":"What exact region or surface must the hexagons cover?","id":"coverage-claim-q01","kind":"boundary"},{"text":"Is the finite representation the complete object or a sample of an asserted infinite tiling?","id":"coverage-claim-q02","kind":"definition"}]}]}]},{"id":"hexagon-geometry","name":"Hexagon geometry","description":"Records the cell shapes and their realisation in a geometric space.","layers":[{"id":"cell-shape","name":"Cell shape","description":"Makes six-sidedness, regularity, and congruence independently assessable.","findings":[{"id":"shape-tests","name":"Shape tests","description":"Record boundary edges, simplicity, side lengths, and angles, with explicit treatment of collinear subdivision vertices.","questions":[{"text":"Does every cell have six geometric sides, and how are collinear boundary segments counted?","id":"shape-tests-q01","kind":"definition"},{"text":"Where regularity is claimed, do side lengths agree and interior angles equal 120 degrees within the declared tolerance?","id":"shape-tests-q02","kind":"measurement"}]}]},{"id":"embedding-and-scale","name":"Embedding and scale","description":"Connects the abstract arrangement to measurable positions, lengths, and areas.","findings":[{"id":"metric-realisation","name":"Metric realisation","description":"Record the ambient metric, scale, and orientation needed to interpret the hexagons geometrically.","questions":[{"text":"What metric and units govern side lengths, cell areas, and centre separations?","id":"metric-realisation-q01","kind":"measurement"},{"text":"Which proposed transformations preserve regular hexagons, and which preserve only a six-sided tiling?","id":"metric-realisation-q02","kind":"action"}]}]}]},{"id":"hexagonal-incidence","name":"Hexagonal incidence","description":"Records how hexagonal faces share edges and meet at vertices.","layers":[{"id":"edge-and-vertex-meetings","name":"Edge and vertex meetings","description":"Checks conformity and the local arrangement around each vertex.","findings":[{"id":"local-incidence-tests","name":"Local incidence tests","description":"Test complete-edge sharing and record incident cells, distinguishing interior locations from boundaries.","questions":[{"text":"Do cells meet along complete shared edges or at vertices, or are partial-edge contacts present?","id":"local-incidence-tests-q01","kind":"boundary"},{"text":"For the regular reference case, do exactly three hexagons meet at every interior vertex?","id":"local-incidence-tests-q02","kind":"measurement"}]}]},{"id":"primal-and-dual-graphs","name":"Primal and dual graphs","description":"Distinguishes the hexagon boundary network from the network of adjacent cells.","findings":[{"id":"graph-role","name":"Graph role","description":"Identify whether graph nodes represent tiling vertices or hexagonal cells before assigning degrees or distances.","questions":[{"text":"Do graph nodes represent polygon corners or whole hexagonal cells?","id":"graph-role-q01","kind":"definition"},{"text":"Does the regular interior exhibit degree three in the vertex-edge graph and degree six in the cell-adjacency graph?","id":"graph-role-q02","kind":"measurement"}]}]}]},{"id":"hex-grid-addressing","name":"Hex-grid addressing","description":"Supports unambiguous cell indexing, neighbourhood operations, and symmetry-aware comparisons.","layers":[{"id":"index-conventions","name":"Index conventions","description":"Records the origin, axes, orientation, and indexing convention.","findings":[{"id":"coordinate-contract","name":"Coordinate contract","description":"Specify how indices map to cells, including offset parity or the cube-coordinate constraint when applicable.","questions":[{"text":"Which coordinate convention, origin, axis directions, and offset parity identify each cell?","id":"coordinate-contract-q01","kind":"definition"},{"text":"Can indices be converted to cell centres and back without changing cell identity?","id":"coordinate-contract-q02","kind":"action"}]}]},{"id":"distance-and-symmetry","name":"Distance and symmetry","description":"Separates movement through adjacent cells from geometric displacement and structural equivalence.","findings":[{"id":"grid-operations","name":"Grid operations","description":"Record the distance interpretation and transformations permitted by the actual tiling, patch, and labels.","questions":[{"text":"Is distance measured as edge-adjacent cell steps, Euclidean centre separation, or a weighted path cost?","id":"grid-operations-q01","kind":"measurement"},{"text":"Which translations, rotations, and reflections preserve this instance, including its boundaries and labels?","id":"grid-operations-q02","kind":"action"}]}]}]},{"id":"coverage-boundaries-and-repair","name":"Coverage, boundaries, and