{ "model": { "rank": 7708, "code": "thing-q714886", "model_id": "vr.tr.regular-polygon", "name": "regular polygon", "purpose": "Let an agent define regular polygons, relay formulas for angles, area and circumradius, explain constructibility and star polygons, and identify specific regular n-gons.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QLT", "industry": "", "version": "", "url": "/models/thing/q714886/", "tier": 2, "score": 54, "payload": { "layer": "wikidata", "aliases": [ "zerogon", "improper regular polygon", "henicosagram", "regular tetraicosagon", "257-gon", "65537-gon", "constructible polygon", "regular star polygon", "regular convex polygon", "regular pentagon", "chiliagram", "eight-pointed star", "six-pointed star", "fourteen-pointed star", "eleven-pointed star", "twenty four-pointed star", "four-pointed star", "twenty one-pointed star", "nine-pointed star", "seven-pointed star" ], "aliasCount": 20, "merged": 20, "knownIn": 54, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q714886", "url": "https://www.wikidata.org/wiki/Q714886", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-13T00:11:44Z", "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": 3, "engine": "claude" }, "metaModel": { "id": "THING-Q714886", "registryId": "vr.tr.regular-polygon", "name": "regular polygon", "version": "0.2.0-wave.3", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QLT" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q714886/", "model": { "registry_id": "vr.tr.regular-polygon", "name": "regular polygon", "purpose": "Let an agent define regular polygons, relay formulas for angles, area and circumradius, explain constructibility and star polygons, and identify specific regular n-gons.", "definition": "A polygon that is both equiangular and equilateral, with all sides and all interior angles equal, including convex forms such as the equilateral triangle, square, regular pentagon and hexagon and star forms such as the pentagram and henicosagram, with n-gons for any n of 3 or more such as the 257-gon and 65537-gon, and degenerate cases such as the digon and the zerogon in some conventions; regular polygons are central to geometry, tilings, symmetry groups and constructibility with compass and straightedge.", "what_it_is_for": "Not applicable; a geometric object.", "affordances": [ "define and classify", "relay formulas", "explain constructibility", "identify specific n-gons" ], "distinguishing_features": [ "Equal sides and angles", "Dihedral symmetry", "Schlafli symbol notation", "Gauss-Wantzel constructibility" ], "appearance": "A closed figure with equal sides and equal angles, convex or star-shaped.", "visual_identification": [ "Equilateral and equiangular polygon", "Convex n-gons and star polygons", "Equilateral or equiangular alone is not regular; regular polytopes generalise to higher dimensions" ], "physical_properties": [ { "quantity": "interior angle", "typical_range": "(n-2)*180/n", "unit": "degrees", "note": "convex" }, { "quantity": "area", "typical_range": "(1/4)*n*s^2*cot(pi/n)", "unit": "formula", "note": "side s" } ], "families_and_kinds": [ "convex regular polygons from the triangle upward", "star polygons such as the pentagram", "constructible polygons including the 17-gon, 257-gon and 65537-gon", "degenerate cases such as the digon and zerogon", "improper regular polygons in some treatments" ], "related_models": [ { "relation": "is a kind of", "target": "equiangular polygon", "why": "in registry terms" }, { "relation": "is a kind of", "target": "equilateral polygon", "why": "in registry terms" }, { "relation": "is a kind of", "target": "regular polytope", "why": "in two dimensions" }, { "relation": "is described by", "target": "Schlafli symbol", "why": "the notation" } ], "identifiers": [ { "scheme": "Schlafli symbol", "value_or_pattern": "{n} or {n/m}", "note": "" } ], "standards_and_regulation": [ "No regulation; mathematical conventions" ], "failure_modes_and_hazards": [ "Confusing equilateral with regular", "Convention disputes on degenerate cases", "Errors in formulas" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "define", "name": "Define", "description": "Definition.", "rationale": "Mathematics.", "layers": [ { "id": "definition", "name": "Definition", "description": "Definition.", "findings": [ { "id": "definition-finding", "name": "Definition", "description": "Definition.", "questions": [ { "text": "What is a regular polygon, and what does the Schlafli symbol encode?", "kind": "definition" }, { "text": "Is the figure regular, or only equilateral or equiangular?", "kind": "boundary" } ] } ] }, { "id": "kinds", "name": "Kinds", "description": "Convex and star forms.", "findings": [ { "id": "kinds-finding", "name": "Kinds", "description": "Kinds.", "questions": [ { "text": "How do convex and star regular polygons differ, and what are the degenerate cases?", "kind": "definition" }, { "text": "Which entry fits the specific n-gon?", "kind": "action" } ] } ] } ] }, { "id": "compute", "name": "Compute", "description": "Formulas.", "rationale": "Mathematics.", "layers": [ { "id": "angles", "name": "Angles", "description": "Angles and radii.", "findings": [ { "id": "angles-finding", "name": "Angles", "description": "Angles.", "questions": [ { "text": "What are the formulas for interior angles, circumradius and inradius?", "kind": "measurement" }, { "text": "Which entry fits polygon geometry?", "kind": "action" } ] } ] }, { "id": "area", "name": "Area", "description": "Area and perimeter.", "findings": [ { "id": "area-finding", "name": "Area", "description": "Area.", "questions": [ { "text": "How are area and perimeter computed, and how do they approach a circle as n grows?", "kind": "measurement" }, { "text": "Which references are standard?", "kind": "provenance" } ] } ] } ] }, { "id": "construct", "name": "Construct", "description": "Constructibility and symmetry.", "rationale": "Mathematics.", "layers": [ { "id": "constructibility", "name": "Constructibility", "description": "Constructibility.", "findings": [ { "id": "constructibility-finding", "name": "Constructibility", "description": "Constructibility.", "questions": [ { "text": "Which regular