Three-gap theorem

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In mathematics, the three-gap theorem, three-distance theorem, or Steinhaus conjecture states that if one places n {\displaystyle n} points on a circle, at angles of θ , 2 θ , 3 θ , . . . {\displaystyle \theta ,2\theta ,3\theta ,...} from the starting point, then 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. Unless θ {\displaystyle \theta } is a rational multiple of π {\displaystyle \pi } , there will also be at least two distinct distances. This result was conjectured by Hugo Steinhaus, and proved in the 1950s by Vera T. Sós, János Surányi, and Stanisław Świerczkowski; more proofs were added by others later. Applications of the three-gap theorem include the study of plant growth and musical tuning systems, and the theory of light reflection within a mirrored square.