{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.pappus-chain","code":"thing-q511431","url":"https://ver.cy/models/thing/q511431/","name":"Pappus chain","alternateNames":[],"kind":"thing","status":"published","version":"0.1.0-research.1","language":"en","classifiers":{"family":"Thing Registry","category":"Cross-cutting context","entryKind":"thing","plane":"XCT","domain":["XCT.QLT"],"industry":[],"navPath":"","tags":[],"facets":{}},"whatItIs":"A Pappus chain is a sequence of circles in an arbelos, each tangent to the outer semicircle and one fixed inner semicircle, with successive circles mutually tangent.","purpose":"Enable an agent to recognise, construct and validate a Pappus chain while keeping its supporting geometry, indexing convention and mathematical justification explicit.","scope":{"in":["The supporting arbelos, selected pair of boundary circles and chain branch","The seed circle, successor rule and indexing convention","Circle centres, radii, tangencies and containment constraints","The Pappus height relation and its applicability conditions","Construction methods, finite representations and validation evidence"],"out":["The arbelos and its properties independent of a chain","General circle packing and Apollonian gasket constructions","Steiner chains and their closure conditions","Pappus's hexagon theorem and centroid theorems","General inversion theory beyond its application to this configuration"],"boundaries":[]},"distinguishingFeatures":["In the classical arbelos construction, the first circle touches all three boundary semicircles; successive circles touch their predecessor and the selected outer and inner boundaries. [Boas, Reflections on the Arbelos](https://haroldpboas.gitlab.io/preprints/arbelos.pdf)","Recognition requires an ordered successor sequence with a fixed supporting pair; an arbitrary collection of tangent circles is insufficient.","The supporting circles meet at a tangency point toward which the classical branch progresses; a closed ring of circles requires a different classification.","With the arbelos incircle indexed as 1, the centre height of circle n equals n times its diameter. This checks the convention and geometry together. [Boas, Reflections on the Arbelos](https://haroldpboas.gitlab.io/preprints/arbelos.pdf)"],"structure":{"bundles":[{"id":"supporting-geometry","name":"Supporting geometry","description":"Identify the arbelos and the persistent boundary pair that define the chain.","layers":[{"id":"arbelos-frame","name":"Arbelos frame","description":"Establish the baseline, diameter subdivision and chosen half-plane.","findings":[{"id":"valid-arbelos","name":"Valid arbelos","description":"Record the outer diameter endpoints and their interior subdivision point so that the three semicircles can be reconstructed.","questions":[{"text":"Which collinear points define the outer diameter and its two positive subdiameters?","id":"valid-arbelos-q01","kind":"definition"},{"text":"Which half-plane contains the semicircles, and which coordinate frame and length scale represent them?","id":"valid-arbelos-q02","kind":"measurement"}]}]},{"id":"persistent-boundaries","name":"Persistent boundaries","description":"Distinguish the two boundaries retained throughout the chosen branch from the third seed boundary.","findings":[{"id":"selected-support-pair","name":"Selected support pair","description":"Identify the outer circle, selected inner circle and their shared tangency point.","questions":[{"text":"Which inner semicircle remains tangent to every member after the incircle?","id":"selected-support-pair-q01","kind":"definition"},{"text":"Do the supporting circles satisfy internal tangency with strictly positive, unequal radii?","id":"selected-support-pair-q02","kind":"boundary"}]}]}]},{"id":"sequence-and-membership","name":"Sequence and membership","description":"Specify how the seed and successive members form one ordered branch.","layers":[{"id":"seed-and-indexing","name":"Seed and indexing","description":"Make the starting member and correspondence between labels and geometry explicit.","findings":[{"id":"index-contract","name":"Index contract","description":"Record whether the incircle is labelled 1 and whether a boundary circle is included as an auxiliary member labelled 0.","questions":[{"text":"Which geometric circle receives the first index, and is an auxiliary boundary circle counted as a member?","id":"index-contract-q01","kind":"definition"},{"text":"How do these labels translate to the indexing used by the chosen formula or reference?","id":"index-contract-q02","kind":"provenance"}]}]},{"id":"successor-selection","name":"Successor selection","description":"Define advancement along the branch without reversal or duplication.","findings":[{"id":"forward-tangent-member","name":"Forward tangent member","description":"Select the admissible circle tangent to the supporting pair and current member, excluding the predecessor and respecting the chosen branch.","questions":[{"text":"Which geometric condition selects the forward solution among the tangent-circle candidates?","id":"forward-tangent-member-q01","kind":"action"},{"text":"Does the proposed member lie in the intended region without overlapping earlier members?","id":"forward-tangent-member-q02","kind":"boundary"}]}]}]},{"id":"metric-invariants","name":"Metric invariants","description":"Connect measured or symbolic member geometry to the conditions defining the chain.","layers":[{"id":"tangency-equations","name":"Tangency equations","description":"Check internal boundary contact and external member contact separately.","findings":[{"id":"contact-residuals","name":"Contact residuals","description":"For outer centre O and radius R, inner centre I and radius