{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.isohedral-figure","code":"thing-q3847067","url":"https://ver.cy/models/thing/q3847067/","name":"isohedral figure","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":"An isohedral figure is a geometric figure with faces, such as a polyhedron or tiling, whose symmetry group acts transitively on those faces, so that any face can be mapped onto any other by a symmetry of the entire figure.","purpose":"Enable an agent to recognise, verify and manipulate an isohedral figure by establishing whether its specified faces form a single orbit under symmetries of the whole figure.","scope":{"in":["The face set on which isohedrality is asserted","Ambient geometry and the permitted whole-figure symmetries","Face orbits and evidence of transitivity","Distinctions between congruent faces and symmetry-equivalent faces","The effect of realisation, decoration and geometric transformations on isohedrality"],"out":["General construction and classification of polyhedra independently of face transitivity","Vertex transitivity and edge transitivity as independent classification tasks","Classification of tile shapes by their ability to admit isohedral tilings","Physical dice fairness, material density and rolling dynamics","Manufacturing tolerances and structural performance of physical replicas"],"boundaries":[]},"distinguishingFeatures":["For any two designated faces, a permitted symmetry must preserve the entire figure and carry one face onto the other.","Congruent faces alone do not establish isohedrality: their equivalence must extend to a symmetry of the whole figure.","Vertex or edge transitivity cannot substitute for a single face orbit.","A combinatorial permutation of faces establishes geometric isohedrality only when realised by an allowed geometric symmetry.","An isohedral tiling concerns equivalence of whole tiles within a tiling; a polyhedron's isohedrality concerns equivalence of its faces."],"structure":{"bundles":[{"id":"isohedral-meaning","name":"Isohedral meaning","description":"Fix the interpretation of the registry term before classifying an instance.","layers":[{"id":"polyhedral-core","name":"Polyhedral core","description":"Anchor the working interpretation in whole-figure face transitivity.","findings":[{"id":"face-transitivity-definition","name":"Face transitivity definition","description":"The working polyhedral interpretation requires the symmetry group to act transitively on faces; the reference definition of an isohedron additionally specifies convexity. Source: [Wolfram MathWorld: Isohedron](https://mathworld.wolfram.com/Isohedron.html).","questions":[{"text":"Which reference supplies the definition used for this instance?","id":"face-transitivity-definition-q01","kind":"provenance"},{"text":"Does that definition require convexity, or explicitly admit nonconvex figures?","id":"face-transitivity-definition-q02","kind":"boundary"}]}]},{"id":"extended-interpretations","name":"Extended interpretations","description":"Prevent changes in the acted-on entities from silently changing the meaning.","findings":[{"id":"faces-versus-tiles","name":"Faces versus tiles","description":"In isohedral tilings, the symmetry action concerns tiles of the whole tiling. Any use of that interpretation here must be explicit. Source: [Wolfram MathWorld: Isohedral Tiling](https://mathworld.wolfram.com/IsohedralTiling.html).","questions":[{"text":"Is the assertion about a polyhedron's faces or a tiling's tiles?","id":"faces-versus-tiles-q01","kind":"definition"},{"text":"For a higher-dimensional figure, does 'face' mean a two-dimensional face or a facet?","id":"faces-versus-tiles-q02","kind":"boundary"}]}]}]},{"id":"face-realisation","name":"Face realisation","description":"Specify the geometric figure and the faces whose equivalence is being assessed.","layers":[{"id":"face-decomposition","name":"Face decomposition","description":"Identify face boundaries and their incidence within the whole figure.","findings":[{"id":"designated-faces","name":"Designated faces","description":"Record the adopted face decomposition, including whether coplanar subdivisions count as separate faces.","questions":[{"text":"What boundary and incidence data identify each designated face?","id":"designated-faces-q01","kind":"definition"},{"text":"Are coplanar subdivisions, holes or intersecting face boundaries admitted by the convention?","id":"designated-faces-q02","kind":"boundary"}]}]},{"id":"metric-realisation","name":"Metric realisation","description":"Distinguish exact geometry from an illustration or approximate mesh.","findings":[{"id":"exactness-and-congruence","name":"Exactness and congruence","description":"Record coordinates or defining constraints and compare face shapes as an initial obstruction test; congruence alone leaves the transitivity question open.","questions":[{"text":"Which exact coordinates or geometric constraints define this realisation?","id":"exactness-and-congruence-q01","kind":"provenance"},{"text":"How many face congruence classes are established, and with what numerical