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6simplex f5 = 7, f4 = 21, C = 35, F = 35, E = 21, V = 7 (χ=0) Coxeter group A6, , order 5040 Bowers name and (acronym) Heptapeton (hop) Vertex figure 5simplex Circumradius 0.645497 Properties convex , isogonal selfdual In geometry , a 6simplex 6simplex is a selfdual regular 6polytope . It has 7 vertices , 21 edges , 35 triangle faces , 35 tetrahedral cells , 21 5cell 4faces, and 7 5simplex 5faces. Its dihedral angle is cos−1(1/6), or approximately 80.41°. CONTENTS * 1 Alternate names * 2 Coordinates * 3 Images * 4 Related uniform 6polytopes * 5 Notes * 6 References * 7 External links ALTERNATE NAMESIt can also be called a HEPTAPETON, or HEPTA6TOPE, as a 7facetted polytope in 6dimensions. The name heptapeton is derived from hepta for seven facets in Greek and peta for having fivedimensional facets, and on [...More...]  "6simplex" 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 (geometry) 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 (geometry)" on: Wikipedia Yahoo 

Cell (mathematics) In solid geometry , a FACE is a flat (planar ) surface that forms part of the boundary of a solid object; a threedimensional solid bounded exclusively by flat faces is a polyhedron . In more technical treatments of the geometry of polyhedra and higherdimensional polytopes , the term is also used to mean an element of any dimension of a more general polytope (in any number of dimensions). CONTENTS* 1 Polygonal face * 1.1 Number of polygonal faces of a polyhedron * 2 kface * 2.1 Cell or 3face * 2.2 Facet or (n1)face * 2.3 Ridge or (n2)face * 2.4 Peak or (n3)face * 3 See also * 4 References * 5 External links POLYGONAL FACEIn elementary geometry, a FACE is a polygon on the boundary of a polyhedron . Other names for a polygonal face include SIDE of a polyhedron, and TILE of a Euclidean plane tessellation . For example, any of the six squares that bound a cube is a face of the cube [...More...]  "Cell (mathematics)" on: Wikipedia Yahoo 

Face (geometry) In solid geometry , a FACE is a flat (planar ) surface that forms part of the boundary of a solid object; a threedimensional solid bounded exclusively by flat faces is a polyhedron . In more technical treatments of the geometry of polyhedra and higherdimensional polytopes , the term is also used to mean an element of any dimension of a more general polytope (in any number of dimensions). CONTENTS* 1 Polygonal face * 1.1 Number of polygonal faces of a polyhedron * 2 kface * 2.1 Cell or 3face * 2.2 Facet or (n1)face * 2.3 Ridge or (n2)face * 2.4 Peak or (n3)face * 3 See also * 4 References * 5 External links POLYGONAL FACEIn elementary geometry, a FACE is a polygon on the boundary of a polyhedron . Other names for a polygonal face include SIDE of a polyhedron, and TILE of a Euclidean plane tessellation . For example, any of the six squares that bound a cube is a face of the cube [...More...]  "Face (geometry)" on: Wikipedia Yahoo 

Edge (geometry) In geometry , an EDGE is a particular type of line segment joining two vertices in a polygon , polyhedron , or higherdimensional polytope . In a polygon, an edge is a line segment on the boundary, and is often called a SIDE. In a polyhedron or more generally a polytope, an edge is a line segment where two faces meet. A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal . CONTENTS * 1 Relation to edges in graphs * 2 Number of edges in a polyhedron * 3 Incidences with other faces * 4 Alternative terminology * 5 See also * 6 References * 7 External links RELATION TO EDGES IN GRAPHSIn graph theory , an edge is an abstract object connecting two graph vertices , unlike polygon and polyhedron edges which have a concrete geometric representation as a line segment [...More...]  "Edge (geometry)" 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 

