{ "model": { "rank": 18095, "code": "thing-q836270", "model_id": "vr.tr.polygonal-number", "name": "polygonal number", "purpose": "Let an agent handle polygonal numbers by polygon, index, formula and properties.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QTY", "industry": "", "version": "", "url": "/models/thing/q836270/", "tier": 2, "score": 30, "payload": { "layer": "wikidata", "aliases": [ "dodecagonal number", "hexagonal number", "Platonic number", "icosipentagonal number", "tridecagonal number", "hendecagonal number", "hexadecagonal number", "icosagonal number", "icosidigonal number", "icosihenagonal number", "icositetragonal number", "icositrigonal number", "myriagonal number", "octadecagonal number", "heptadecagonal number", "enneadecagonal number", "pentadecagonal number", "tetradecagonal number", "octagonal number", "nonagonal number", "decagonal number", "pentagonal number", "heptagonal number" ], "aliasCount": 23, "merged": 23, "knownIn": 30, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q836270", "url": "https://www.wikidata.org/wiki/Q836270", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-11T06:19:08Z", "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": 1, "engine": "claude" }, "metaModel": { "id": "THING-Q836270", "registryId": "vr.tr.polygonal-number", "name": "polygonal number", "version": "0.2.0-wave.1", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QTY" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q836270/", "model": { "registry_id": "vr.tr.polygonal-number", "name": "polygonal number", "purpose": "Let an agent handle polygonal numbers by polygon, index, formula and properties.", "definition": "A number that can be represented by dots arranged in a regular polygon, such as triangular, square, pentagonal and hexagonal numbers, given by a formula for each polygon.", "what_it_is_for": "Recreational and classical number theory, and teaching patterns.", "affordances": [ "compute them by formula", "test whether a number is polygonal", "draw dot patterns", "study relations between types" ], "distinguishing_features": [ "Figurate numbers on polygons", "Formula depends on number of sides", "Triangular numbers are the simplest", "Related to centred polygonal numbers" ], "appearance": "Dot arrangements forming polygons; sequences of numbers.", "visual_identification": [ "Dot patterns", "Names such as hexagonal or dodecagonal number", "OEIS sequences" ], "physical_properties": [], "families_and_kinds": [ "triangular numbers", "square numbers", "pentagonal and hexagonal numbers", "higher polygonal numbers such as dodecagonal", "centred polygonal numbers" ], "related_models": [ { "relation": "is a kind of", "target": "figurate number", "why": "category" }, { "relation": "is a kind of", "target": "integer", "why": "values" }, { "relation": "forms", "target": "sequence", "why": "series of values" }, { "relation": "is studied by", "target": "mathematics", "why": "number theory" } ], "identifiers": [ { "scheme": "OEIS A-number", "value_or_pattern": "per polygon type", "note": "sequences, such as A000217 triangular" } ], "standards_and_regulation": [], "failure_modes_and_hazards": [ "Off-by-one indexing", "Mixing ordinary and centred polygonal numbers" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "definition", "name": "Definition", "description": "Which polygonal number.", "rationale": "Polygon and index define it.", "layers": [ { "id": "type", "name": "Type", "description": "Number of sides.", "findings": [ { "id": "type-finding", "name": "Type", "description": "Polygon type.", "questions": [ { "text": "For which polygon (number of sides)?", "kind": "definition" }, { "text": "Which index?", "kind": "definition" } ] } ] }, { "id": "formula", "name": "Formula", "description": "General term.", "findings": [ { "id": "formula-finding", "name": "Formula", "description": "Formula.", "questions": [ { "text": "What formula gives the nth number?", "kind": "definition" }, { "text": "Does indexing start at 0 or 1?", "kind": "boundary" } ] } ] } ] }, { "id": "properties", "name": "Properties", "description": "Patterns.", "rationale": "Patterns relate types.", "layers": [ { "id": "relations", "name": "Relations", "description": "Between types.", "findings": [ { "id": "relations-finding", "name": "Relations", "description": "Relations.", "questions": [ { "text": "How do polygonal numbers relate to each other?", "kind": "definition" }, { "text": "Which theorems apply?", "kind": "provenance" } ] } ] }, { "id": "test", "name": "Test", "description": "Is it polygonal.", "findings": [ { "id": "test-finding", "name": "Test", "description": "Membership test.", "questions": [ { "text": "Is a given number polygonal of this type?", "kind": "measurement" }, { "text": "How is that tested?", "kind": "definition" } ] } ] } ] }, { "id": "representation", "name": "Representation", "description": "Dots and sequences.", "rationale": "Visuals help learning.", "layers": [ { "id": "dots", "name": "Dot patterns", "description": "Figures.", "findings": [ { "id": "dots-finding", "name": "Dots", "description": "Dot representation.", "questions": [ { "text": "How is it drawn as dots?", "kind": "action" }, { "text": "Is the figure regular?", "kind": "boundary" } ] } ] }, { "id": "sequence", "name": "Sequence", "description": "OEIS.", "findings": [ { "id": "sequence-finding", "name": "Sequence", "description": "Sequence reference.", "questions": [ { "text": "Which OEIS sequence lists it?", "kind": "provenance" }, { "text": "What are the first terms?", "kind": "measurement" } ] } ] } ] }, { "id": "history", "name": "History", "description": "Classical study.", "rationale": "Polygonal numbers are ancient.", "layers": [ { "id": "origins", "name": "Origins", "description": "Greek mathematics.", "findings": [ { "id": "origins-finding", "name": "Origins", "description": "History.", "questions": [ { "text": "Who studied them historically?", "kind": "provenance" }, { "text": "Which sources?", "kind": "provenance" } ] } ] }, { "id": "teaching", "name": "Teaching", "description": "Classroom