Three-gap theorem

This result was conjectured by Hugo Steinhaus, and proved in the 1950s by Vera T. Sós, János Surányi [hu], 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.

The three-gap theorem can be stated geometrically in terms of points on a circle.

An equivalent and more algebraic form involves the fractional parts of multiples of a real number.

divide the unit interval into subintervals with at most three different lengths.

[2][3][4] In the study of phyllotaxis, the arrangements of leaves on plant stems, it has been observed that each successive leaf on the stems of many plants is turned from the previous leaf by the golden angle, approximately 137.5°.

It has been suggested that this angle maximizes the sun-collecting power of the plant's leaves.

For instance, the fact that golden ratio is a badly approximable number implies that points spaced at this angle along the Fermat spiral (as they are in some models of plant growth) form a Delone set; intuitively, this means that they are uniformly spaced.

, and similarly other common musical intervals other than the octave do not correspond to rational angles.

[9] A tuning system is a collection of tones used to compose and play music.

Some other tuning systems do not space their tones equally, but instead generate them by some number of consecutive multiples of a given interval.

[10] Instead of being spaced at angles of exactly 1/12 of a circle, as the tones of equal temperament would be, the tones of the Pythagorean tuning are separated by intervals of two different angles, close to but not exactly 1/12 of a circle, representing two different types of semitones.

[11] If the Pythagorean tuning system were extended by one more perfect fifth, to a set of 13 tones, then the sequence of intervals between its tones would include a third, much shorter interval, the Pythagorean comma.

[12] In this context, the three-gap theorem can be used to describe any tuning system that is generated in this way by consecutive multiples of a single interval.

Some of these tuning systems (like equal temperament) may have only one interval separating the closest pairs of tones, and some (like the Pythagorean tuning) may have only two different intervals separating the tones, but the three-gap theorem implies that there are always at most three different intervals separating the tones.

[13][14] A Sturmian word is infinite sequences of two symbols (for instance, "H" and "V") describing the sequence of horizontal and vertical reflections of a light ray within a mirrored square, starting along a line of irrational slope.

One proof of this result involves partitioning the y-intercepts of the starting lines (modulo 1) into n + 1 subintervals within which the initial n elements of the sequence are the same, and applying the three-gap theorem to this partition.

[15][16] The three-gap theorem was conjectured by Hugo Steinhaus, and its first[17] proofs were found in the late 1950s by Vera T. Sós,[18] János Surányi [hu],[19] and Stanisław Świerczkowski.

[20] Later researchers published additional proofs,[21] generalizing this result to higher dimensions[22][23][24][25], and connecting it to topics including continued fractions,[4][26] symmetries and geodesics of Riemannian manifolds,[27] ergodic theory,[28] and the space of planar lattices.

[3] Mayero (2000) formalizes a proof using the Coq interactive theorem prover.

[29] In the three-gap theorem, there is a constant bound on the ratios between the three gaps, if and only if θ/2π is a badly approximable number.

[7] A closely related but earlier theorem, also called the three-gap theorem, is that if A is any arc of the circle, then the integer sequence of multiples of θ that land in A has at most three lengths of gaps between sequence values.