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Distance Set
In geometry, the distance set of a collection of points is the Set (mathematics), set of distances between distinct pairs of points. Thus, it can be seen as the generalization of a Minkowski difference, difference set, the set of distances (and their negations) in collections of numbers. Several problems and results in geometry concern distance sets, usually based on the principle that a large collection of points must have a large distance set (for varying definitions of "large"): *Falconer's conjecture is the statement that, for a collection of points in d-dimensional space that has Hausdorff dimension larger than d/2, the corresponding distance set has nonzero Lebesgue measure. Although partial results are known, the conjecture remains unproven. *The Erdős–Ulam problem asks whether it is possible to have a dense set in the Euclidean plane whose distance set consists only of rational numbers. Again, it remains unsolved. *Fermat's theorem on sums of two squares characterizes t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Erdős–Anning Theorem
The Erdős–Anning theorem states that, whenever an Infinite set, infinite number of points in the plane all have integer distances, the points lie on a straight line. The same result holds in higher dimensional Euclidean spaces. The theorem cannot be strengthened to give a finite bound on the number of points: there exist arbitrarily large finite sets of points that are not on a line and have integer distances. The theorem is named after Paul Erdős and Norman H. Anning, who published a proof of it in 1945. Erdős later supplied a simpler proof, which can also be used to check whether a point set forms an Erdős–Diophantine graph, an inextensible system of integer points with integer distances. The Erdős–Anning theorem inspired the Erdős–Ulam problem on the existence of dense set, dense point sets with rational distances. Rationality versus integrality Although there can be no infinite non-Collinearity, collinear set of points with integer distances, there are infini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Chord Theorem
In mathematical analysis, the universal chord theorem states that if a function ''f'' is continuous on 'a'',''b''and satisfies f(a) = f(b) , then for every natural number n, there exists some x \in ,b such that f(x) = f\left(x + \frac\right) . History The theorem was published by Paul Lévy in 1934 as a generalization of Rolle's theorem. Statement of the theorem Let H(f) = \ denote the chord set of the function ''f''. If ''f'' is a continuous function and h \in H(f) , then \frac h n \in H(f) for all natural numbers ''n''. Case of ''n'' = 2 The case when ''n'' = 2 can be considered an application of the Borsuk–Ulam theorem to the real line. It says that if f(x) is continuous on some interval I = ,b with the condition that f(a) = f(b) , then there exists some x \in ,b such that f(x) = f\left(x + \frac\right) . In less generality, if f : ,1\rightarrow \R is continuous and f(0) = f(1) , then there exists x \in \left ,\frac\right/math> that satisfies ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isosceles Set
In discrete geometry, an isosceles set is a set of points with the property that every three of them form an isosceles triangle. More precisely, each three points should determine at most two distances; this also allows degenerate isosceles triangles formed by three equally-spaced points on a line. History The problem of finding the largest isosceles set in a Euclidean space of a given dimension was posed in 1946 by Paul Erdős. In his statement of the problem, Erdős observed that the largest such set in the Euclidean plane has six points. In his 1947 solution, Leroy Milton Kelly showed more strongly that the unique six-point planar isosceles set consists of the vertices and center of a regular pentagon. In three dimensions, Kelly found an eight-point isosceles set, six points of which are the same; the remaining two points lie on a line perpendicular to the pentagon through its center, at the same distance as the pentagon vertices from the center. This three-dimensional example ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octahedron
In geometry, an octahedron (: octahedra or octahedrons) is any polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Many types of irregular octahedra also exist, including both convex set, convex and non-convex shapes. Combinatorially equivalent to the regular octahedron The following polyhedra are combinatorially equivalent to the regular octahedron. They all have six vertices, eight triangular faces, and twelve edges that correspond one-for-one with the features of it: * Triangular antiprisms: Two faces are equilateral, lie on parallel planes, and have a common axis of symmetry. The other six triangles are isosceles. The regular octahedron is a special case in which the six lateral triangles are also equilateral. * Tetragonal bipyramids, in which at least one of the equatorial quadrilaterals lies on a plane. The regular octahedron is a special case in which all thr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manhattan Distance
Taxicab geometry or Manhattan geometry is geometry where the familiar Euclidean distance is ignored, and the distance between two point (geometry), points is instead defined to be the sum of the absolute differences of their respective Cartesian coordinates, a distance function (or Metric (mathematics), metric) called the ''taxicab distance'', ''Manhattan distance'', or ''city block distance''. The name refers to the island of Manhattan, or generically any planned city with a rectangular grid of streets, in which a taxicab can only travel along grid directions. In taxicab geometry, the distance between any two points equals the length of their shortest grid path. This different definition of distance also leads to a different definition of the length of a curve, for which a line segment between any two points has the same length as a grid path between those points rather than its Euclidean length. The taxicab distance is also sometimes known as ''rectilinear distance'' or distanc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kusner's Conjecture
