{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.figurate-number","code":"thing-q1130510","url":"https://ver.cy/models/thing/q1130510/","name":"figurate number","alternateNames":[],"kind":"thing","status":"research-draft","version":"0.2.0-wave.1","language":"en","classifiers":{"family":"Thing Registry","category":"","entryKind":"","plane":"","domain":["XCT.QTY"],"industry":[],"navPath":"","tags":[],"facets":{}},"whatItIs":"A number that can be represented by a regular geometric arrangement of equally spaced points, such as triangular, square, pentagonal, centred and polyhedral numbers.","purpose":"Let an agent handle figurate numbers by shape, dimension, formula and sequence identifiers.","scope":{"in":[],"out":[],"boundaries":[]},"distinguishingFeatures":["Numbers from geometric arrangements","Polygonal, centred and polyhedral families","Closed formulas","Catalogued in OEIS"],"structure":{"bundles":[{"id":"family","name":"Family","description":"Which figurate number.","layers":[{"id":"shape","name":"Shape","description":"Polygon or polyhedron.","findings":[{"id":"shape-finding","name":"Shape","description":"Family.","questions":[{"text":"Which shape and dimension define the family?","kind":"definition"},{"text":"Is it centred?","kind":"definition"}]}]},{"id":"sequence","name":"Sequence","description":"OEIS.","findings":[{"id":"sequence-finding","name":"Sequence","description":"Sequence.","questions":[{"text":"What are the first terms?","kind":"measurement"},{"text":"What is its OEIS number?","kind":"provenance"}]}]}]},{"id":"formula","name":"Formula","description":"nth term.","layers":[{"id":"term","name":"Term","description":"Closed form.","findings":[{"id":"term-finding","name":"Term","description":"Formula.","questions":[{"text":"What is the formula for the nth term?","kind":"definition"},{"text":"Which index starts the sequence?","kind":"boundary"}]}]},{"id":"test","name":"Test","description":"Membership.","findings":[{"id":"test-finding","name":"Test","description":"Membership test.","questions":[{"text":"Is this number in the family?","kind":"boundary"},{"text":"How is that tested?","kind":"action"}]}]}]},{"id":"relations","name":"Relations","description":"Identities.","layers":[{"id":"identities","name":"Identities","description":"Sums.","findings":[{"id":"identities-finding","name":"Identities","description":"Identities.","questions":[{"text":"Which identities link families?","kind":"definition"},{"text":"For example sums of triangular numbers?","kind":"definition"}]}]},{"id":"theorems","name":"Theorems","description":"Results.","findings":[{"id":"theorems-finding","name":"Theorems","description":"Theorems.","questions":[{"text":"Which theorems apply, such as Fermat polygonal number theorem?","kind":"provenance"},{"text":"Who proved them?","kind":"provenance"}]}]}]},{"id":"visual","name":"Visual","description":"Dot patterns.","layers":[{"id":"drawing","name":"Drawing","description":"Patterns.","findings":[{"id":"drawing-finding","name":"Drawing","description":"Dot patterns.","questions":[{"text":"How can the pattern be drawn?","kind":"action"},{"text":"How does it grow?","kind":"definition"}]}]},{"id":"teaching","name":"Teaching","description":"Learning.","findings":[{"id":"teaching-finding","name":"Teaching","description":"Teaching.","questions":[{"text":"How can figurate numbers be taught?","kind":"action"},{"text":"Which activities help?","kind":"action"}]}]}]}]},"agentConduct":{"may":["compute the nth term","test whether a number is figurate","find sequences in OEIS","visualise with dot patterns"],"mustNot":[],"requiresHuman":[]},"ethics":{"considerations":[],"affectedParties":[]},"owners":{"steward":"","roles":[],"masterSystems":[]},"relations":[{"target":"positive integer","type":"related","note":"category"},{"target":"polygon and polyhedron","type":"related","note":"shapes"},{"target":"OEIS","type":"related","note":"database"},{"target":"number theory","type":"related","note":"field"}],"interaction":{"identity":{"applicability":"required","items":["OEIS A-number: A000217 for triangular numbers sequence ids"]},"properties":{"applicability":"not-applicable","items":[]},"recognition":{"applicability":"optional","items":["Dot patterns","Names such as triangular or tetrahedral number","Pronic numbers are products of consecutive integers","Dots arranged in triangles, squares, polygons or polyhedra."]},"capabilities":{"applicability":"required","items":["compute the nth term","test whether a number is figurate","find sequences in OEIS","visualise with dot patterns"]},"hazards":{"applicability":"required","items":["Off-by-one indexing conventions","Confusing centred and plain forms"]},"interfaces":{"applicability":"required","items":[]},"context":{"applicability":"required","items":["Number theory, combinatorics and recreational mathematics.","polygonal numbers","centred polygonal numbers","polyhedral numbers such as tetrahedral","polytopic numbers such as pentatope","pronic numbers"]}},"sources":[],"openQuestions":["Should each family be a separate entry?","How should OEIS ids be linked?","How should indexing conventions be recorded?","How is that tested?","How can the pattern be drawn?","How can figurate numbers be taught?"],"resources":{"spec":"/models/things/publications/thing-q1130510/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q1130510/spec.json"],"providers":["Claude"],"researchStatus":"unreviewed","generatedAt":"2026-09-11T07:21:51Z","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":"missing","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":0.719}}