Cauchy's Rigidity Theorem states that if two convex polyhedra have corresponding faces that are congruent to one another, then the two polyhedra must themselves be congruent, meaning a convex polyhedron cannot be flexed into a different shape while every one of its faces stays rigid. Named for Augustin-Louis Cauchy, it stands in contrast to non-convex polyhedra, some of which were later shown to flex continuously while preserving every face, a phenomenon Cauchy's own proof does not permit for the convex case.
Facts
StatementIf two 3-dimensional convex polytopes are combinatorially equivalent, with isomorphic face lattices, then they are congruent. 1 Classification
Statement Form Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Source Cauchy's theorem (geometry) (Wikipedia)
Named After
Derived from the theorem's own name (unambiguous possessive-token match to exactly one live mathematician entity, w-bfill-g5-0924 browse backfill)
Proved By
Sources
1. Cauchy's theorem (geometry) (Wikipedia)
Statement section
Let P and Q be combinatorially equivalent 3-dimensional convex polytopes; that is, they are convex polytopes with isomorphic face lattices.
History section
This version of the result was proved by Cauchy in 1813 based on earlier work by Lagrange.
In Branch: Geometry, Lead sentence
Cauchy's theorem is a theorem in geometry, named after Augustin Cauchy.
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