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Schläfli Symbol
In geometry , the SCHLäFLI SYMBOL is a notation of the form {p,q,r,...} that defines regular polytopes and tessellations . The Schläfli symbol
Schläfli symbol
is named after the 19th-century Swiss mathematician Ludwig Schläfli , who made important contributions in geometry and other areas
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Dodecahedron
In geometry , a DODECAHEDRON (Greek δωδεκάεδρον, from δώδεκα dōdeka "twelve" + ἕδρα hédra "base", "seat" or "face") is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron , which is a Platonic solid
Platonic solid
. There are also three regular star dodecahedra , which are constructed as stellations of the convex form. All of these have icosahedral symmetry , order 120. The pyritohedron is an irregular pentagonal dodecahedron, having the same topology as the regular one but pyritohedral symmetry while the tetartoid has tetrahedral symmetry . The rhombic dodecahedron , seen as a limiting case of the pyritohedron, has octahedral symmetry . The elongated dodecahedron and trapezo-rhombic dodecahedron variations, along with the rhombic dodecahedra, are space-filling . There are a large number of other dodecahedra
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Pentagon
In geometry , a PENTAGON (from the Greek πέντε pente and γωνία gonia, meaning five and 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
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Vertex (geometry)
In geometry , a VERTEX (plural: VERTICES or VERTEXES) is a point where two or more curves , lines , or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices. CONTENTS* 1 Definition * 1.1 Of an angle * 1.2 Of a polytope * 1.3 Of a plane tiling * 2 Principal vertex * 2.1 Ears * 2.2 Mouths * 3 Number of vertices of a polyhedron * 4 Vertices in computer graphics * 5 References * 6 External links DEFINITIONOF AN ANGLE A vertex of an angle is the endpoint where two line segments or rays come together. The vertex of an angle is the point where two rays begin or meet, where two line segments join or meet, where two lines intersect (cross), or any appropriate combination of rays, segments and lines that result in two straight "sides" meeting at one place
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Geometry
GEOMETRY (from the Ancient Greek : γεωμετρία; _geo-_ "earth", _-metron_ "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer . Geometry arose independently in a number of early cultures as a practical way for dealing with lengths , areas , and volumes . Geometry began to see elements of formal mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into an axiomatic form by Euclid , whose treatment, Euclid\'s _Elements_ , set a standard for many centuries to follow. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC
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List Of Regular Polytopes
This page lists the regular polytopes and regular polytope compounds in Euclidean , spherical and hyperbolic spaces. The Schläfli symbol describes every regular tessellation of an n-sphere, Euclidean and hyperbolic spaces. A Schläfli symbol describing an n-polytope equivalently describes a tessellation of an (n−1)-sphere. In addition, the symmetry of a regular polytope or tessellation is expressed as a Coxeter group , which Coxeter
Coxeter
expressed identically to the Schläfli symbol, except delimiting by square brackets, a notation that is called Coxeter
Coxeter
notation . Another related symbol is the Coxeter-Dynkin diagram which represents a symmetry group with no rings, and the represents regular polytope or tessellation with a ring on the first node
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Ludwig Schläfli
LUDWIG SCHLäFLI (15 January 1814 – 20 March 1895) was a Swiss mathematician, specialising in geometry and complex analysis (at the time called function theory) who was one of the key figures in developing the notion of higher-dimensional spaces. The concept of multidimensionality has come to play a pivotal role in physics , and is a common element in science fiction. CONTENTS* 1 Life and career * 1.1 Youth and education * 1.2 Teaching * 1.3 Later life * 2 Higher dimensions * 3 Polytopes * 4 See also * 5 References * 6 External links LIFE AND CAREERYOUTH AND EDUCATIONLudwig spent most of his life in Switzerland
Switzerland
. He was born in Grasswil (now part of Seeberg ), his mother's hometown. The family then moved to the nearby Burgdorf , where his father worked as a tradesman . His father wanted Ludwig to follow in his footsteps, but Ludwig was not cut out for practical work
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Recursive Definition
