{ "model": { "rank": 14239, "code": "thing-q747980", "model_id": "vr.tr.polytope", "name": "polytope", "purpose": "Let an agent explain polytopes and their definitions and structure, describe regular and special polytopes, support combinatorial and computational questions, and support learning in higher-dimensional geometry.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QLT", "industry": "", "version": "", "url": "/models/thing/q747980/", "tier": 2, "score": 36, "payload": { "layer": "wikidata", "aliases": [], "aliasCount": 0, "merged": 0, "knownIn": 36, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q747980", "url": "https://www.wikidata.org/wiki/Q747980", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-12T04:08:37Z", "providers": [ "Claude" ], "breadth": "written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify", "pass": 2, "wave": 2, "engine": "claude" }, "metaModel": { "id": "THING-Q747980", "registryId": "vr.tr.polytope", "name": "polytope", "version": "0.2.0-wave.2", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QLT" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q747980/", "model": { "registry_id": "vr.tr.polytope", "name": "polytope", "purpose": "Let an agent explain polytopes and their definitions and structure, describe regular and special polytopes, support combinatorial and computational questions, and support learning in higher-dimensional geometry.", "definition": "A geometric object with flat sides, generalising the polygon in two dimensions and the polyhedron in three dimensions to any number of dimensions, defined either as the convex hull of finitely many points or as a bounded intersection of half-spaces, with faces of every dimension, and including regular polytopes such as simplices, hypercubes and cross-polytopes as well as irregular and non-convex forms; polytopes are studied in geometry, combinatorics and optimisation.", "what_it_is_for": "Flat-sided geometric objects in any dimension.", "affordances": [ "explain definitions and structure", "describe regular polytopes", "support combinatorial questions", "support learning" ], "distinguishing_features": [ "Flat faces", "Vertex and half-space descriptions", "Face lattice", "Regular families" ], "appearance": "Not physical beyond three dimensions; polygons and polyhedra as low-dimensional cases.", "visual_identification": [ "Flat-sided object in n dimensions", "Polygons and polyhedra generalised", "Curved bodies are not polytopes; a manifold need not have flat sides" ], "physical_properties": [], "families_and_kinds": [ "polygons and polyhedra", "regular polytopes such as simplices and hypercubes", "convex polytopes", "non-convex and star polytopes", "polytopes in optimisation and combinatorics" ], "related_models": [ { "relation": "is a kind of", "target": "geometric shape", "why": "category" }, { "relation": "is related to", "target": "simplex", "why": "a family of regular polytopes" }, { "relation": "is related to", "target": "hypercube", "why": "a family of regular polytopes" }, { "relation": "is related to", "target": "convex set", "why": "convex polytopes as convex sets" } ], "identifiers": [], "standards_and_regulation": [ "Mathematical conventions", "No regulation" ], "failure_modes_and_hazards": [ "Confusing polytope and polyhedron terminology", "Face counting errors", "Assuming convexity" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "understand", "name": "Understand", "description": "What polytopes are.", "rationale": "Mathematics.", "layers": [ { "id": "definition", "name": "Definition", "description": "Definitions.", "findings": [ { "id": "definition-finding", "name": "Definition", "description": "Definitions.", "questions": [ { "text": "How are polytopes defined by vertices or half-spaces, and how do polygons and polyhedra fit as cases?", "kind": "definition" }, { "text": "Which dimension or family is meant?", "kind": "boundary" } ] } ] }, { "id": "structure", "name": "Structure", "description": "Faces and structure.", "findings": [ { "id": "structure-finding", "name": "Structure", "description": "Structure.", "questions": [ { "text": "What are faces, the face lattice and Euler-type relations for polytopes?", "kind": "definition" }, { "text": "Which entry fits the Euler characteristic?", "kind": "action" } ] } ] } ] }, { "id": "families", "name": "Families", "description": "Regular and special polytopes.", "rationale": "Reference.", "layers": [ { "id": "regular", "name": "Regular", "description": "Regular polytopes.", "findings": [ { "id": "regular-finding", "name": "Regular", "description": "Regular.", "questions": [ { "text": "What are the regular polytopes in each dimension, and why are there only a few beyond three dimensions?", "kind": "provenance" }, { "text": "Which entry fits a specific regular polytope?", "kind": "action" } ] } ] }, { "id": "other", "name": "Other", "description": "Other polytopes.", "findings": [ { "id": "other-finding", "name": "Other", "description": "Other.", "questions": [ { "text": "What are uniform, star and non-convex polytopes, and how are they classified?", "kind": "provenance" }, { "text": "Which entry fits polyhedra?", "kind": "action" } ] } ] } ] }, { "id": "apply", "name": "Apply", "description": "Applications.", "rationale": "Practice.", "layers": [ { "id": "optimisation", "name": "Optimisation", "description": "Optimisation.", "findings": [ { "id": "optimisation-finding", "name": "Optimisation", "description": "Optimisation.", "questions": [ { "text": "How do polytopes describe feasible regions in linear programming and combinatorial optimisation?", "kind": "provenance" }, { "text": "Which entry fits linear programming?", "kind": "action" } ] } ] }, { "id": "compute", "name": "Compute", "description": "Computation.", "findings": [ { "id": "compute-finding", "name": "Compute", "description": "Computation.", "questions": [ { "text": "How are polytopes computed, enumerated and visualised in software?", "kind": "action" }, { "text": "Which entry fits computational geometry?", "kind": "action" } ] } ] } ] }, { "id": "learn", "name": "Learn", "description": "History and teaching.", "rationale": "Education.", "layers": [ { "id": "history", "name": "History", "description": "History.", "findings": [ { "id": "history-finding", "name": "History", "description": "History.", "questions": [ { "text": "How