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Distance Set
In geometry, the distance set of a collection of points is the set of distances between distinct pairs of points. Thus, it can be seen as the generalization of a 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 the numbers in the distance set of the two ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Erdős–Anning Theorem
The Erdős–Anning theorem states that an infinite number of points in the plane can have mutual integer distances only if all the points lie on a straight line. It is named after Paul Erdős and Norman H. Anning, who published a proof of it in 1945. Rationality versus integrality Although there can be no infinite non-collinear set of points with integer distances, there are infinite non-collinear sets of points whose distances are rational numbers. For instance, the subset of points on a unit circle obtained by repeatedly rotating by the sharp angle in a 3–4–5 right triangle has this property. It forms a dense set in the circle. The (still unsolved) Erdős–Ulam problem asks whether there can exist a set of points at rational distances from each other that forms a dense set for the whole Euclidean plane. For any finite set ''S'' of points at rational distances from each other, it is possible to find a similar set of points at integer distances from each other, by expandi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shape Analysis (digital Geometry)
This article describes shape analysis to analyze and process geometric shapes. Description ''Shape analysis'' is the (mostly) automatic analysis of geometric shapes, for example using a computer to detect similarly shaped objects in a database or parts that fit together. For a computer to automatically analyze and process geometric shapes, the objects have to be represented in a digital form. Most commonly a boundary representation is used to describe the object with its boundary (usually the outer shell, see also 3D model). However, other volume based representations (e.g. constructive solid geometry) or point based representations (point clouds) can be used to represent shape. Once the objects are given, either by modeling (computer-aided design), by scanning (3D scanner) or by extracting shape from 2D or 3D images, they have to be simplified before a comparison can be achieved. The simplified representation is often called a ''shape descriptor'' (or fingerprint, signature). The ... [...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 wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octahedron
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. A regular octahedron is the dual polyhedron of a cube. It is a rectified tetrahedron. It is a square bipyramid in any of three orthogonal orientations. It is also a triangular antiprism in any of four orientations. An octahedron is the three-dimensional case of the more general concept of a cross polytope. A regular octahedron is a 3-ball in the Manhattan () metric. Regular octahedron Dimensions If the edge length of a regular octahedron is ''a'', the radius of a circumscribed sphere (one that touches the octahedron at all vertices) is :r_u = \frac a \approx 0.707 \cdot a and the radius of an inscribed sphere (tangent to each of the octahedron's faces) is :r_i = \frac a \approx 0.408\cdot a while the midradius, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manhattan Distance
A taxicab geometry or a Manhattan geometry is a geometry whose usual distance function or Metric (mathematics), metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of their Cartesian coordinates. The taxicab metric is also known as rectilinear distance, ''L''1 distance, ''L''1 distance or \ell_1 norm (see Lp space, ''Lp'' space), Snake (video game), snake distance, city block distance, Manhattan distance or Manhattan length. The latter names refer to the rectilinear street layout on the island of Manhattan, where the shortest path a taxi travels between two points is the sum of the absolute values of distances that it travels on avenues and on streets. The geometry has been used in regression analysis since the 18th century, and is often referred to as Lasso (statistics), LASSO. The geometric interpretation dates to non-Euclidean geometry of the 19th century and is due to Hermann Minkowski. In \mat ... [...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 a d-dimensional Euclidean space is d+1, achieved by a regular simplex, and the equilateral dimension of a d-dimensional vector space with the Chebyshev distance (L^\infty norm) is 2^d, achieved by 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 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=\left(, x_1, ^p+, x_2, ^p+\cdots+, x_d, ^p\right)^. The equilateral dimension of L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most 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 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 branches of mathematics. Many types of mathematical objects have a natural notion of distance and t ... [...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 a d-dimensional Euclidean space is d+1, achieved by a regular simplex, and the equilateral dimension of a d-dimensional vector space with the Chebyshev distance (L^\infty norm) is 2^d, achieved by 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 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=\left(, x_1, ^p+, x_2, ^p+\cdots+, x_d, ^p\right)^. The equilateral dimension of L ... [...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, when ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Golomb Ruler
In mathematics, a Golomb ruler is a 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 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 if it does, it is called a '' perfect'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nets Katz
Nets Hawk Katz is the IBM Professor of Mathematics at the California Institute of Technology. He was a professor of Mathematics at Indiana University Bloomington until March 2013. 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 C is a constant that depends on A. This result was followed by the subsequent work of Bourgain, Sergei Konyagin and Glibichuk, establishing that every ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