uncertainty?","id":"exactness-and-congruence-q02","kind":"measurement"}]}]}]},{"id":"symmetry-action","name":"Symmetry action","description":"Record transformations that preserve the whole figure and their action on faces.","layers":[{"id":"admissible-symmetries","name":"Admissible symmetries","description":"Fix the geometric and decoration constraints on transformations.","findings":[{"id":"symmetry-group-convention","name":"Symmetry group convention","description":"Record whether all isometries are allowed and whether colours, labels or orientations are part of the preserved figure.","questions":[{"text":"Must a symmetry preserve only geometry, or also face colours, labels and orientations?","id":"symmetry-group-convention-q01","kind":"definition"},{"text":"Are orientation-reversing isometries permitted in the asserted classification?","id":"symmetry-group-convention-q02","kind":"boundary"}]}]},{"id":"face-orbits","name":"Face orbits","description":"Operationalise transitivity through the induced permutations of faces.","findings":[{"id":"orbit-computation","name":"Orbit computation","description":"For each verified symmetry generator, record its face permutation and compute the resulting orbits. Distinguish a known subgroup from the full permitted symmetry group.","questions":[{"text":"Which whole-figure isometries generate the tested action, and how does each permute faces?","id":"orbit-computation-q01","kind":"measurement"},{"text":"How many face orbits result, and is the tested group known to be complete?","id":"orbit-computation-q02","kind":"measurement"}]}]}]},{"id":"transitivity-evidence","name":"Transitivity evidence","description":"Distinguish a proof, a counterexample and an incomplete symmetry search.","layers":[{"id":"positive-certificate","name":"Positive certificate","description":"Make a claim of isohedrality reproducible.","findings":[{"id":"transitive-subgroup-witness","name":"Transitive subgroup witness","description":"A verified subgroup acting transitively suffices for a positive certificate; identifying every symmetry is unnecessary for that conclusion.","questions":[{"text":"What verified transformations carry a representative face to every designated face?","id":"transitive-subgroup-witness-q01","kind":"measurement"},{"text":"How can an agent check that these transformations preserve the whole geometric realisation?","id":"transitive-subgroup-witness-q02","kind":"action"}]}]},{"id":"negative-and-unresolved-results","name":"Negative and unresolved results","description":"Require a genuine obstruction before rejecting isohedrality.","findings":[{"id":"orbit-separation-certificate","name":"Orbit separation certificate","description":"Support non-isohedrality with a complete permitted-group computation or an invariant separating faces; otherwise retain an unresolved result.","questions":[{"text":"Which invariant prevents two identified faces from being exchanged by any permitted symmetry?","id":"orbit-separation-certificate-q01","kind":"measurement"},{"text":"Could undiscovered symmetries merge the currently computed face orbits?","id":"orbit-separation-certificate-q02","kind":"boundary"},{"text":"What additional computation or exact geometric data would settle the unresolved claim?","id":"orbit-separation-certificate-q03","kind":"action"}]}]}]},{"id":"classification-and-change","name":"Classification and change","description":"Control comparisons with neighbouring symmetry properties and reassess modified figures.","layers":[{"id":"neighbouring-symmetry-properties","name":"Neighbouring symmetry properties","description":"Keep face transitivity distinct from other forms of geometric uniformity.","findings":[{"id":"independent-transitivity-claims","name":"Independent transitivity claims","description":"Record vertex, edge and flag transitivity separately, and do not treat abstract incidence automorphisms as geometric isometries without evidence.","questions":[{"text":"Which vertex, edge or flag transitivity claims have been independently established?","id":"independent-transitivity-claims-q01","kind":"boundary"},{"text":"Does the available certificate concern geometric isometries or only incidence-preserving permutations?","id":"independent-transitivity-claims-q02","kind":"provenance"}]}]},{"id":"modified-realisations","name":"Modified realisations","description":"Track whether a construction preserves the face set and its transitive symmetry action.","findings":[{"id":"post-transformation-verification","name":"Post-transformation verification","description":"After deformation, subdivision, decoration or dual construction, identify the resulting face set and verify the relevant symmetry action before transferring an isohedral classification.","questions":[{"text":"Which previously verified face-mapping symmetries remain isometries preserving the modified figure?","id":"post-transformation-verification-q01","kind":"measurement"},{"text":"Does the operation change the designated faces or split their symmetry orbit?","id":"post-transformation-verification-q02","kind":"boundary"},{"text":"What new certificate is required before the modified figure can be classified as