did polytope theory develop from Euclid through Schlafli and Coxeter?", "kind": "provenance" }, { "text": "Which references are standard?", "kind": "provenance" } ] } ] }, { "id": "teach", "name": "Teach", "description": "Teaching.", "findings": [ { "id": "teach-finding", "name": "Teach", "description": "Teaching.", "questions": [ { "text": "How can polytopes be taught by analogy from polygons and polyhedra?", "kind": "action" }, { "text": "Which misconceptions arise?", "kind": "provenance" } ] } ] } ] } ] }, "openQuestions": [ "Should regular polytopes be a separate entry?", "How should polytope databases be linked?", "How should optimisation resources be linked?" ], "statistics": { "bundles": 4, "layers": 8, "findings": 8, "questions": 16 } }, "draft": { "generator": "vr.draft.v3", "status": "draft-generated", "researched": false, "archetype": "abstract concept", "method": "Written from the archetype playbook - what this kind of thing needs beyond identity and provenance - and from the structure that recurred across 6,333 models already researched by two engines. Applied to this entry by rule. No source was read for this thing and no claim here is researched. This entry carries no facets of its own, so they were inferred from its domain - a guess about a whole domain applied to one thing.", "facetsInferred": true, "nextPass": "A researcher replaces this draft with a sourced specification. Treat every sentence below as a proposal to argue with.", "purpose": "Give an agent a durable, checkable way to recognise a polytope, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent explain polytopes and their definitions and structure, describe regular and special polytopes, support combinatorial and computational questions, and support learning in higher-dimensional geometry.", "characteristics": { "substance": "abstract", "origin": "conceptual", "agency": "inert" }, "whatYouCanDoWithIt": [ "observed and measured" ], "distinguishingFeatures": [ "Described in 36 Wikipedia languages, which is a measure of how widely the thing is known, not of how important it is." ], "openQuestionsForResearch": [ "Which of the bundles below does a real task actually need, and which are ceremony?", "What does this thing have that the facets do not capture at all?", "Which neighbouring kind is most often confused with a polytope, and on what evidence are they told apart?" ], "whatItIsMadeOf": "an abstraction with no physical instance", "physicalCharacter": [ "Does nothing on its own; everything it does, something else did to it.", "These come from the domain this entry sits in rather than from the entry itself, so treat them as a first guess about the whole domain applied to one thing." ], "whatCanBeDoneWithIt": [ "observe it, measure it, record its state" ], "howItIsRecognised": [ "Nothing to see. What is recognised is an instance of it, and which instances count is exactly what is argued about." ], "relatedModels": [], "standing": "Described in 36 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is.", "structure": { "bundles": [ { "id": "identity-and-classification", "name": "Identity, naming and classification", "description": "How an agent tells one polytope from another, and a polytope from things that resemble it.", "rationale": "Recognition comes before every other claim. Without stable identity nothing else in the model can be trusted to be about the same thing twice.", "layers": [ { "id": "naming-and-identifiers", "name": "Names and identifiers", "description": "The names this thing goes by and the identifiers that survive translation and time.", "findings": [ { "id": "preferred-name-and-aliases", "name": "Preferred name, aliases and local names", "description": "Which name to use, which names mean the same thing, and which merely sound similar.", "questions": [ { "id": "preferred-name-and-aliases-q01", "text": "What identifies and describes the name of a polytope, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "preferred-name-and-aliases-q02", "text": "Who or what asserted this about the name of a polytope, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "preferred-name-and-aliases-q03", "text": "What may an agent decide or do once the name of a polytope is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "stable-identifiers", "name": "Stable identifiers and external keys", "description": "Identifiers that keep pointing at this kind of thing across systems and languages.", "questions": [ { "id": "stable-identifiers-q01", "text": "What identifies and describes an identifier for a polytope, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stable-identifiers-q02", "text": "Who or what asserted this about an identifier for a polytope, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stable-identifiers-q03", "text": "What may an agent decide or do once an identifier for a polytope is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "classification-and-granularity", "name": "Classification and granularity", "description": "Where a polytope sits among kinds, and how finely a task needs to cut it.", "findings": [ { "id": "kind-and-parents", "name": "Kind, parents and neighbouring kinds", "description": "The classes this thing belongs to and the ones it is next to.", "questions": [ { "id": "kind-and-parents-q01", "text": "What identifies and describes the kind of a polytope, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "kind-and-parents-q02", "text": "Who or what asserted this about the kind of a polytope, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "kind-and-parents-q03", "text": "What may an agent decide or do once the kind of a polytope is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "distinguishing-features", "name": "Distinguishing features", "description": "What separates a polytope from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a polytope, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "distinguishing-features-q02", "text": "Who or what asserted this about what distinguishes a polytope, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "distinguishing-features-q03", "text": "What may an agent decide or do once what distinguishes a polytope is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "state-and-lifecycle", "name": "State and lifecycle", "description": "The states a polytope passes through and