In mathematics, the equilateral dimension of a metric space is the maximum size of any subset of the space whose points are all at equal distances to each other. Equilateral dimension has also been called " metric dimension", but the term "metric dimension" also has many other inequivalent usages. The equilateral dimension of Euclidean space is d+1, achieved by the vertices of a regular simplex, and the equilateral dimension of a vector space with the Chebyshev distance (L^\infty norm) is 2^d, achieved by the vertices of a hypercube. However, the equilateral dimension of a space with the Manhattan distance (L^1 norm) is not known. Kusner's conjecture, named after Robert B. Kusner, states that it is exactly 2d, achieved by the vertices of a cross polytope. Lebesgue spaces The equilateral dimension has been particularly studied for Lebesgue spaces, finite-dimensional normed vector spaces with the L^p norm \ \, x\, _p=\bigl(, x_1, ^p+, x_2, ^p+\cdots+, x_d, ^p\bigr)^. The equila ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), function called a metric or distance function. Metric spaces are a general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a Conceptual metaphor , metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are a tool used in many different bra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equilateral Dimension
In mathematics, the equilateral dimension of a metric space is the maximum size of any subset of the space whose points are all at equal distances to each other. Equilateral dimension has also been called " metric dimension", but the term "metric dimension" also has many other inequivalent usages. The equilateral dimension of Euclidean space is d+1, achieved by the vertices of a regular simplex, and the equilateral dimension of a vector space with the Chebyshev distance (L^\infty norm) is 2^d, achieved by the vertices of a hypercube. However, the equilateral dimension of a space with the Manhattan distance (L^1 norm) is not known. Kusner's conjecture, named after Robert B. Kusner, states that it is exactly 2d, achieved by the vertices of a cross polytope. Lebesgue spaces The equilateral dimension has been particularly studied for Lebesgue spaces, finite-dimensional normed vector spaces with the L^p norm \ \, x\, _p=\bigl(, x_1, ^p+, x_2, ^p+\cdots+, x_d, ^p\bigr)^. The equila ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sophie Piccard
Sophie Piccard (1904–1990) was a Russian-Swiss mathematician who became the first female full professor (professor ordinarius) in Switzerland. Her research concerned set theory, group theory, linear algebra, and the history of mathematics.. Early life and education Piccard was born on September 27, 1904, in Saint Petersburg, with a French Huguenot mother and a Swiss father. She earned a diploma in Smolensk in 1925, where her father, Eugène-Ferdinand Piccard, was a university professor and her mother a language teacher at the lycée. Soon afterwards she moved to Switzerland with her parents, escaping the unrest in Russia that her mother, Eulalie Piccard, would become known for writing about. Sophie Piccard's Russian degree was worthless in Switzerland, and she earned another from the University of Lausanne in 1927, going on to complete a doctorate there in 1929 under the supervision of Dmitry Mirimanoff. Career and later life She worked outside of mathematics until 1936, whe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Golomb Ruler
In mathematics, a Golomb ruler is a set (mathematics), set of marks at integer positions along a ruler such that no two pairs of marks are the same distance apart. The number of marks on the ruler is its ''order'', and the largest distance between two of its marks is its ''length''. Translation and reflection of a Golomb ruler are considered trivial, so the smallest mark is customarily put at 0 and the next mark at the smaller of its two possible values. Golomb rulers can be viewed as a one-dimensional special case of Costas arrays. The Golomb ruler was named for Solomon W. Golomb and discovered independently by and . Sophie Piccard also published early research on these sets, in 1939, stating as a theorem the claim that two Golomb rulers with the same distance set must be congruence (geometry), congruent. This turned out to be false for six-point rulers, but true otherwise. There is no requirement that a Golomb ruler be able to measure ''all'' distances up to its length, but ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nets Katz
Nets Hawk Katz is the W.L. Moody Professor of Mathematics at Rice University. He was a professor of mathematics at Indiana University Bloomington until March 2013 and the IBM Professor of Mathematics at the California Institute of Technology until 2023. He is currently the W. L. Moody Professor of Mathematics at Rice University. Katz earned a B.A. in mathematics from Rice University in 1990 at the age of 17. He received his Ph.D. in 1993 under Dennis DeTurck at the University of Pennsylvania, with a dissertation titled "Noncommutative Determinants and Applications". He is the author of several important results in combinatorics (especially additive combinatorics), harmonic analysis and other areas. In 2003, jointly with Jean Bourgain and Terence Tao, he proved that any subset of \Z/p\Z grows substantially under either addition or multiplication. More precisely, if A is a set such that \max(, A \cdot A, , , A+A, ) \leq K, A, , then A has size at most K^C or at least p/K^C where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
