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5simplex In fivedimensional geometry , a 5simplex 5simplex is a selfdual regular 5polytope 5polytope . It has six vertices , 15 edges , 20 triangle faces , 15 tetrahedral cells , and 6 5cell 5cell facets . It has a dihedral angle of cos−1(1/5), or approximately 78.46°. CONTENTS * 1 Alternate names * 2 Regular hexateron cartesian coordinates * 3 Projected images * 4 Related uniform 5polytopes * 5 Other forms * 6 Notes * 7 References * 8 External links ALTERNATE NAMESIt can also be called a HEXATERON, or HEXA5TOPE, as a 6facetted polytope in 5dimensions. The name hexateron is derived from hexa for having six facets and teron (with ter being a corruption of tetra ) for having fourdimensional facets. By Jonathan Bowers, a hexateron is given the acronym HIX [...More...]  "5simplex" on: Wikipedia Yahoo 

Dihedral Symmetry In mathematics , a DIHEDRAL GROUP is the group of symmetries of a regular polygon , which includes rotations and reflections . Dihedral groups are among the simplest examples of finite groups , and they play an important role in group theory , geometry , and chemistry . The notation for the dihedral group of order n differs in geometry and abstract algebra . In geometry , Dn or Dihn refers to the symmetries of the ngon , a group of order 2n. In abstract algebra , Dn refers to the dihedral group of order n. The geometric convention is used in this article [...More...]  "Dihedral Symmetry" on: Wikipedia Yahoo 

Stereographic Projection In geometry , the STEREOGRAPHIC PROJECTION is a particular mapping (function ) that projects a sphere onto a plane . The projection is defined on the entire sphere, except at one point: the projection point. Where it is defined, the mapping is smooth and bijective . It is conformal , meaning that it preserves angles at which curves meet. It is neither isometric nor areapreserving: that is, it preserves neither distances nor the areas of figures. Intuitively, then, the stereographic projection is a way of picturing the sphere as the plane, with some inevitable compromises. Because the sphere and the plane appear in many areas of mathematics and its applications, so does the stereographic projection; it finds use in diverse fields including complex analysis , cartography , geology , and photography . In practice, the projection is carried out by computer or by hand using a special kind of graph paper called a STEREOGRAPHIC NET, shortened to STEREONET, or WULFF NET [...More...]  "Stereographic Projection" on: Wikipedia Yahoo 

Schlegel Diagram In geometry , a SCHLEGEL DIAGRAM is a projection of a polytope from R d {displaystyle R^{d}} into R d 1 {displaystyle R^{d1}} through a point beyond one of its facets or faces . The resulting entity is a polytopal subdivision of the facet in R d 1 {displaystyle R^{d1}} that is combinatorially equivalent to the original polytope. Named for Victor Schlegel , who in 1886 introduced this tool for studying combinatorial and topological properties of polytopes. In dimensions 3 and 4, a Schlegel diagram Schlegel diagram is a projection of a polyhedron into a plane figure and a projection of a 4polytope to 3space , respectively. As such, Schlegel diagrams are commonly used as a means of visualizing fourdimensional polytopes [...More...]  "Schlegel Diagram" on: Wikipedia Yahoo 

Coxeter HAROLD SCOTT MACDONALD "DONALD" COXETER, FRS, FRSC, CC (February 9, 1907 – March 31, 2003) was a Britishborn Canadian geometer. Coxeter is regarded as one of the greatest geometers of the 20th century. He was born in London London but spent most of his adult life in Canada Canada . He was always called Donald, from his third name MacDonald. CONTENTS * 1 Biography * 2 Awards * 3 Works * 4 See also * 5 References * 6 Further reading * 7 External links BIOGRAPHYIn his youth, Coxeter composed music and was an accomplished pianist at the age of 10. He felt that mathematics and music were intimately related, outlining his ideas in a 1962 article on "Mathematics and Music" in the Canadian Music Journal. He worked for 60 years at the University of Toronto University of Toronto and published twelve books . He was most noted for his work on regular polytopes and higherdimensional geometries [...More...]  "Coxeter" on: Wikipedia Yahoo 