repair","description":"Assesses whether a candidate arrangement satisfies its declared tiling contract and how it may be changed.","layers":[{"id":"patches-and-seams","name":"Patches and seams","description":"Records treatment of finite edges, clipped cells, holes, and periodic identifications.","findings":[{"id":"boundary-contract","name":"Boundary contract","description":"Distinguish absent neighbours at a patch edge from identified neighbours across a periodic seam.","questions":[{"text":"Are boundary cells retained whole, clipped to a region, or identified with cells across a periodic seam?","id":"boundary-contract-q01","kind":"boundary"},{"text":"Do seam identifications preserve the intended cell geometry and incidence without unintended self-adjacency?","id":"boundary-contract-q02","kind":"measurement"}]}]},{"id":"validation-and-editing","name":"Validation and editing","description":"Connects detected violations to operations that respect the chosen interpretation.","findings":[{"id":"tiling-validity-evidence","name":"Tiling validity evidence","description":"Record evidence for six-sided cells, coverage, disjoint interiors, and any additional regularity requirements at the extent actually checked.","questions":[{"text":"What proof or computation establishes gap-free coverage and disjoint cell interiors over the declared domain?","id":"tiling-validity-evidence-q01","kind":"provenance"},{"text":"Can a detected defect be repaired while preserving six-sided cells, the required regularity, and the boundary contract?","id":"tiling-validity-evidence-q02","kind":"action"},{"text":"Does validation cover only a finite patch, or also establish the correctness of its infinite extension rule?","id":"tiling-validity-evidence-q03","kind":"boundary"}]}]}]},{"id":"evidence-and-external-alignment","name":"Evidence and external alignment","description":"What the world already says about this thing, gathered so the model can be checked against it.","layers":[{"id":"reported-evidence","name":"Reported evidence","description":"Findings from the breadth pass, kept separate from the structural claims.","findings":[{"id":"evidence-confidence-notes","name":"Check these first","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these check these first hold for the sense of hexagonal tiling this model covers, and on what evidence?","id":"evidence-confidence-notes-q01","kind":"provenance"}]},{"id":"evidence-kinds","name":"Kinds and varieties","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these kinds and varieties hold for the sense of hexagonal tiling this model covers, and on what evidence?","id":"evidence-kinds-q01","kind":"provenance"}]},{"id":"evidence-identifiers","name":"Identifiers and schemes","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these identifiers and schemes hold for the sense of hexagonal tiling this model covers, and on what evidence?","id":"evidence-identifiers-q01","kind":"provenance"}]},{"id":"evidence-real-world-use","name":"Real-world use","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these real-world use hold for the sense of hexagonal tiling this model covers, and on what evidence?","id":"evidence-real-world-use-q01","kind":"provenance"}]},{"id":"evidence-measurements","name":"Typical measurements","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these typical measurements hold for the sense of hexagonal tiling this model covers, and on what evidence?","id":"evidence-measurements-q01","kind":"provenance"}]},{"id":"evidence-failure-modes-and-hazards","name":"Failure modes and hazards","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these failure modes and hazards hold for the sense of hexagonal tiling this model covers, and on what evidence?","id":"evidence-failure-modes-and-hazards-q01","kind":"provenance"}]},{"id":"evidence-neighbours","name":"Neighbouring kinds and how to tell them apart","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these neighbouring kinds and how to tell them apart hold for the sense of hexagonal tiling this model covers, and on what evidence?","id":"evidence-neighbours-q01","kind":"provenance"}]}]}]}]},"agentConduct":{"may":["Explain the hexagonal tiling, its symmetry and its occurrence in nature and design.","Generate hexagonal grids for maps, games or layouts on request."],"mustNot":["Claim that natural hexagons such as honeycomb are perfect tilings when they are approximate.","Present tiling patterns from a culture's art as anonymous or ownerless designs when their origin is known.","Use flawed grid geometry in a technical design without disclosing it."],"requiresHuman":["Using a grid for planning or survey work whose errors could affect people's property."]