the events that move it between them.", "rationale": "Most decisions about a thing depend on what state it is in now, which is a claim with a time on it, not a property.", "layers": [ { "id": "lifecycle-stages", "name": "Lifecycle stages", "description": "From coming into existence to ceasing to be one of these.", "findings": [ { "id": "stages-and-transitions", "name": "Stages and transitions", "description": "The stages worth naming and what moves a polytope between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a polytope, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stages-and-transitions-q02", "text": "Who or what asserted this about the lifecycle of a polytope, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stages-and-transitions-q03", "text": "What may an agent decide or do once the lifecycle of a polytope is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "observations-and-status", "name": "Observations and current status", "description": "What is observed about a polytope, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a polytope is recorded so that it can be superseded rather than overwritten.", "questions": [ { "id": "observation-record-q01", "text": "What identifies and describes an observation of a polytope, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "observation-record-q02", "text": "Who or what asserted this about an observation of a polytope, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "observation-record-q03", "text": "What may an agent decide or do once an observation of a polytope is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "definitions-in-contest", "name": "Definitions and who holds them", "description": "What polytope is taken to mean, and by whom.", "rationale": "When a field disagrees about a concept, the disagreement is the content. A model that picks one definition silently destroys the information.", "layers": [ { "id": "competing-definitions", "name": "Competing definitions", "description": "The main readings and the traditions behind them.", "findings": [ { "id": "definition-map", "name": "Definitions and their holders", "description": "Each definition with the school or body that holds it.", "questions": [ { "id": "definition-map-q01", "text": "Which definitions of polytope are in use, and which tradition or body holds each?", "kind": "definition" }, { "id": "definition-map-q02", "text": "What turns on the difference between them in practice?", "kind": "boundary" } ] } ] }, { "id": "operationalisation", "name": "Operationalisation", "description": "How it is measured or applied when it has to be.", "findings": [ { "id": "operational-record", "name": "Measures and proxies", "description": "Instruments and indicators used to stand in for it.", "questions": [ { "id": "operational-record-q01", "text": "How is polytope operationalised or measured in practice, and by what instrument?", "kind": "measurement" }, { "id": "operational-record-q02", "text": "What does that operationalisation leave out, and when does that matter?", "kind": "boundary" } ] } ] } ] }, { "id": "instances-and-use", "name": "Instances, use and consequence", "description": "What counts as an instance of polytope and what follows from calling something that.", "rationale": "Applying a concept is an act with consequences, so a model must say what the label licenses and what it does not.", "layers": [ { "id": "instances", "name": "What counts as an instance", "description": "Clear cases, borderline cases and non-cases.", "findings": [ { "id": "instance-tests", "name": "Tests for an instance", "description": "What would settle whether something falls under it.", "questions": [ { "id": "instance-tests-q01", "text": "What would settle whether something is an instance of polytope?", "kind": "boundary" }, { "id": "instance-tests-q02", "text": "Which borderline cases are argued about, and on what grounds?", "kind": "definition" } ] } ] }, { "id": "consequence", "name": "Consequence of application", "description": "Rights, duties or decisions that follow from the label.", "findings": [ { "id": "consequence-record", "name": "What the label licenses", "description": "What an agent may do once something is classified this way.", "questions": [ { "id": "consequence-record-q01", "text": "What follows practically once something is treated as polytope?", "kind": "action" }, { "id": "consequence-record-q02", "text": "What must an agent not infer from the label alone?", "kind": "action" } ] } ] } ] }, { "id": "provenance-and-evidence", "name": "Provenance, evidence and time", "description": "Where every claim about a polytope came from and when it held.", "rationale": "A claim without a source and a time cannot be superseded, only overwritten, and an agent that overwrites loses the ability to explain itself.", "layers": [ { "id": "source-and-authority", "name": "Source and authority", "description": "Who said it, on what evidence, and how strongly.", "findings": [ { "id": "claim-provenance", "name": "Claim provenance and confidence", "description": "The authority behind each claim about a polytope and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a polytope, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "claim-provenance-q02", "text": "Who or what asserted this about a claim about a polytope, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "claim-provenance-q03", "text": "What may an agent decide or do once a claim about a polytope is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "time-and-versions", "name": "Time, versions and supersession", "description": "When a claim was true, when it was learnt, and what replaced it.", "findings": [ { "id": "validity-and-supersession", "name": "Validity period and supersession", "description": "How an old claim about a polytope is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a polytope, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "validity-and-supersession-q02", "text": "Who or what asserted this about the validity of a claim about a polytope, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "validity-and-supersession-q03", "text": "What may an agent decide or do once the validity of a claim about a polytope is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] } ] }, "statistics": { "bundles": 5, "layers": 10, "findings": 12, "questions": 32 } } } }