Orthographic Projection ORTHOGRAPHIC PROJECTION (sometimes ORTHOGONAL PROJECTION), is a means of representing threedimensional objects in two dimensions . It is a form of parallel projection , in which all the projection lines are orthogonal to the projection plane , resulting in every plane of the scene appearing in affine transformation on the viewing surface. The obverse of an orthographic projection is an oblique projection , which is a parallel projection in which the projection lines are not orthogonal to the projection plane. The term orthographic is sometimes reserved specifically for depictions of objects where the principal axes or planes of the object are also parallel with the projection plane, but these are better known as multiview projections [...More...]  "Orthographic Projection" on: Wikipedia Yahoo 

Hyperplane In geometry a HYPERPLANE is a subspace of one dimension less than its ambient space . If a space is 3dimensional then its hyperplanes are the 2dimensional planes , while if the space is 2dimensional, its hyperplanes are the 1dimensional lines . This notion can be used in any general space in which the concept of the dimension of a subspace is defined. In different settings, the objects which are hyperplanes may have different properties. For instance, a hyperplane of an ndimensional affine space is a flat subset with dimension n − 1. By its nature, it separates the space into two half spaces . But a hyperplane of an ndimensional projective space does not have this property [...More...]  "Hyperplane" on: Wikipedia Yahoo 

Dihedral Angle A DIHEDRAL ANGLE is the angle between two intersecting planes. In chemistry it is the angle between planes through two sets of three atoms, having two atoms in common. In solid geometry it is defined as the union of a line and two halfplanes that have this line as a common edge . In higher dimension, a dihedral angle represents the angle between two hyperplanes . CONTENTS * 1 Definitions * 2 Dihedral angles in stereochemistry * 3 Dihedral angles of proteins * 3.1 Converting from dihedral angles to Cartesian coordinates in chains * 4 Calculation of a dihedral angle * 5 Dihedral angles in polyhedra * 6 See also * 7 References * 8 External links DEFINITIONSA dihedral angle is an angle between two intersecting planes on a third plane perpendicular to the line of intersection [...More...]  "Dihedral Angle" on: Wikipedia Yahoo 

Facet (mathematics) In geometry , a FACET is a feature of a polyhedron , polytope , or related geometric structure, generally of dimension one less than the structure itself. * In threedimensional geometry a FACET of a polyhedron is any polygon whose corners are vertices of the polyhedron, and is not a face . To FACET a polyhedron is to find and join such facets to form the faces of a new polyhedron; this is the reciprocal process to stellation and may also be applied to higherdimensional polytopes . * In polyhedral combinatorics and in the general theory of polytopes , a FACET of a polytope of dimension n is a face that has dimension n − 1. Facets may also be called (n − 1)faces. In threedimensional geometry, they are often called "faces" without qualification. * A FACET of a simplicial complex is a maximal simplex, that is a simplex that is not a face of another simplex of the complex [...More...]  "Facet (mathematics)" on: Wikipedia Yahoo 

Tetra NUMERAL or NUMBER PREFIXES are prefixes derived from numerals or occasionally other numbers . In English and other European languages, they are used to coin numerous series of words, such as unicycle – bicycle – tricycle, dyad – triad – decade, biped – quadruped, September – October – November – December, decimal – hexadecimal, sexagenarian – octogenarian, centipede – millipede, etc. There are two principal systems, taken from Latin Latin and Greek , each with several subsystems; in addition, Sanskrit Sanskrit occupies a marginal position. There is also an international set of metric prefixes , which are used in the metric system , and which for the most part are either distorted from the forms below or not based on actual number words [...More...]  "Tetra" on: Wikipedia Yahoo 