a, require |P_n-O|=R-r_n, |P_n-I|=a+r_n and |P_n-P_(n-1)|=r_n+r_(n-1) where applicable.","questions":[{"text":"Do the recorded centres and radii satisfy all applicable contact equations with the correct tangency signs?","id":"contact-residuals-q01","kind":"measurement"},{"text":"Are these equalities exact, or what scale-relative residual tolerance is accepted?","id":"contact-residuals-q02","kind":"measurement"}]}]},{"id":"height-theorem","name":"Height theorem","description":"Apply the classical height relation only after checking its baseline and indexing assumptions.","findings":[{"id":"height-index-consistency","name":"Height-index consistency","description":"Record the perpendicular centre height h_n and test h_n=2nr_n under incircle-first indexing.","questions":[{"text":"Is height measured perpendicular to the arbelos diameter line, with the incircle indexed as 1?","id":"height-index-consistency-q01","kind":"boundary"},{"text":"Does h_n/(2r_n) equal the declared index exactly or within the stated numerical tolerance?","id":"height-index-consistency-q02","kind":"measurement"}]}]}]},{"id":"construction-and-proof","name":"Construction and proof","description":"Record reproducible ways to generate members and justify their properties.","layers":[{"id":"inversion-construction","name":"Inversion construction","description":"Use inversion at the supporting circles' common tangency point to obtain parallel boundaries and an equal-radius circle sequence. [Boas, Reflections on the Arbelos](https://haroldpboas.gitlab.io/preprints/arbelos.pdf)","findings":[{"id":"invertible-construction-record","name":"Invertible construction record","description":"Retain the inversion centre, inversion radius, transformed seed and direction needed to reconstruct the original branch.","questions":[{"text":"Which inversion maps the supporting pair to parallel lines, and where does the seed map?","id":"invertible-construction-record-q01","kind":"action"},{"text":"Does inverse transformation recover the intended branch and its declared indices?","id":"invertible-construction-record-q02","kind":"boundary"}]}]},{"id":"formula-justification","name":"Formula justification","description":"Associate coordinate or radius formulas with their parameter meanings and supporting derivations.","findings":[{"id":"parameter-compatible-formulas","name":"Parameter-compatible formulas","description":"Require explicit correspondence between formula symbols, arbelos dimensions, branch and member labels.","questions":[{"text":"Which derivation or source supports the formula, and what normalization and indexing does it assume?","id":"parameter-compatible-formulas-q01","kind":"provenance"},{"text":"Does substitution verify the seed and the general tangency conditions rather than only a few plotted members?","id":"parameter-compatible-formulas-q02","kind":"action"}]}]}]},{"id":"extent-and-boundaries","name":"Extent and boundaries","description":"Distinguish an infinite chain from its finite representation and delimit admissible extensions.","layers":[{"id":"tail-representation","name":"Tail representation","description":"Record how a finite construction represents continued approach to the supporting tangency point.","findings":[{"id":"finite-prefix-contract","name":"Finite prefix contract","description":"Identify the final represented index and the rule governing subsequent members.","questions":[{"text":"What is the largest represented index, and what construction or formula specifies the omitted tail?","id":"finite-prefix-contract-q01","kind":"definition"},{"text":"Is generation stopped by a geometric error bound, a display threshold or numerical loss of distinctness?","id":"finite-prefix-contract-q02","kind":"action"}]}]},{"id":"extension-admissibility","name":"Extension admissibility","description":"Keep changes of convention distinct from changes to the mathematical configuration.","findings":[{"id":"classical-versus-extended-instance","name":"Classical versus extended instance","description":"Classify reflected branches, auxiliary members and broader circle-chain constructions through explicit definitions rather than their shared name.","questions":[{"text":"Is the candidate a classical arbelos branch, a reflection of one or an extension requiring a separate definition?","id":"classical-versus-extended-instance-q01","kind":"boundary"},{"text":"Which recognition tests and theorem assumptions remain valid after the proposed extension?","id":"classical-versus-extended-instance-q02","kind":"boundary"}]}]}]},{"id":"evidence-and-external-alignment","name":"Evidence and external alignment","description":"What the world already says about this thing, gathered so the model can be checked against it.","layers":[{"id":"reported-evidence","name":"Reported evidence","description":"Findings from the breadth pass, kept separate from the structural claims.","findings":[{"id":"evidence-confidence-notes","name":"Check these first","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these check these first hold for the sense of Pappus chain this model covers, and on what evidence?","id":"evidence-confidence-notes-q01","kind":"provenance"}]},{"id":"evidence-real-world-use","name":"Real-world use","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these real-world use hold for the sense of Pappus chain this model covers, and on what evidence?","id":"evidence-real-world-use-q01","kind":"provenance"}]},{"id":"evidence-measurements","name":"Typical measurements","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these typical measurements hold for the sense of Pappus chain this model covers, and on what evidence?","id":"evidence-measurements-q01","kind":"provenance"}]},{"id":"evidence-failure-modes-and-hazards","name":"Failure