In mathematics, the three-gap theorem, three-distance theorem, or Steinhaus conjecture states that if one places points on a circle, at angles of , , , ... 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 is a rational multiple of , 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, , 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.
Statement
The three-gap theorem can be stated geometrically in terms of points on a circle. In this form, it states that if one places points on a circle, at angles of from the starting point, then there will be at most three distinct distances between pairs of points in adjacent positions around the circle. An equivalent and more algebraic form involves the fractional parts of multiples of a real number. It states that, for any positive real number and integer , the fractional parts of the numbers divide the unit interval into subintervals with at most three different lengths. The two problems are equivalent under a linear correspondence between the unit interval and the circumference of the circle, and a correspondence between the real number and the
Applications
Plant growth
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. If one looks end-on at a plant stem that has grown in this way, there will be at most three distinct angles between two leaves that are consecutive in the cyclic order given by this end-on view.
For example, in the figure, the largest of these three angles occurs three times, between the leaves numbered 3 and 6, between leaves 4 and 7, and between leaves 5 and 8. The second-largest angle occurs five times, between leaves 6 and 1, 9 and 4, 7 and 2, 10 and 5, and 8 and 3. And the smallest angle occurs only twice, between leaves 1 and 9 and between leaves 2 and 10. The phenomenon of having three types of distinct gaps depends only on fact that the growth pattern uses a constant rotation angle, and not on the relation of this angle to the golden ratio; the same phenomenon would happen for any other rotation angle, and not just for the golden angle. However, other properties of this growth pattern do depend on the golden ratio. 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.
Music theory
In music theory, a musical interval describes the ratio in frequency between two musical tones. Intervals are commonly considered consonant or harmonious when they are the ratio of two small integers; for instance, the octave corresponds to the ratio 2:1, while the perfect fifth corresponds to the ratio 3:2. Two tones are commonly considered to be equivalent when they differ by a whole number of octaves; this equivalence can be represented geometrically by the chromatic circle, the points of which represent classes of equivalent tones. Mathematically, this circle can be described as the unit circle in the complex plane, and the point on this circle that represents a given tone can be obtained by the mapping the frequency to the complex number . An interval with ratio corresponds to the angle between points on this circle, meaning that two musical tones differ by the given interval when their two points on the circle differ by this angle. For instance, this formula gives (a whole circle) as the angle corresponding to an octave. Because 3/2 is not a rational power of two, the angle on the chromatic circle that represents a perfect fifth is not a rational multiple of , and similarly other common musical intervals other than the octave do not correspond to rational angles.
