The Three-Gap Theorem, also called the three-distance theorem or the Steinhaus conjecture, states that if n points are placed on a circle at angles theta, two theta, three theta, and so on from a starting point, the points in adjacent positions around the circle are separated by at most three distinct distances, and whenever there are three distances the largest always equals the sum of the other two. Hugo Steinhaus conjectured the result, and it was proved in the 1950s by Vera Sos, Janos Suranyi, and Stanislaw Swierczkowski, with further proofs added later by others. Applications include the study of plant growth patterns, musical tuning systems, and the reflection of light within a mirrored square.
Facts
Classification
Statement Form Statement Form Statement Form StatementIf n points are placed on a circle at angles theta, 2 theta, 3 theta, ... from the starting point, there are at most three distinct distances between pairs of points in adjacent positions around the circle, and when there are three distances the largest equals the sum of the other two. 1 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.
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.
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.
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.
Sources
1. Three-gap theorem (Wikipedia)
Intro, sentences 1-2Quote, Intro, sentences 1-2
there will be at most three distinct distances between pairs of points in adjacent positions around the circle. When there are three distances, the largest of the three always equals the sum of the other two.
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