polygons are constructible, and what does the Gauss-Wantzel theorem say?", "kind": "provenance" }, { "text": "Which entry fits constructible polygon?", "kind": "action" } ] } ] }, { "id": "symmetry", "name": "Symmetry", "description": "Symmetry and tilings.", "findings": [ { "id": "symmetry-finding", "name": "Symmetry", "description": "Symmetry.", "questions": [ { "text": "What symmetry groups do regular polygons have, and which tile the plane?", "kind": "provenance" }, { "text": "Which entry fits tessellation?", "kind": "action" } ] } ] } ] }, { "id": "context", "name": "Context", "description": "History and applications.", "rationale": "Context.", "layers": [ { "id": "history", "name": "History", "description": "History.", "findings": [ { "id": "history-finding", "name": "History", "description": "History.", "questions": [ { "text": "How were regular polygons studied from Euclid to Gauss?", "kind": "provenance" }, { "text": "Which sources are cited?", "kind": "provenance" } ] } ] }, { "id": "applications", "name": "Applications", "description": "Applications.", "findings": [ { "id": "applications-finding", "name": "Applications", "description": "Applications.", "questions": [ { "text": "Where do regular polygons appear in design, nature and engineering?", "kind": "provenance" }, { "text": "Which entry fits the specific application?", "kind": "action" } ] } ] } ] } ] }, "openQuestions": [ "Should star polygons be a separate entry?", "How should reference formulas be linked?", "The registry entry has merged aliases naming specific n-gons and edge cases; should they be split off?" ], "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 regular polygon, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent define regular polygons, relay formulas for angles, area and circumradius, explain constructibility and star polygons, and identify specific regular n-gons.", "characteristics": { "substance": "abstract", "origin": "conceptual", "agency": "inert" }, "whatYouCanDoWithIt": [ "observed and measured" ], "distinguishingFeatures": [ "Names folded into this entry, which a task may need to split apart again: zerogon, improper regular polygon, henicosagram, regular tetraicosagon, 257-gon, 65537-gon, constructible polygon, regular star polygon, regular convex polygon, regular pentagon, chiliagram, eight-pointed star.", "20 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 54 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 regular polygon, 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": [ { "relation": "covers", "note": "Finer kinds folded into this entry because telling them apart needs a task that asks for it. Each is a model waiting to be split out when one does.", "targets": [ "zerogon", "improper regular polygon", "henicosagram", "regular tetraicosagon", "257-gon", "65537-gon", "constructible polygon", "regular star polygon", "regular convex polygon", "regular pentagon", "chiliagram", "eight-pointed star" ] } ], "standing": "Described in 54 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 20 finer distinctions are held inside this entry as names rather than as separate models.", "structure": { "bundles": [ { "id": "identity-and-classification", "name": "Identity, naming and classification", "description": "How an agent tells one regular polygon from another, and a regular polygon 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 regular polygon, 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 regular polygon, 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 regular polygon 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 regular polygon, 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 regular polygon, 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 regular polygon 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 regular polygon 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 regular polygon, 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 regular polygon, 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 regular polygon 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 regular polygon from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a regular polygon, 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 regular polygon, 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 regular polygon 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 regular polygon 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 regular polygon between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a regular polygon, 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 regular polygon, 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 regular polygon 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 regular polygon, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a regular polygon 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 regular polygon, 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 regular polygon, 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 regular polygon 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 regular polygon 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 regular polygon 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 regular polygon 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 regular polygon 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 regular polygon?", "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 regular polygon?", "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 regular polygon 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 regular polygon and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a regular polygon, 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 regular polygon, 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 regular polygon 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 regular polygon is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a regular polygon, 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 regular polygon, 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 regular polygon 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 } } } }