Details

The result was conjectured by H. Steinhaus and proved in 1958, independently, by V. T. Sós, J. Surányi and S. \:015awierczkowski. A short proof by Frank Liang can be found at either reference. Shiu [1] answers the question of which angles and values lead to only two gaps as opposed to three.

Snapshot 1: θ is the golden angle, about 137.5°. Here the three gaps are always in proportion to the golden mean.

Snapshot 2: θ is , which is close to . When there are 22 sectors, the last point almost agrees with the first. There are three gaps, but one is close to 0 while the other two are close to each other.

Snapshot 3: θ is and is 49, a case that is explained in [1]. There are only two gaps if and only if .

References

[1] P. Shiu, "A Footnote to the Three Gaps Theorem," The American Mathematical Monthly, 125(3), 2018 pp. 264–266. doi:10.1080/00029890.2018.1412210.

[2] Wikipedia. "Three-Gap Theorem" (May 7, 2020) en.wikipedia.org/wiki/Three-gap_theorem.