Theorem

Beckman-Quarles Theorem

Geometry

The Beckman-Quarles Theorem states that any transformation of the Euclidean plane, or of any higher-dimensional Euclidean space, that preserves the distance of exactly one unit between points must in fact preserve all distances, and so must be a full rigid isometry, even though no continuity assumption is made anywhere in its statement or proof. Named for Frank Beckman and Donald Quarles, it is a striking rigidity result showing that a single fixed distance is already enough information to force an unrestricted map to be an isometry.

Facts

Statement

If a transformation of the Euclidean plane or a higher-dimensional Euclidean space preserves unit distances, then it preserves all Euclidean distances. 1

Proof Year

1953 1

Classification

Statement Form

Characterization Theorem 1

Connections

Has Statement Form

Characterization

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

Geometry

Source Beckman Quarles theorem (Wikipedia)

Sources

1. Beckman Quarles theorem (Wikipedia)

Statement and proof idea
if a transformation of the Euclidean plane or a higher-dimensional Euclidean space preserves unit distances, then it preserves all Euclidean distances.
History
The Beckman-Quarles theorem was first published by Frank S. Beckman and Donald A. Quarles Jr. in 1953.
In Branch: Geometry, Lead sentence

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Beckman-Quarles Theorem

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