In mathematics, the **three-gap theorem**, **three-distance theorem**, or **Steinhaus conjecture** states that if one places *n* points on a circle, at angles of *θ*, 2*θ*, 3*θ* ... 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.^{ [1] } Unless *θ* is a rational multiple of π, 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. Its applications include the study of plant growth and musical tuning systems, and the theory of Sturmian words.

In phyllotaxis (the theory of plant growth), 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.^{ [2] } 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.^{ [3] } 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. (This phenomenon has nothing to do with the golden ratio; the same property, of having only three distinct gaps between consecutive points on a circle, happens for any other rotation angle, and not just for the golden angle.)^{ [3] }

In music theory, this theorem implies that if a tuning system is generated by some number of consecutive multiples of a given interval, reduced to a cyclic sequence by considering two tones to be equivalent when they differ by whole numbers of octaves, then there are at most three different intervals between consecutive tones of the scale.^{ [4] }^{ [5] } For instance, the Pythagorean tuning is constructed in this way from multiples of a perfect fifth. It has only two distinct intervals representing its semitones,^{ [6] } but if it were extended by one more step then the sequence of intervals between its tones would include a third shorter interval, the Pythagorean comma.^{ [7] }

In the theory of Sturmian words, the theorem implies that the words of a given length *n* that appear within a given Sturmian word have at most three distinct frequencies. If there are three frequencies, then one of them must equal the sum of the other two.^{ [8] }

The three-gap theorem was conjectured by Hugo Steinhaus, and its first proofs were published in the late 1950s by Vera T. Sós,^{ [9] } János Surányi ,^{ [10] } and Stanisław Świerczkowski.^{ [11] } Several later proofs have also been published.^{ [12] }^{ [13] }^{ [14] }^{ [15] }^{ [16] }

The following simple proof is due to Frank Liang. Define a gap (an arc of the circle between adjacent points of the given set) to be *rigid* if rotating that gap by an angle of *θ* does not produce another gap of the same length. Each rotation by *θ* increases the position of the gap endpoints in the placement ordering of the points, and such an increase cannot be repeated indefinitely, so every gap has the same length as a rigid gap. But the only ways for a gap to be rigid are for one of its two endpoints to be the last point in the placement sequence (so that the corresponding point is missing from the rotated gap) or for another point to land in its rotated copy. An endpoint can only be missing if the gap is one of the two gaps on either side of the last point in the placement ordering. And a point can only land within the rotated copy if it is the first point in the placement ordering. So there can be at most three rigid gaps, and at most three lengths of gaps. Additionally, when there are three, the rotated copy of a rigid gap that has the first point in it is partitioned by that point into two smaller gaps, so in this case the longest gap length is the sum of the other two.^{ [17] }^{ [18] }

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 gaps between sequence values. Again, if there are three gaps then one is the sum of the other two.^{ [19] }^{ [20] }

**Euclidean space** is the fundamental space of classical geometry. Originally it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension, including the three-dimensional space and the *Euclidean plane*. It was introduced by the Ancient Greek mathematician Euclid of Alexandria, and the qualifier *Euclidean* is used to distinguish it from other spaces that were later discovered in physics and modern mathematics.

A **prime number** is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a product in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order.

A **sphere** is a geometrical object in three-dimensional space that is the surface of a ball.

A **twin prime** is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair. In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term *twin prime* is used for a pair of twin primes; an alternative name for this is **prime twin** or **prime pair**.

In mathematics, the **Sato–Tate conjecture** is a statistical statement about the family of elliptic curves *E _{p}* over the finite field with

In mathematics, **Ribet's theorem** is a statement in number theory concerning properties of Galois representations associated with modular forms. It was proposed by Jean-Pierre Serre and proven by Ken Ribet. The proof of the epsilon conjecture was a significant step towards the proof of Fermat's Last Theorem. As shown by Serre and Ribet, the Taniyama–Shimura conjecture and the epsilon conjecture together imply that Fermat's Last Theorem is true.

