{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.regular-polyhedron","code":"thing-q735071","url":"https://ver.cy/models/thing/q735071/","name":"regular polyhedron","alternateNames":["hosohedron","polygonal dihedron","petrial cube","petrial octahedron","petrial tetrahedron","petrial dodecahedron","petrial icosahedron","petrial great dodecahedron","petrial small stellated dodecahedron","petrial great icosahedron","petrial great stellated dodecahedron","apeirogonal hosohedron"],"kind":"thing","status":"research-draft","version":"0.2.0-wave.3","language":"en","classifiers":{"family":"Thing Registry","category":"Cross-cutting context","entryKind":"thing","plane":"XCT","domain":["XCT.QLT"],"industry":[],"navPath":"","tags":[],"facets":{}},"whatItIs":"A polyhedron whose faces are congruent regular polygons and whose vertices are all alike, so that its symmetry group acts transitively on its flags; the convex regular polyhedra are the five Platonic solids, the non-convex ones are the four Kepler-Poinsot polyhedra, and the definition extends to degenerate cases such as hosohedra and dihedra and to abstract and skew forms such as the Petrie duals.","purpose":"Let an agent define regular polyhedra, list the Platonic and Kepler-Poinsot solids with their properties, explain the extended and abstract cases, relay proofs of the classification, and distinguish regular from uniform and semiregular polyhedra.","scope":{"in":[],"out":[],"boundaries":[]},"distinguishingFeatures":["Flag-transitive symmetry","Finite classification","Duality pairs","Extensions to degenerate and abstract cases"],"structure":{"bundles":[{"id":"define","name":"Define","description":"What a regular polyhedron is.","layers":[{"id":"definition","name":"Definition","description":"Definition and list.","findings":[{"id":"definition-finding","name":"Definition","description":"Definition.","questions":[{"text":"What defines a regular polyhedron, and which nine exist in the classical sense?","kind":"definition"},{"text":"Is the solid regular, uniform or semiregular?","kind":"boundary"}]}]},{"id":"extended","name":"Extended","description":"Extended cases.","findings":[{"id":"extended-finding","name":"Extended","description":"Extended.","questions":[{"text":"How do hosohedra, dihedra, skew and abstract polyhedra extend the definition?","kind":"definition"},{"text":"Which entry fits the specific polyhedron?","kind":"action"}]}]}]},{"id":"properties","name":"Properties","description":"Properties and formulas.","layers":[{"id":"counts","name":"Counts","description":"Faces, edges, vertices.","findings":[{"id":"counts-finding","name":"Counts","description":"Counts.","questions":[{"text":"What are the face, edge and vertex counts, angles and symmetry groups of each?","kind":"action"},{"text":"What are the values for the polyhedron in question?","kind":"measurement"}]}]},{"id":"duality","name":"Duality","description":"Duality.","findings":[{"id":"duality-finding","name":"Duality","description":"Duality.","questions":[{"text":"How do the regular polyhedra pair as duals?","kind":"definition"},{"text":"Which entry fits dual polyhedron?","kind":"action"}]}]}]},{"id":"theory","name":"Theory","description":"Classification.","layers":[{"id":"proof","name":"Proof","description":"Classification proof.","findings":[{"id":"proof-finding","name":"Proof","description":"Proof.","questions":[{"text":"How is the classification proved, from Euclid to Euler and Schlafli?","kind":"provenance"},{"text":"Which references are standard?","kind":"provenance"}]}]},{"id":"higher","name":"Higher","description":"Higher dimensions.","findings":[{"id":"higher-finding","name":"Higher","description":"Higher.","questions":[{"text":"How does the classification extend to regular polytopes in higher dimensions?","kind":"provenance"},{"text":"Which entry fits regular polytope?","kind":"action"}]}]}]},{"id":"context","name":"Context","description":"History and use.","layers":[{"id":"history","name":"History","description":"History.","findings":[{"id":"history-finding","name":"History","description":"History.","questions":[{"text":"How were regular polyhedra studied from Plato and Euclid to Kepler