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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 exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
2d Isosceles Set
D, or d, is the fourth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''dee'' (pronounced ), plural ''dees''. History The Semitic letter Dāleth may have developed from the logogram for a fish or a door. There are many different Egyptian hieroglyphs that might have inspired this. In Semitic, Ancient Greek and Latin, the letter represented ; in the Etruscan alphabet the letter was archaic, but still retained (see letter B). The equivalent Greek letter is Delta, Δ. Architecture The minuscule (lower-case) form of 'd' consists of a lower-story left bowl and a stem ascender. It most likely developed by gradual variations on the majuscule (capital) form 'D', and today now composed as a stem with a full lobe to the right. In handwriting, it was common to start the arc to the left of the vertical stroke, resulting in a serif at the top of the arc. This serif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Hull
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. For a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset. Convex hulls of open sets are open, and convex hulls of compact sets are compact. Every compact convex set is the convex hull of its extreme points. The convex hull operator is an example of a closure operator, and every antimatroid can be represented by applying this closure operator to finite sets of points. The algorithmic problems of finding the convex hull of a finite set of points in the plane or other low-dimensional Euclidean spaces, and its dual problem of intersecting half-spaces, are fundamental problems of co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eindhoven University Of Technology
The Eindhoven University of Technology ( nl, Technische Universiteit Eindhoven), abbr. TU/e, is a public technical university in the Netherlands, located in the city of Eindhoven. In 2020–21, around 14,000 students were enrolled in its BSc and MSc programs and around 1350 students were enrolled in its PhD and PDEng programs. In 2021, the TU/e employed around 3900 people. Eindhoven University of Technology has been ranked in the top 200 universities in three major ranking systems. The 2019 QS World University Rankings place Eindhoven 99th in the world, 34th in Europe, and 3rd in the Netherlands. TU/e is the Dutch member of thEuroTech Universities Alliance a strategic partnership of universities of science & technology in Europe: Technical University of Denmark (DTU), École Polytechnique Fédérale de Lausanne (EPFL), École Polytechnique (L’X), The Technion, Eindhoven University of Technology (TU/e), and Technical University of Munich (TUM). History The Eindhoven Unive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ultrametric Space
In mathematics, an ultrametric space is a metric space in which the triangle inequality is strengthened to d(x,z)\leq\max\left\. Sometimes the associated metric is also called a non-Archimedean metric or super-metric. Although some of the theorems for ultrametric spaces may seem strange at a first glance, they appear naturally in many applications. Formal definition An ultrametric on a set is a real-valued function :d\colon M \times M \rightarrow \mathbb (where denote the real numbers), such that for all : # ; # (''symmetry''); # ; # if then ; # (strong triangle inequality or ultrametric inequality). An ultrametric space is a pair consisting of a set together with an ultrametric on , which is called the space's associated distance function (also called a metric). If satisfies all of the conditions except possibly condition 4 then is called an ultrapseudometric on . An ultrapseudometric space is a pair consisting of a set and an ultrapseudometric on . In the case ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamming Space
In statistics and coding theory, a Hamming space (named after American mathematician Richard Hamming) is usually the set of all 2^N binary strings of length ''N''. It is used in the theory of coding signals and transmission. More generally, a Hamming space can be defined over any alphabet (set) ''Q'' as the set of words of a fixed length ''N'' with letters from ''Q''.Cohen et al., ''Covering Codes'', p. 15 If ''Q'' is a finite field, then a Hamming space over ''Q'' is an ''N''-dimensional vector space over ''Q''. In the typical, binary case, the field is thus GF(2) (also denoted by Z2). In coding theory, if ''Q'' has ''q'' elements, then any subset ''C'' (usually assumed of cardinality at least two) of the ''N''-dimensional Hamming space over ''Q'' is called a q-ary code of length N; the elements of ''C'' are called codewords. In the case where ''C'' is a linear subspace of its Hamming space, it is called a linear code. A typical example of linear code is the Hamming code. C ... [...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 an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular 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, whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypersimplex
In polyhedral combinatorics, the hypersimplex \Delta_ is a convex polytope that generalizes the simplex. It is determined by two integers d and k, and is defined as the convex hull of the d-dimensional vectors whose coefficients consist of k ones and d-k zeros. Equivalently, \Delta_ can be obtained by slicing the d-dimensional unit hypercube ,1d with the hyperplane of equation x_1+\cdots+x_d=k and, for this reason, it is a (d-1)-dimensional polytope when 0 Properties The number of vertices of is . The graph formed by the vertices and edges of the hypersimplex is the .Alternative constructions An alternative construction (for ) is to take the convex hull ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Pentagon
In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simple or self-intersecting. A self-intersecting ''regular pentagon'' (or ''star pentagon'') is called a pentagram. Regular pentagons A '' regular pentagon'' has Schläfli symbol and interior angles of 108°. A '' regular pentagon'' has five lines of reflectional symmetry, and rotational symmetry of order 5 (through 72°, 144°, 216° and 288°). The diagonals of a convex regular pentagon are in the golden ratio to its sides. Given its side length t, its height H (distance from one side to the opposite vertex), width W (distance between two farthest separated points, which equals the diagonal length D) and circumradius R are given by: :\begin H &= \frac~t \approx 1.539~t, \\ W= D &= \frac~t\approx 1.618~t, \\ W &= \sqr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Geometry
Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object. Discrete geometry has a large overlap with convex geometry and computational geometry, and is closely related to subjects such as finite geometry, combinatorial optimization, digital geometry, discrete differential geometry, geometric graph theory, toric geometry, and combinatorial topology. History Although polyhedra and tessellations had been studied for many years by people such as Kepler and Cauchy, modern discrete geometry has its origins in the late 19th century. Early topic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leroy Milton Kelly
Leroy Milton Kelly (May 8, 1914 – February 21, 2002 for Leroy M Kelly: Holt, Michigan.) was an American whose research primarily concerned . In 1986 he settled a conjecture of by proving that n points in complex 3-space, not all lying on a plane, determine an [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of parallel lines, and also metrical notions of distance, circles, and angle measurement. The set \mathbb^2 of pairs of real numbers (the real coordinate plane) augmented by appropriate structure often serves as the canonical example. History Books I through IV and VI of Euclid's Elements dealt with two-dimensional geometry, developing such notions as similarity of shapes, the Pythagorean theorem (Proposition 47), equality of angles and areas, parallelism, the sum of the angles in a triangle, and the three cases in which triangles are "equal" (have the same area), among many other topics. Later, the plane was described in a so-called '' Cartesian coordinate system'', a coordinate system that specifies each point uniquely in a plan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