A RECURSIVE DEFINITION (or INDUCTIVE DEFINITION) in mathematical logic and computer science is used to define the elements in a set in terms of other elements in the set (Aczel 1978:740ff). A recursive definition of a function defines values of the functions for some inputs in terms of the values of the same function for other inputs. For example, the factorial function n! is defined by the rules 0! = 1. (n+1)! = (n+1)·n!. This definition is valid for all n, because the recursion eventually reaches the BASE CASE of 0. The definition may also be thought of as giving a procedure describing how to construct the function n!, starting from n = 0 and proceeding onwards with n = 1, n = 2, n = 3 etc.. The recursion theorem states that such a definition indeed defines a function. The proof uses mathematical induction . An inductive definition of a set describes the elements in a set in terms of other elements in the set
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Regular Polygon
Regular polygons Edges and vertices _n_ Schläfli symbol {_n_} Coxeter–Dynkin diagram Symmetry group Dn , order 2n Dual polygon Self-dual Area
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Convex Set
In convex geometry , a CONVEX SET is a subset of an affine space that is closed under convex combinations . More specifically, in a Euclidean space
Euclidean space
, a CONVEX REGION is a region where, for every pair of points within the region, every point on the straight line segment that joins the pair of points is also within the region. For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex. The boundary of a convex set is always a convex curve . The intersection of all convex sets containing a given subset A of Euclidean space
Euclidean space
is called the convex hull of A. It is the smallest convex set containing A. A convex function is a real-valued function defined on an interval with the property that its epigraph (the set of points on or above the graph of the function) is a convex set
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Equilateral Triangle
In geometry , an EQUILATERAL TRIANGLE is a triangle in which all three sides are equal. In the familiar Euclidean geometry , equilateral triangles are also equiangular ; that is, all three internal angles are also congruent to each other and are each 60°. They are regular polygons , and can therefore also be referred to as regular triangles
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Square (geometry)
In geometry , a SQUARE is a regular quadrilateral , which means that it has four equal sides and four equal angles (90-degree angles, or right angles ). It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted {displaystyle square } ABCD
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Regular Pentagon
In geometry , a PENTAGON (from the Greek πέντε pente and γωνία gonia, meaning five and 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
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Star Polygon
In geometry , a STAR POLYGON is a type of non-convex polygon . Only the REGULAR STAR POLYGONS have been studied in any depth; star polygons in general appear not to have been formally defined. Branko Grünbaum identified two primary definitions used by Kepler , one being the regular star polygons with intersecting edges that don't generate new vertices, and the second being simple isotoxal concave polygons . The first usage is included in polygrams which includes polygons like the pentagram but also compound figures like the hexagram
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Irreducible Fraction
An IRREDUCIBLE FRACTION (or FRACTION IN LOWEST TERMS or REDUCED FRACTION) is a fraction in which the numerator and denominator are integers that have no other common divisors than 1 (and -1, when negative numbers are considered). In other words, a fraction a⁄b is irreducible if and only if a and b are coprime , that is, if a and b have a greatest common divisor of 1. In higher mathematics , "IRREDUCIBLE FRACTION" may also refer to rational fractions such that the numerator and the denominator are coprime polynomials . Every positive rational number can be represented as an irreducible fraction in exactly one way. An equivalent definition is sometimes useful: if a, b are integers, then the fraction a⁄b is irreducible if and only if there is no other equal fraction c⁄d such that c < a or d < b, where a means the absolute value of a. (Two fractions a⁄b and c⁄d are equal or equivalent if and only if ad = bc.) For example, 1⁄4, 5⁄6, and −101⁄100 are all irreducible fractions
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Pentagram
A PENTAGRAM (sometimes known as a PENTALPHA or PENTANGLE or a STAR PENTAGON ) is the shape of a five-pointed star drawn with five straight strokes. Pentagrams were used symbolically in ancient Greece and Babylonia
Babylonia
, and are used today as a symbol of faith by many Wiccans , akin to the use of the cross by Christians and the Star of David
Star of David
by Jews. The pentagram has magical associations, and many people who practice Neopagan faiths wear jewelry incorporating the symbol. Christians once more commonly used the pentagram to represent the five wounds of Jesus . The pentagram has associations with Freemasonry and is also utilized by other belief systems. The word pentagram comes from the Greek word πεντάγραμμον (pentagrammon), from πέντε (pente), "five" + γραμμή (grammē), "line"
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