and Coxeter?","kind":"provenance"},{"text":"Which entry fits the history of geometry?","kind":"action"}]}]},{"id":"use","name":"Use","description":"Applications.","findings":[{"id":"use-finding","name":"Use","description":"Use.","questions":[{"text":"Where do regular polyhedra appear in crystals, viruses, art and games?","kind":"provenance"},{"text":"Which entry fits a specific application?","kind":"action"}]}]}]}]},"agentConduct":{"may":["define and list regular polyhedra","relay properties and formulas","explain the classification proof","distinguish from uniform and semiregular polyhedra"],"mustNot":[],"requiresHuman":[]},"ethics":{"considerations":[],"affectedParties":[]},"owners":{"steward":"","roles":[],"masterSystems":[]},"relations":[{"target":"Q23821397","type":"parent","note":"registry parent class"},{"target":"Q2745944","type":"parent","note":"registry parent class"},{"target":"Q3847026","type":"parent","note":"registry parent class"},{"target":"Q7045824","type":"parent","note":"registry parent class"},{"target":"regular polytope","type":"related","note":"in three dimensions"},{"target":"isohedron","type":"related","note":"face-transitive"},{"target":"Platonic solid","type":"related","note":"the convex cases"},{"target":"Archimedean solid","type":"related","note":"semiregular"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.regular-polyhedron","Wikidata: Q735071 (https://www.wikidata.org/wiki/Q735071)","Schlafli symbol: {p,q} for each polyhedron"]},"properties":{"applicability":"not-applicable","items":["convex regular polyhedra: 5 count - Platonic solids","non-convex regular polyhedra: 4 count - Kepler-Poinsot"]},"recognition":{"applicability":"optional","items":["Congruent regular faces, identical vertices","Five convex and four star forms","Archimedean solids have regular faces but are not regular","Highly symmetric solids: tetrahedron, cube, octahedron, dodecahedron, icosahedron, and star forms."]},"capabilities":{"applicability":"required","items":["define and list regular polyhedra","relay properties and formulas","explain the classification proof","distinguish from uniform and semiregular polyhedra"]},"hazards":{"applicability":"required","items":["Confusing regular with uniform or semiregular","Ignoring the extended definitions","Errors in vertex and face counts"]},"interfaces":{"applicability":"required","items":["No regulation; definitions follow standard geometry"]},"context":{"applicability":"required","items":["A foundational object in geometry and symmetry.","Platonic solids","Kepler-Poinsot polyhedra","degenerate regular polyhedra: hosohedra and dihedra","regular skew and Petrie dual polyhedra such as the petrial cube","abstract regular polyhedra","regular tilings as infinite analogues"]}},"sources":[],"openQuestions":["Should each polyhedron be a separate entry?","How should Schlafli symbols be linked?","The registry entry has merged aliases naming degenerate and Petrie dual cases; should they be split off?","Which entry fits the specific polyhedron?","What are the face, edge and vertex counts, angles and symmetry groups of each?","Which entry fits dual polyhedron?"],"resources":{"spec":"/models/things/publications/thing-q735071/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q735071/spec.json"],"providers":["Claude"],"researchStatus":"unreviewed","generatedAt":"2026-09-12T21:54:36Z","builder":"tools/build_cards.py@1.0.0"},"completeness":{"sections":{"classifiers":"filled","whatItIs":"filled","purpose":"filled","distinguishingFeatures":"filled","structure":"filled","agentConduct":"derived","ethics":"missing","owners":"missing","relations":"filled","interaction.identity":"filled","interaction.properties":"not-applicable","interaction.recognition":"filled","interaction.capabilities":"filled","interaction.hazards":"filled","interaction.interfaces":"filled","interaction.context":"filled","sources":"missing"},"notes":{"agentConduct":"Affordances listed as what may be done; prohibitions collected from the text.","interaction.properties":"Plane XCT: no invented physical properties.","sources":"Written from model knowledge without web access; claims are unverified."},"score":0.781}}