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 Hugo Dyonizy Steinhaus ( ; ; January 14, 1887 – February 25, 1972) was a Polish mathematician and educator. Steinhaus obtained his PhD under David Hilbert at Göttingen University in 1911 and later became a professor at the Jan Kazimierz Unive ...

, and proved in the 1950s by

Vera T. Sós Vera T. Sós (born September 11, 1930) is a Hungarian mathematician, specializing in number theory and combinatorics. She was a student and close collaborator of both Paul Erdős and Alfréd Rényi. She also collaborated frequently with her husb ...

, , and

Stanisław Świerczkowski Stanisław (Stash) Świerczkowski (16 July 1932 – 30 September 2015) was a Polish mathematician famous for his solutions to two iconic problems posed by Hugo Steinhaus: the three-gap theorem and the Non-Tetratorus Theorem. Early life and edu ...

; 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

n

points on a circle, at angles of

\theta, 2\theta, \dots, n\theta

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 part The fractional part or decimal part of a non‐negative real number x is the excess beyond that number's integer part. If the latter is defined as the largest integer not greater than , called floor of or \lfloor x\rfloor, its fractional part can ...

s of multiples of a

real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...

. It states that, for any positive real number

\alpha

and integer

n

, the fractional parts of the numbers

\alpha, 2\alpha, \dots, n\alpha

divide the

unit interval In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis, ...

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

\alpha

and the

Applications

Plant growth

In the study of

phyllotaxis In botany, phyllotaxis () or phyllotaxy is the arrangement of leaf, leaves on a plant stem. Phyllotactic spirals form a distinctive class of patterns in nature. Leaf arrangement The basic leaf#Arrangement on the stem, arrangements of leaves ...

, 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 In geometry, the golden angle is the smaller of the two angles created by sectioning the circumference of a circle according to the golden ratio; that is, into two arcs such that the ratio of the length of the smaller arc to the length of the l ...

, 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. Sunflower spiral

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 In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...

; 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 In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria. The first problem was to know how well a real number can be approximated by r ...

implies that points spaced at this angle along the Fermat spiral (as they are in some models of plant growth) form a

Delone set In the mathematical theory of metric spaces, ε-nets, ε-packings, ε-coverings, uniformly discrete sets, relatively dense sets, and Delone sets (named after Boris Delone) are several closely related definitions of well-spaced sets of points, a ...

; intuitively, this means that they are uniformly spaced.

Music theory

music theory Music theory is the study of the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory". The first is the "rudiments", that are needed to understand music notation (ke ...

, a

musical interval In music theory, an interval is a difference in pitch between two sounds. An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and vertical or ha ...

describes the ratio in

frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...

between two

musical tone Traditionally in Western music, a musical tone is a steady periodic sound. A musical tone is characterized by its duration, pitch, intensity (or loudness), and timbre (or quality). The notes used in music can be more complex than musical ton ...

s. Intervals are commonly considered consonant or harmonious when they are the ratio of two small integers; for instance, the

octave In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...

corresponds to the ratio 2:1, while the

perfect fifth In music theory, a perfect fifth is the Interval (music), musical interval corresponding to a pair of pitch (music), pitches with a frequency ratio of 3:2, or very nearly so. In classical music from Western culture, a fifth is the interval fro ...

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 chromatic circle is a clock diagram for displaying relationships among the 12 equal-tempered pitch classes making up the familiar chromatic scale on a circle. Explanation If one starts on any equal-tempered pitch and repeatedly ascends by t ...

, the points of which represent classes of equivalent tones. Mathematically, this circle can be described as the

unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...

in the

complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...

, and the point on this circle that represents a given tone can be obtained by the mapping the frequency

\nu

to the complex number

\exp(2\pi i\log_2\nu)

. An interval with ratio

\rho

corresponds to the angle

2\pi\log_2\rho

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

2\pi

(a whole circle) as the angle corresponding to an octave. Because 3/2 is not a rational

power of two A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer as the exponent. In a context where only integers are considered, is restricted to non-negative ...

, the angle on the chromatic circle that represents a perfect fifth is not a rational multiple of

2\pi

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

tuning system In music, there are two common meanings for tuning: * Tuning practice, the act of tuning an instrument or voice. * Tuning systems, the various systems of pitches used to tune an instrument, and their theoretical bases. Tuning practice Tun ...

is a collection of tones used to compose and play music. For instance, the

equal temperament An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, wh ...

commonly used for the

piano The piano is a stringed keyboard instrument in which the strings are struck by wooden hammers that are coated with a softer material (modern hammers are covered with dense wool felt; some early pianos used leather). It is played using a keyboa ...

is a tuning system, consisting of 12 tones equally spaced around the chromatic circle. Some other tuning systems do not space their tones equally, but instead generate them by some number of consecutive multiples of a given interval. An example is the

Pythagorean tuning Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2.Bruce Benward and Marilyn Nadine Saker (2003). ''Music: In Theory and Practice'', seventh edition, 2 vols. (Boston: Mc ...

, which is constructed in this way from twelve tones, generated as the consecutive multiples of a perfect fifth in the

circle of fifths In music theory, the circle of fifths is a way of organizing the 12 chromatic pitches as a sequence of perfect fifths. (This is strictly true in the standard 12-tone equal temperament system — using a different system requires one interval ...