},"ethics":{"considerations":["Traditional tiling patterns belong to the heritage of the cultures that made them."],"affectedParties":["Students","Designers","Communities whose motifs are used"]},"owners":{"steward":"Nobody owns it; it is a concept of geometry.","roles":[],"masterSystems":[]},"relations":[{"target":"Q18062082","type":"parent","note":"registry parent class"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.hexagonal-tiling","Wikidata: Q3063631 (https://www.wikidata.org/wiki/Q3063631)"]},"properties":{"applicability":"not-applicable","items":["Interpretation of hexagonal: regular Euclidean honeycomb | irregular six-sided-cell tiling | combinatorial hexagonal cellulation","Underlying domain: Euclidean plane | bounded planar region | periodic quotient | explicitly specified other surface","Edge-to-edge conformity: verified | violated | untested","Regular-cell side length: positive length in declared embedding units; not applicable without regular congruent cells","Vertex configuration: 6.6.6 for the regular reference case; otherwise recorded incident-face sequences","Cell adjacency: shared-edge adjacency with boundary and periodic identifications recorded","Coordinate convention: axial | cube | offset | explicit geometric coordinates | other declared system","Coverage validity: verified over declared domain | gap detected | overlap detected | unresolved"]},"recognition":{"applicability":"optional","items":["A plane covered by equal regular hexagons, three meeting at each corner like a honeycomb."]},"capabilities":{"applicability":"required","items":["Classify a candidate as a regular honeycomb, an irregular hexagonal tiling, or an insufficiently specified arrangement.","Generate cells or extract a finite patch under explicit scale, orientation, and boundary conventions.","Convert cell indices and compute edge-sharing neighbours under a declared coordinate system.","Measure cell areas, geometric separations, and graph distances without conflating these quantities.","Detect gaps, overlaps, malformed cells, and incidence violations.","Apply transformations or periodic identifications and recheck the properties they are intended to preserve."]},"hazards":{"applicability":"required","items":["Coordinate mistakes in hexagonal grids misplace cells in maps and simulations."]},"interfaces":{"applicability":"not-applicable","items":[]},"context":{"applicability":"required","items":["Regular hexagonal tiling is studied in geometry as an edge-to-edge covering of the plane without gaps or overlaps.","It is used in game boards, spatial grids and surface patterns where six equivalent neighboring cells are useful.","Physical applications approximate the mathematical tiling within boundaries and manufacturing tolerances."]}},"sources":[{"title":"Wikidata item Q3063631: hexagonal tiling","url":"https://www.wikidata.org/wiki/Q3063631","note":"identity and sense of the item"},{"title":"Wikipedia: Hexagonal tiling","url":"https://en.wikipedia.org/wiki/Hexagonal_tiling","note":"general description of the item"}],"openQuestions":["Does the registry intend hexagonal tiling to mean only the regular Euclidean honeycomb, or also tilings by irregular hexagons?","Should abstract hexagonal cellulations and tilings on non-Euclidean surfaces be variants of this entry or linked neighbouring concepts?","Which authoritative references should establish the definition, notation, and admissible irregular variants?","How should clipped boundary cells and collinear edge subdivisions affect classification as a hexagonal tiling?","What numerical tolerances and validation evidence should be required for approximate digital representations?"],"resources":{"spec":"/models/things/publications/thing-q3063631/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q3063631/spec.json"],"providers":["Codex"],"researchStatus":"unreviewed","generatedAt":"2026-09-12T10:15:29Z","builder":"tools/build_cards.py@1.0.0"},"completeness":{"sections":{"classifiers":"filled","whatItIs":"filled","purpose":"filled","distinguishingFeatures":"filled","structure":"filled","agentConduct":"filled","ethics":"filled","owners":"filled","relations":"filled","interaction.identity":"filled","interaction.properties":"not-applicable","interaction.recognition":"filled","interaction.capabilities":"filled","interaction.hazards":"filled","interaction.interfaces":"not-applicable","interaction.context":"filled","sources":"filled"},"notes":{"_gate":"Published through the thing publication gate 1.0 on 2026-10-07T12:16:32+00:00: completeness 1.00; sense: Wikidata description; 2 live sources; review by codex: pass","interaction.properties":"Plane XCT: no invented physical properties.","interaction.interfaces":"No standard or regulation applies specifically to this mathematical tiling independently of its application."},"score":1.0}}