{ "model": { "rank": 13087, "code": "thing-q740207", "model_id": "vr.tr.isometry", "name": "isometry", "purpose": "Let an agent explain isometries, relay definitions, classifications and uses from mathematics sources, describe the named forms, and distinguish isometries from similarities that scale distances, conformal maps that preserve angles, homeomorphisms and isometric projection in drawing.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QLT", "industry": "", "version": "", "url": "/models/thing/q740207/", "tier": 2, "score": 38, "payload": { "layer": "wikidata", "aliases": [ "Riemannian isometry", "geometric motion", "Euclidean plane isometry" ], "aliasCount": 3, "merged": 3, "knownIn": 38, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q740207", "url": "https://www.wikidata.org/wiki/Q740207", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-13T22:13:41Z", "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": 4, "engine": "claude" }, "metaModel": { "id": "THING-Q740207", "registryId": "vr.tr.isometry", "name": "isometry", "version": "0.2.0-wave.4", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QLT" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q740207/", "model": { "registry_id": "vr.tr.isometry", "name": "isometry", "purpose": "Let an agent explain isometries, relay definitions, classifications and uses from mathematics sources, describe the named forms, and distinguish isometries from similarities that scale distances, conformal maps that preserve angles, homeomorphisms and isometric projection in drawing.", "definition": "A distance-preserving map between metric spaces, so that the distance between any two points equals the distance between their images, including Euclidean plane isometries, which are translations, rotations, reflections and glide reflections, rigid or geometric motions in higher dimensions, and Riemannian isometries that preserve the metric tensor of manifolds; isometries of a space form its isometry group, central to symmetry and to Klein s Erlangen programme of 1872.", "what_it_is_for": "Describing symmetry and congruence.", "affordances": [ "explain the definition", "relay classifications", "describe named forms", "distinguish related maps" ], "distinguishing_features": [ "Distance preservation", "Group structure", "Congruence", "Metric spaces generally" ], "appearance": "Not a physical object; seen as shapes moved without distortion.", "visual_identification": [ "Distance-preserving map", "Riemannian isometry, geometric motion, Euclidean plane isometry", "Similarities scale distances; conformal maps preserve angles; homeomorphisms preserve topology; isometric projection is a drawing method" ], "physical_properties": [ { "quantity": "plane isometry types", "typical_range": "translation, rotation, reflection, glide reflection", "unit": "list", "note": "" }, { "quantity": "Erlangen programme", "typical_range": "1872", "unit": "year", "note": "Felix Klein" }, { "quantity": "Euclidean group", "typical_range": "isometries of Euclidean space", "unit": "note", "note": "" } ], "families_and_kinds": [ "Euclidean isometries", "orientation-preserving rigid motions", "reflections", "Riemannian isometries", "isometries of normed spaces" ], "related_models": [ { "relation": "is a kind of", "target": "geometric transformation", "why": "in registry terms" }, { "relation": "forms", "target": "isometry group", "why": "" }, { "relation": "is contrasted with", "target": "similarity (geometry)", "why": "" }, { "relation": "is contrasted with", "target": "conformal map", "why": "" } ], "identifiers": [], "standards_and_regulation": [ "No regulation" ], "failure_modes_and_hazards": [ "Confusing isometry with isometric projection", "Assuming all isometries preserve orientation", "Confusing isometric embeddings and surjective isometries" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "understand", "name": "Understand", "description": "What an isometry is.", "rationale": "Definition.", "layers": [ { "id": "definition", "name": "Definition", "description": "Definition.", "findings": [ { "id": "definition-finding", "name": "Definition", "description": "Definition.", "questions": [ { "text": "What is an isometry, and how does it differ from similarities, conformal maps, homeomorphisms and isometric projection?", "kind": "definition" }, { "text": "Is the question about plane geometry, metric spaces, manifolds or drawing?", "kind": "boundary" } ] } ] }, { "id": "forms", "name": "Forms", "description": "Named forms.", "findings": [ { "id": "forms-finding", "name": "Forms", "description": "Forms.", "questions": [ { "text": "What are Euclidean plane isometries, rigid motions and Riemannian isometries?", "kind": "definition" }, { "text": "Which entry fits the specific form?", "kind": "action" } ] } ] } ] }, { "id": "geometry", "name": "Geometry", "description": "Classification.", "rationale": "Science.", "layers": [ { "id": "plane", "name": "Plane", "description": "Plane isometries.", "findings": [ { "id": "plane-finding", "name": "Plane", "description": "Plane.", "questions": [ { "text": "Why is every plane isometry one of four types?", "kind": "provenance" }, { "text": "Which references are standard?", "kind": "provenance" } ] } ] }, { "id": "groups", "name": "Groups", "description": "Isometry groups.", "findings": [ { "id": "groups-finding", "name": "Groups", "description": "Groups.", "questions": [ { "text": "How do isometry groups describe symmetry, such as wallpaper