Greek Language GREEK ( Modern Greek : ελληνικά , elliniká, "Greek", ελληνική γλώσσα ( listen ), ellinikí glóssa, "Greek language") is an independent branch of the IndoEuropean family of languages, native to Greece Greece and other parts of the Eastern Mediterranean . It has the longest documented history of any living IndoEuropean language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the major part of its history; other systems, such as Linear B and the Cypriot syllabary , were used previously. The alphabet arose from the Phoenician script and was in turn the basis of the Latin Latin , Cyrillic Cyrillic , Armenian , Coptic , Gothic and many other writing systems [...More...]  "Greek Language" 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 

Coxeter Plane In mathematics , the COXETER NUMBER h is the order of a COXETER ELEMENT of an irreducible Coxeter group Coxeter group . It is named after H.S.M. Coxeter . CONTENTS * 1 Definitions * 2 Group order * 3 Coxeter elements * 4 Coxeter plane * 5 See also * 6 Notes * 7 References DEFINITIONSNote that this article assumes a finite Coxeter group. For infinite Coxeter groups, there are multiple conjugacy classes of Coxeter elements, and they have infinite order. There are many different ways to define the Coxeter number h of an irreducible root system. A COXETER ELEMENT is a product of all simple reflections. The product depends on the order in which they are taken, but different orderings produce conjugate elements , which have the same order . * The Coxeter number is the number of roots divided by the rank [...More...]  "Coxeter Plane" 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 

Cartesian Coordinate A CARTESIAN COORDINATE SYSTEM is a coordinate system that specifies each point uniquely in a plane by a pair of numerical COORDINATES, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length . Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin , usually at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin [...More...]  "Cartesian Coordinate" on: Wikipedia Yahoo 

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 [...More...]  "Vertex (geometry)" on: Wikipedia Yahoo 

Peta PETA /ˈpɛ.tə/ is a decimal unit prefix in the metric system denoting multiplication by 1015 (1000000000000000). It was adopted as an SI prefix SI prefix in the International System of Units International System of Units in 1975, and has the symbol P. Peta is derived from the Greek πέντε, meaning "five". It denotes the fifth power of 1000 (10005). It is similar to the prefix penta ("five"), but without the letter n (on the analogy of the prefix tera (from the Greek for "monster") for 10004 looking like tetra ("four") with a letter missing) [...More...]  "Peta" on: Wikipedia Yahoo 

Vertex Figure In geometry , a VERTEX FIGURE, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. CONTENTS* 1 Definitions – theme and variations * 1.1 As a flat slice * 1.2 As a spherical polygon * 1.3 As the set of connected vertices * 1.4 Abstract definition * 2 General properties * 3 Dorman Luke construction * 4 Regular polytopes * 5 An example vertex figure of a honeycomb * 6 Edge figure * 7 See also * 8 References * 8.1 Notes * 8.2 Bibliography * 9 External links DEFINITIONS – THEME AND VARIATIONSTake some vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw lines across the connected faces, joining adjacent points. When done, these lines form a complete circuit, i.e. a polygon, around the vertex. This polygon is the vertex figure. More precise formal definitions can vary quite widely, according to circumstance [...More...]  "Vertex Figure" on: Wikipedia Yahoo 

Isogonal Figure In geometry , a polytope (a polygon , polyhedron or tiling, for example) is ISOGONAL or VERTEXTRANSITIVE if, loosely speaking, all its vertices are equivalent. That implies that each vertex is surrounded by the same kinds of face in the same or reverse order, and with the same angles between corresponding faces. Technically, we say that for any two vertices there exists a symmetry of the polytope mapping the first isometrically onto the second. Other ways of saying this are that the group of automorphisms of the polytope is transitive on its vertices, or that the vertices lie within a single symmetry orbit . All vertices of a finite ndimensional isogonal figure exist on an (n1)sphere . The term ISOGONAL has long been used for polyhedra. VERTEXTRANSITIVE is a synonym borrowed from modern ideas such as symmetry groups and graph theory [...More...]  "Isogonal Figure" on: Wikipedia Yahoo 