In mathematics, a sequence {*s*_{1}, *s*_{2}, *s*_{3}, ...} of real numbers is said to be **equidistributed**, or **uniformly distributed**, if the proportion of terms falling in a subinterval is proportional to the length of that interval. Such sequences are studied in Diophantine approximation theory and have applications to Monte Carlo integration.

In mathematics, a **Sturmian word**, 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 billiards on a square table. The struck ball will successively hit the vertical and horizontal edges labelled 0 and 1 generating a sequence of letters. This sequence is a Sturmian word.

In number theory, the **Elliott–Halberstam conjecture** is a conjecture about the distribution of prime numbers in arithmetic progressions. It has many applications in sieve theory. It is named for Peter D. T. A. Elliott and Heini Halberstam, who stated the conjecture in 1968.

In number theory, the **Erdős–Straus conjecture** states that for all integers *n* ≥ 2, the rational number 4/*n* can be expressed as the sum of three positive unit fractions. Paul Erdős and Ernst G. Straus formulated the conjecture in 1948. It is one of many conjectures by Erdős.

In mathematics, the **equidistribution theorem** is the statement that the sequence

A **Pierpont prime** is a prime number of the form

**Rotation around a fixed axis** or **about a fixed axis of revolution** or **motion with respect to a fixed axis of rotation** is a special case of rotational motion. The fixed axis hypothesis excludes the possibility of an axis changing its orientation, and cannot describe such phenomena as wobbling or precession. According to Euler's rotation theorem, simultaneous rotation along a number of stationary axes at the same time is impossible. If two rotations are forced at the same time, a new axis of rotation will appear.

In mathematics, **Kruskal's tree theorem** states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding. The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved within ATR_{0} (a form of arithmetical transfinite recursion), and a finitary application of the theorem gives the existence of the notoriously fast-growing **TREE function**.

A **prime gap** is the difference between two successive prime numbers. The *n*-th prime gap, denoted *g*_{n} or *g*(*p*_{n}) is the difference between the (*n* + 1)-th and the *n*-th prime numbers, i.e.

In mathematics, the **axis–angle representation** of a rotation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector **e** indicating the direction of an axis of rotation, and an angle *θ* describing the magnitude of the rotation about the axis. Only two numbers, not three, are needed to define the direction of a unit vector **e** rooted at the origin because the magnitude of **e** is constrained. For example, the elevation and azimuth angles of **e** suffice to locate it in any particular Cartesian coordinate frame.

In the mathematical theory of dynamical systems, an **irrational rotation** is a map

In mathematics, the **Riemann hypothesis** is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Many consider it to be the most important unsolved problem in pure mathematics. It is of great interest in number theory because it implies results about the distribution of prime numbers. It was proposed by Bernhard Riemann (1859), after whom it is named.

In geometric measure theory, **Falconer's conjecture**, named after Kenneth Falconer, is an unsolved problem concerning the sets of Euclidean distances between points in compact -dimensional spaces. Intuitively, it states that a set of points that is large in its Hausdorff dimension must determine a set of distances that is large in measure. More precisely, if is a compact set of points in -dimensional Euclidean space whose Hausdorff dimension is strictly greater than , then the conjecture states that the set of distances between pairs of points in must have nonzero Lebesgue measure.