use.", "findings": [ { "id": "teaching-finding", "name": "Teaching", "description": "Teaching use.", "questions": [ { "text": "How are they used in teaching?", "kind": "definition" }, { "text": "What activities work well?", "kind": "action" } ] } ] } ] } ] }, "openQuestions": [ "Should each polygonal type be a separate entry?", "How should indexing conventions be stated?", "The registry entry includes Platonic number as an alias; is that a separate concept?" ], "statistics": { "bundles": 4, "layers": 8, "findings": 8, "questions": 16 } }, "draft": { "generator": "vr.draft.v3", "status": "draft-generated", "researched": false, "archetype": "unit, quantity or measure", "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 polygonal number, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent handle polygonal numbers by polygon, index, formula and properties.", "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: dodecagonal number, hexagonal number, Platonic number, icosipentagonal number, tridecagonal number, hendecagonal number, hexadecagonal number, icosagonal number, icosidigonal number, icosihenagonal number, icositetragonal number, icositrigonal number.", "23 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 30 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 polygonal number, 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: a unit is a convention. What is seen is an instrument reading in it." ], "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": [ "dodecagonal number", "hexagonal number", "Platonic number", "icosipentagonal number", "tridecagonal number", "hendecagonal number", "hexadecagonal number", "icosagonal number", "icosidigonal number", "icosihenagonal number", "icositetragonal number", "icositrigonal number" ] } ], "standing": "Described in 30 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 23 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 polygonal number from another, and a polygonal number 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 polygonal number, 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 polygonal number, 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 polygonal number 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 polygonal number, 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 polygonal number, 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 polygonal number 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 polygonal number 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 polygonal number, 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 polygonal number, 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 polygonal number 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 polygonal number from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a polygonal number, 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 polygonal number, 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 polygonal number 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 polygonal number 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 polygonal number between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a polygonal number, 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 polygonal number, 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 polygonal number 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 polygonal number, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a polygonal number 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 polygonal number, 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 polygonal number, 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 polygonal number is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "definition-and-realisation", "name": "Definition, dimension and realisation", "description": "What polygonal number is defined as and how it is realised in practice.", "rationale": "A unit is not the quantity it measures, and its definition has a history that older data still depends on.", "layers": [ { "id": "definition", "name": "Definition and dimension", "description": "The defining relation and the quantity it measures.", "findings": [ { "id": "definition-record", "name": "Definition, body and date", "description": "What defines it now, and what defined it before.", "questions": [ { "id": "definition-record-q01", "text": "How is polygonal number defined, by which body, and since when?", "kind": "definition" }, { "id": "definition-record-q02", "text": "What was the previous definition, and what data still depends on it?", "kind": "provenance" } ] } ] }, { "id": "conversion", "name": "Conversion and compatibility", "description": "Relations to other units, exact or approximate.", "findings": [ { "id": "conversion-record", "name": "Conversions and their exactness", "description": "Factors, and whether each is exact by definition.", "questions": [ { "id": "conversion-record-q01", "text": "What conversions relate polygonal number to other units, and which are exact by definition?", "kind": "measurement" }, { "id": "conversion-record-q02", "text": "Where does rounding a conversion change a decision, and how should that be recorded?", "kind": "action" } ] } ] } ] }, { "id": "provenance-and-evidence", "name": "Provenance, evidence and time", "description": "Where every claim about a polygonal number 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 polygonal number and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a polygonal number, 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 polygonal number, 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 polygonal number 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 polygonal number is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a polygonal number, 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 polygonal number, 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 polygonal number is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] } ] }, "statistics": { "bundles": 4, "layers": 8, "findings": 10, "questions": 28 } } } }