isohedral?","id":"post-transformation-verification-q03","kind":"action"}]}]}]},{"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 isohedral figure 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 isohedral figure this model covers, and on what evidence?","id":"evidence-kinds-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 isohedral figure 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 isohedral figure 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 isohedral figure 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 isohedral figure this model covers, and on what evidence?","id":"evidence-neighbours-q01","kind":"provenance"}]}]}]}]},"agentConduct":{"may":["Define, classify and draw isohedral figures and tilings from mathematical sources.","Check whether a figure is face-transitive in calculations."],"mustNot":["Present a figure as isohedral when its faces are not all equivalent under its symmetries.","Misattribute discoveries in tiling theory."],"requiresHuman":["Publishing classification results as new mathematics."]},"ethics":{"considerations":["Accurate mathematics underpins design, crystallography and teaching."],"affectedParties":["Students","Designers","Mathematicians"]},"owners":{"steward":"Nobody owns the concept; it belongs to mathematics.","roles":[],"masterSystems":[]},"relations":[{"target":"Q747980","type":"parent","note":"registry parent class"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.isohedral-figure","Wikidata: Q3847067 (https://www.wikidata.org/wiki/Q3847067)"]},"properties":{"applicability":"not-applicable","items":["Geometric interpretation: polyhedral face transitivity | explicitly specified extension | unresolved","Ambient geometry: Euclidean | spherical | hyperbolic | other specified geometry","Designated face set: Reference to a face decomposition with face dimension and incidence data","Symmetry convention: full isometry group | orientation-preserving subgroup | decoration-preserving subgroup | other specified subgroup","Face orbit count: Positive integer or infinite cardinal when established; otherwise unknown","Face congruence class count: Positive integer when established; otherwise unknown","Verification state: unassessed | candidate | proved isohedral | proved non-isohedral | unresolved","Verification certificate: Reference to exact symmetry generators, orbit proof or obstruction"]},"recognition":{"applicability":"optional","items":["A solid or tiling in which every face is the same shape and is placed alike, such as a cube or a rhombic dodecahedron."]},"capabilities":{"applicability":"required","items":["Test whether exact whole-figure symmetries act transitively on the designated faces.","Construct a face correspondence from one representative face to every other face.","Reject candidate isohedrality using a verified symmetry invariant that separates faces.","Identify which face orbits split when reflections or decoration-changing symmetries are excluded.","Reassess isohedrality after changing coordinates, subdividing faces or applying a geometric construction."]},"hazards":{"applicability":"required","items":["Mistaking congruent faces for symmetry-equivalent faces can produce incorrect classifications and invalidate geometric arguments.","Assuming face transitivity guarantees structural strength or manufacturing suitability can lead to unreliable physical designs."]},"interfaces":{"applicability":"not-applicable","items":[]},"context":{"applicability":"required","items":["Isohedral figures are studied in geometry, symmetry theory and the classification of polyhedra and tilings.","They are used in geometric modelling and pattern design when every face must be equivalent under a symmetry of the entire figure."]}},"sources":[{"title":"Wikidata item Q3847067: isohedral figure","url":"https://www.wikidata.org/wiki/Q3847067","note":"identity and sense of the item"},{"title":"Wikipedia: Isohedral figure","url":"https://en.wikipedia.org/wiki/Isohedral_figure","note":"general description of the item"}],"openQuestions":["Does the registry intend 'isohedral figure' to cover only polyhedra, or also tilings and higher-dimensional figures?","Does the intended definition require convexity, and which nonconvex or self-intersecting realisations, if any, are admitted?","Should unqualified isohedrality always use the full geometric symmetry group, with orientation and decoration restrictions recorded as separate claims?","Which convention should determine the designated face set when coplanar subdivisions or higher-dimensional faces are present?","Which existing Vercy model, if any, already owns this concept and should serve as its single source of truth?"],"resources":{"spec":"/models/things/publications/thing-q3847067/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q3847067/spec.json"],"providers":["Codex"],"researchStatus":"unreviewed","generatedAt":"2026-09-12T10:37:57Z","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 2026-10-07T10:10:30+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 specific standard, law or official classification is applicable to the mathematical property of face transitivity."},"score":1.0}}