modes and hazards","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these failure modes and hazards hold for the sense of Pappus chain this model covers, and on what evidence?","id":"evidence-failure-modes-and-hazards-q01","kind":"provenance"}]},{"id":"evidence-neighbours","name":"Neighbouring kinds and how to tell them apart","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these neighbouring kinds and how to tell them apart hold for the sense of Pappus chain this model covers, and on what evidence?","id":"evidence-neighbours-q01","kind":"provenance"}]}]}]}]},"agentConduct":{"may":["Explain the Pappus chain and its classical theorem on circle heights.","Draw or compute chains from stated parameters and check tangency numerically."],"mustNot":["Present an unproved claim about the chain as a theorem.","Attribute results to the wrong mathematician or period without evidence.","Copy figures or proofs from published works without credit."],"requiresHuman":["Publishing a new result about the chain under a person's name."]},"ethics":{"considerations":["Accurate attribution respects the history of mathematics and its scholars."],"affectedParties":["Students","Mathematicians","Historians of mathematics"]},"owners":{"steward":"Nobody owns it; it is a concept of geometry.","roles":[],"masterSystems":[]},"relations":[{"target":"Q815741","type":"parent","note":"registry parent class"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.pappus-chain","Wikidata: Q511431 (https://www.wikidata.org/wiki/Q511431)"]},"properties":{"applicability":"not-applicable","items":["Construction convention: classical arbelos branch | explicitly specified extension","Supporting circle pair: outer circle and selected internally tangent inner circle","Arbelos proportion: dimensionless ratio of selected inner radius to outer radius, strictly between 0 and 1","Index origin and direction: declared seed label, successor direction and permitted integer indices","Member geometry: centre coordinates and positive radius in a declared Euclidean frame and length unit","Height-to-diameter ratio: dimensionless perpendicular centre height divided by circle diameter","Representation extent: infinite construction rule | finite prefix with rule | finite candidate sample","Geometric verification: symbolically verified | numerically checked with tolerance | unverified | inconsistent"]},"recognition":{"applicability":"optional","items":["A row of ever-smaller circles inside an arbelos, each touching its neighbours and two of the semicircles."]},"capabilities":{"applicability":"required","items":["Construct a finite prefix from a valid arbelos and declared branch.","Extend the sequence by selecting the forward tangent-circle solution.","Check supporting tangencies, consecutive tangencies and containment.","Translate between indexing conventions before comparing formulas.","Use inversion to derive or verify the chain construction.","Render a finite approximation with its omitted tail and numerical tolerance identified."]},"hazards":{"applicability":"required","items":["Floating-point drawing errors accumulate along the chain and give false tangency."]},"interfaces":{"applicability":"not-applicable","items":[]},"context":{"applicability":"required","items":["A Pappus chain is encountered in Euclidean geometry when studying tangent circles inside an arbelos.","It is used in geometric constructions, proofs and teaching examples involving circle tangency, inversion and relationships between circle sizes and positions.","Computational drawings use explicit boundary and indexing conventions to check that the intended chain has been constructed."]}},"sources":[{"title":"Wikidata item Q511431: Pappus chain","url":"https://www.wikidata.org/wiki/Q511431","note":"identity and sense of the item"},{"title":"Wikipedia: Pappus chain","url":"https://en.wikipedia.org/wiki/Pappus_chain","note":"general description of the item"}],"openQuestions":["Should this registry entry cover only classical arbelos branches or also broader configurations called Pappus chains in specialist literature?","Should a boundary circle indexed as 0 be represented as a chain member or solely as construction support?","Which canonical indexing and branch convention should Vercy use when exchanging formulas between sources?","Which source-backed formulas should be adopted for member coordinates and rigorous finite-tail error bounds?","Should similarity-related configurations be treated as the same abstract instance or as distinct geometric realizations?"],"resources":{"spec":"/models/things/publications/thing-q511431/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q511431/spec.json"],"providers":["Codex"],"researchStatus":"unreviewed","generatedAt":"2026-09-12T11:52:55Z","builder":"tools/build_cards.py@1.0.0"},"completeness":{"sections":{"classifiers":"filled","whatItIs":"filled","purpose":"filled","distinguishingFeatures":"filled","structure":"filled","agentConduct":"filled","ethics":"filled","owners":"filled","relations":"filled","interaction.identity":"filled","interaction.properties":"not-applicable","interaction.recognition":"filled","interaction.capabilities":"filled","interaction.hazards":"filled","interaction.interfaces":"not-applicable","interaction.context":"filled","sources":"filled"},"notes":{"_gate":"Published through the thing publication gate 1.0 on 2026-10-07T10:10:31+00:00: completeness 1.00; sense: Wikidata description; 2 live sources; review by codex: pass","interaction.properties":"Plane XCT: no invented physical properties.","interaction.interfaces":"No dedicated standard, law or official classification applies to the Pappus chain as a mathematical construction."},"score":1.0}}