groups?", "kind": "provenance" }, { "text": "Which sources are cited?", "kind": "provenance" } ] } ] } ] }, { "id": "advanced", "name": "Advanced", "description": "Manifolds and spaces.", "rationale": "Science.", "layers": [ { "id": "riemann", "name": "Riemannian", "description": "Riemannian isometries.", "findings": [ { "id": "riemann-finding", "name": "Riemannian", "description": "Riemannian.", "questions": [ { "text": "How are isometries defined on Riemannian manifolds, and what are Killing vector fields?", "kind": "provenance" }, { "text": "Which entry fits Killing vector field?", "kind": "action" } ] } ] }, { "id": "mazur", "name": "Normed spaces", "description": "Mazur-Ulam theorem.", "findings": [ { "id": "mazur-finding", "name": "Normed spaces", "description": "Normed spaces.", "questions": [ { "text": "What does the Mazur-Ulam theorem say about isometries of normed spaces?", "kind": "provenance" }, { "text": "Which entry fits Mazur-Ulam theorem?", "kind": "action" } ] } ] } ] }, { "id": "context", "name": "Context", "description": "History and uses.", "rationale": "Context.", "layers": [ { "id": "klein", "name": "Klein", "description": "Erlangen programme.", "findings": [ { "id": "klein-finding", "name": "Klein", "description": "Klein.", "questions": [ { "text": "How did Klein classify geometries by their transformation groups?", "kind": "provenance" }, { "text": "Which entry fits Erlangen program?", "kind": "action" } ] } ] }, { "id": "applications", "name": "Applications", "description": "Applications.", "findings": [ { "id": "applications-finding", "name": "Applications", "description": "Applications.", "questions": [ { "text": "How are rigid motions used in robotics and computer graphics?", "kind": "provenance" }, { "text": "Which entry fits rigid body transformation?", "kind": "action" } ] } ] } ] } ] }, "openQuestions": [ "Should Euclidean plane isometry be a separate entry?", "How should mathematics references be linked?", "Should isometric embedding be disambiguated?" ], "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 isometry, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent explain isometries, relay definitions, classifications and uses from mathematics sources, describe the named forms, and distinguish isometries from similarities that scale distances, conformal maps that preserve angles, homeomorphisms and isometric projection in drawing.", "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: Riemannian isometry, geometric motion, Euclidean plane isometry.", "3 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 38 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 isometry, 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": [ "Riemannian isometry", "geometric motion", "Euclidean plane isometry" ] } ], "standing": "Described in 38 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 3 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 isometry from another, and a isometry 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 isometry, 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 isometry, 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 isometry 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 isometry, 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 isometry, 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 isometry 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 isometry 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 isometry, 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 isometry, 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 isometry 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 isometry from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a isometry, 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 isometry, 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 isometry 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 isometry 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 isometry between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a isometry, 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 isometry, 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 isometry 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 isometry, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a isometry 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 isometry, 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 isometry, 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 isometry 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 isometry 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 isometry 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 isometry 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 isometry 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 isometry?", "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 isometry?", "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 isometry 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 isometry and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a isometry, 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 isometry, 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 isometry 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 isometry is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a isometry, 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 isometry, 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 isometry 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 } } } }