**Stanisław (Stash) Świerczkowski** 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.

- ↑ Allouche, Jean-Paul; Shallit, Jeffrey (2003), "2.6 The Three-Distance Theorem",
*Automatic Sequences: Theory, Applications, Generalizations*, Cambridge University Press, pp. 53–55, ISBN 9780521823326 - ↑ Adam, John A. (2011),
*A Mathematical Nature Walk*, Princeton University Press, pp. 35–41, ISBN 9781400832903 - 1 2 van Ravenstein, Tony (1987), "Number sequences and phyllotaxis",
*Bulletin of the Australian Mathematical Society*,**36**(2): 333, doi: 10.1017/s0004972700026605 - ↑ Carey, Norman (2007), "Coherence and sameness in well-formed and pairwise well-formed scales",
*Journal of Mathematics and Music*,**1**(2): 79–98, doi:10.1080/17459730701376743 - ↑ Narushima, Terumi (2017),
*Microtonality and the Tuning Systems of Erv Wilson: Mapping the Harmonic Spectrum*, Routledge Studies in Music Theory, Routledge, pp. 90–91, ISBN 9781317513421 - ↑ Strohm, Reinhard; Blackburn, Bonnie J., eds. (2001),
*Music as Concept and Practice in the Late Middle Ages, Volume 3, Part 1*, New Oxford history of music, Oxford University Press, p. 252, ISBN 9780198162056 - ↑ Benson, Donald C. (2003),
*A Smoother Pebble: Mathematical Explorations*, Oxford University Press, p. 51, ISBN 9780198032977 - ↑ Lothaire, M. (2002), "Sturmian Words",
*Algebraic Combinatorics on Words*, Cambridge: Cambridge University Press, ISBN 978-0-521-81220-7, Zbl 1001.68093 - ↑ Sós, V. T. (1958), "On the distribution mod 1 of the sequence ",
*Ann. Univ. Sci. Budapest, Eötvös Sect. Math.*,**1**: 127–134 - ↑ Surányi, J. (1958), "Über die Anordnung der Vielfachen einer reelen Zahl mod 1",
*Ann. Univ. Sci. Budapest, Eötvös Sect. Math.*,**1**: 107–111 - ↑ Świerczkowski, S. (1959), "On successive settings of an arc on the circumference of a circle",
*Fundamenta Mathematicae*,**46**(2): 187–189, doi: 10.4064/fm-46-2-187-189 , MR 0104651 - ↑ Halton, John H. (1965), "The distribution of the sequence ",
*Proc. Cambridge Philos. Soc.*,**61**(3): 665–670, doi:10.1017/S0305004100039013, MR 0202668 - ↑ Slater, Noel B. (1967), "Gaps and steps for the sequence ",
*Proc. Cambridge Philos. Soc.*,**63**(4): 1115–1123, doi:10.1017/S0305004100042195, MR 0217019 - ↑ van Ravenstein, Tony (1988), "The three-gap theorem (Steinhaus conjecture)",
*Journal of the Australian Mathematical Society*, Series A,**45**(3): 360–370, doi: 10.1017/S1446788700031062 , MR 0957201 - ↑ Mayero, Micaela (2000), "The three gap theorem (Steinhaus conjecture)",
*Types for Proofs and Programs: International Workshop, TYPES'99, Lökeberg, Sweden, June 12–16, 1999, Selected Papers*, Lecture Notes in Computer Science,**1956**, Springer, pp. 162–173, arXiv: cs/0609124 , doi:10.1007/3-540-44557-9_10, ISBN 978-3-540-41517-6 - ↑ Marklof, Jens; Strömbergsson, Andreas (2017), "The three gap theorem and the space of lattices",
*American Mathematical Monthly*,**124**(8): 741–745, arXiv: 1612.04906 , doi:10.4169/amer.math.monthly.124.8.741, hdl:1983/b5fd0feb-e42d-48e9-94d8-334b8dc24505, MR 3706822 - ↑ Liang, Frank M. (1979), "A short proof of the distance theorem",
*Discrete Mathematics*,**28**(3): 325–326, doi:10.1016/0012-365X(79)90140-7, MR 0548632 - ↑ Shiu, Peter (2018), "A footnote to the three gaps theorem",
*American Mathematical Monthly*,**125**(3): 264–266, doi:10.1080/00029890.2018.1412210, MR 3768035 - ↑ Slater, N. B. (1950), "The distribution of the integers for which ",
*Proc. Cambridge Philos. Soc.*,**46**(4): 525–534, doi:10.1017/S0305004100026086, MR 0041891 - ↑ Florek, K. (1951), "Une remarque sur la répartition des nombres ",
*Colloq. Math.*,**2**: 323–324

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