Dihedron A DIHEDRON is a type of polyhedron , made of two polygon faces which share the same set of edges. In threedimensional Euclidean space Euclidean space , it is degenerate if its faces are flat, while in threedimensional spherical space , a dihedron with flat faces can be thought of as a lens, an example of which is the fundamental domain of a lens space L(p,q). Dihedra have also been called BIHEDRA, FLAT POLYHEDRA, or DOUBLY COVERED POLYGONS. A regular dihedron is the dihedron formed by two regular polygons , which may be described by the Schläfli symbol Schläfli symbol {n,2}. As a spherical polyhedron, each polygon of such a dihedron fills a hemisphere , with a regular ngon on a great circle equator between them. The dual of a ngonal dihedron is the ngonal hosohedron , where n digon faces share two vertices [...More...]  "Dihedron" on: Wikipedia Yahoo 

Fourdimensional Space A FOURDIMENSIONAL SPACE or 4D SPACE is the simplest possible generalization of the observation that one only needs three numbers, called dimensions, to describe the sizes or locations of objects in the everyday world. For example, the volume of a rectangular box is found by measuring its length (often labeled x), width (y), and depth (z). More than two millennia ago Greek philosophers explored in detail the many implications of this uniformity, culminating in Euclid\'s Elements . However, it was not until recent times that some mathematicians generalized the concept of dimensions to include more than three. The idea of adding a fourth dimension began with JosephLouis Lagrange JosephLouis Lagrange in the mid 1700s and culminated in a precise formalization of the concept in 1854 by Bernhard Riemann Bernhard Riemann [...More...]  "Fourdimensional Space" on: Wikipedia Yahoo 

Coxeter–Dynkin Diagram In geometry , a COXETER–DYNKIN DIAGRAM (or COXETER DIAGRAM, COXETER GRAPH) is a graph with numerically labeled edges (called BRANCHES) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes ). It describes a kaleidoscopic construction: each graph "node" represents a mirror (domain facet ) and the label attached to a branch encodes the dihedral angle order between two mirrors (on a domain ridge ). An unlabeled branch implicitly represents order3. Each diagram represents a Coxeter group Coxeter group , and Coxeter groups are classified by their associated diagrams. Dynkin diagrams are closely related objects, which differ from Coxeter diagrams in two respects: firstly, branches labeled "4" or greater are directed , while Coxeter diagrams are undirected ; secondly, Dynkin diagrams must satisfy an additional (crystallographic ) restriction, namely that the only allowed branch labels are 2, 3, 4, and 6 [...More...]  "Coxeter–Dynkin Diagram" on: Wikipedia Yahoo 

Coxeter Notation In geometry , COXETER NOTATION (also COXETER SYMBOL) is a system of classifying symmetry groups , describing the angles between with fundamental reflections of a Coxeter group Coxeter group in a bracketed notation, with modifiers to indicate certain subgroups. The notation is named after H. S. M. Coxeter H. S. M. Coxeter , and has been more comprehensively defined by Norman Johnson [...More...]  "Coxeter Notation" on: Wikipedia Yahoo 

Group Order In group theory , a branch of mathematics , the term order is used in two unrelated senses: * The ORDER of a group is its cardinality , i.e., the number of elements in its set . Also, the ORDER, sometimes PERIOD, of an element a of a group is the smallest positive integer m such that am = e (where e denotes the identity element of the group, and am denotes the product of m copies of a). If no such m exists, a is said to have infinite order. * The ordering relation of a partially or totally ordered group .This article is about the first sense of order. The order of a group G is denoted by ord(G) or G and the order of an element a is denoted by ord(a) or a. CONTENTS * 1 Example * 2 Order and structure * 3 Counting by order of elements * 4 In relation to homomorphisms * 5 Class equation * 6 Open questions * 7 See also * 8 References EXAMPLEEXAMPLE. The symmetric group S3 has the following multiplication table [...More...]  "Group Order" on: Wikipedia Yahoo 

Hosohedron In geometry , an ngonal HOSOHEDRON is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices. A regular ngonal hosohedron has Schläfli symbol Schläfli symbol {2, n}, with each spherical lune having internal angle 2π/n radians (360/n degrees) [...More...]  "Hosohedron" on: Wikipedia Yahoo 