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, 2θ, 3θ ... 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 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.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. 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.)
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.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, 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.
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.
The three-gap theorem was conjectured by Hugo Steinhaus, and its first proofs were published in the late 1950s by Vera T. Sós,János Surányi , and Stanisław Świerczkowski. Several later proofs have also been published.
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.
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.
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.
Angle trisection is a classical problem of compass and straightedge constructions of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and a compass.
In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic. The center of the circle and its radius are called the circumcenter and the circumradius respectively. Other names for these quadrilaterals are concyclic quadrilateral and chordal quadrilateral, the latter since the sides of the quadrilateral are chords of the circumcircle. Usually the quadrilateral is assumed to be convex, but there are also crossed cyclic quadrilaterals. The formulas and properties given below are valid in the convex case.
A chord of a circle is a straight line segment whose endpoints both lie on the circle. The infinite line extension of a chord is a secant line, or just secant. More generally, a chord is a line segment joining two points on any curve, for instance, an ellipse. A chord that passes through a circle's center point is the circle's diameter. The word chord is from the Latin chorda meaning bowstring.
In mathematics, the Sato–Tate conjecture is a statistical statement about the family of elliptic curves Ep over the finite field with p elements, with p a prime number, obtained from an elliptic curve E over the rational number field, by the process of reduction modulo a prime for almost all p. If Np denotes the number of points on Ep and defined over the field with p elements, the conjecture gives an answer to the distribution of the second-order term for Np. That is, by Hasse's theorem on elliptic curves we have
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
Rotation around a fixed axis 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.
A prime gap is the difference between two successive prime numbers. The n-th prime gap, denoted gn or g(pn) is the difference between the (n + 1)-th and the n-th prime numbers, i.e.
In the mathematical theory of dynamical systems, an irrational rotation is a map
The circle packing theorem describes the possible tangency relations between circles in the plane whose interiors are disjoint. A circle packing is a connected collection of circles whose interiors are disjoint. The intersection graph of a circle packing is the graph having a vertex for each circle, and an edge for every pair of circles that are tangent. If the circle packing is on the plane, or, equivalently, on the sphere, then its intersection graph is called a coin graph; more generally, intersection graphs of interior-disjoint geometric objects are called tangency graphs or contact graphs. Coin graphs are always connected, simple, and planar. The circle packing theorem states that these are the only requirements for a graph to be a coin graph:
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.