. The irrational angle formed on the chromatic circle by a perfect fifth is close to 7/12 of a circle, and therefore the twelve tones of the Pythagorean tuning are close to, but not the same as, the twelve tones of equal temperament, which could be generated in the same way using an angle of exactly 7/12 of a circle. 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

semitone A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between two adjacent no ...

s. 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 In musical tuning, the Pythagorean comma (or ditonic comma), named after the ancient mathematician and philosopher Pythagoras, is the small interval (or comma) existing in Pythagorean tuning between two enharmonically equivalent notes such as C ...

. 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.

Mirrored reflection

Sturmian word In mathematics, a Sturmian word (Sturmian sequence or billiard sequence), named after Jacques Charles François Sturm, is a certain kind of infinitely long sequence of characters. Such a sequence can be generated by considering a game of English ...

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. Equivalently, the same sequence describes the sequence of horizontal and vertical lines of the integer grid that are crossed by the starting line. One property that all such sequences have is that, for any positive integer , the sequence has exactly distinct consecutive subsequences of length . Each subsequence occurs infinitely often with a certain frequency, and the three-gap theorem implies that these subsequences occur with at most three distinct frequencies. If there are three frequencies, then the largest frequency must equal the sum of the other two. One proof of this result involves partitioning the -intercepts of the starting lines (modulo 1) into subintervals within which the initial elements of the sequence are the same, and applying the three-gap theorem to this partition.

History and proof

The three-gap theorem was conjectured by

, and its first proofs were found in the late 1950s by

, , and

. Later researchers published additional proofs,These proofs are briefly surveyed and classified by , from which the following classification of these proofs and many of their references are taken. generalizing this result to higher dimensions, and connecting it to topics including

continued fraction In mathematics, a continued fraction is an expression (mathematics), expression obtained through an iterative process of representing a number as the sum of its integer part and the multiplicative inverse, reciprocal of another number, then writ ...

s, symmetries and

geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...

s of

Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...

ergodic theory Ergodic theory (Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expres ...

, and the space of planar lattices. formalizes a proof using the

Coq Coq is an interactive theorem prover first released in 1989. It allows for expressing mathematical assertions, mechanically checks proofs of these assertions, helps find formal proofs, and extracts a certified program from the constructive proof ...

interactive theorem prover. The following simple proof is due to Frank Liang. Let be the rotation angle generating a set of points as some number of consecutive multiples of on a circle. Define a ''gap'' to be an arc of the circle that extends between two adjacent points of the given set, and define a gap to be ''rigid'' if its endpoints occur later in the sequence of multiples of than any other gap of the same length. From this definition, it follows that every gap has the same length as a rigid gap. If is a rigid gap, then is not a gap, because it has the same length and would be one step later. The only ways for this to happen are for one of the endpoints of to be the last point in the sequence of multiples of (so that the corresponding endpoint of is missing) or for one of the given points to land within , preventing it from being a gap. A point can only land within if it is the first point in the sequence of multiples of , because otherwise its predecessor in the sequence would land within , contradicting the assumption that is a gap. So there can be at most three rigid gaps, the two on either side of the last point and the one in which the predecessor of the first point (if it were part of the sequence) would land. Because there are at most three rigid gaps, there are at most three lengths of gaps.

Related results

Liang's proof additionally shows that, when there are exactly three gap lengths, the longest gap length is the sum of the other two. For, in this case, the rotated copy that has the first point in it is partitioned by that point into two smaller gaps, which must be the other two gaps. Liang also proves a more general result, the "

3d

distance theorem", according to which the union of

d

different arithmetic progressions on a circle has at most

3d

different gap lengths. In the three-gap theorem, there is a constant bound on the ratios between the three gaps, if and only if is a

. A closely related but earlier theorem, also called the three-gap theorem, is that if is any arc of the circle, then the

integer sequence In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers. An integer sequence may be specified ''explicitly'' by giving a formula for its ''n''th term, or ''implicitly'' by giving a relationship between its terms. For ...

of multiples of that land in has at most three lengths of gaps between sequence values. Again, if there are three gap lengths then one is the sum of the other two.

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Bulletin of the Australian Mathematical Society The Australian Mathematical Society (AustMS) was founded in 1956 and is the national society of the mathematics profession in Australia. One of the Society's listed purposes is to promote the cause of mathematics in the community by representing t ...

, page = 333 , title = Number sequences and phyllotaxis , volume = 36, doi-access = free {{citation , last = van Ravenstein , first = Tony , issue = 3 , journal =

Journal of the Australian Mathematical Society The Australian Mathematical Society (AustMS) was founded in 1956 and is the national society of the mathematics profession in Australia. One of the Society's listed purposes is to promote the cause of mathematics in the community by representing t ...

, series = Series A , mr = 957201 , pages = 360–370 , title = The three-gap theorem (Steinhaus conjecture) , volume = 45 , year = 1988, doi = 10.1017/S1446788700031062 , doi-access = free Diophantine approximation Theorems in number theory Articles containing proofs Mathematics of music