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5-cell
In geometry , the 5-CELL is a four-dimensional object bounded by 5 tetrahedral cells . It is also known as a C5, PENTACHORON, PENTATOPE, PENTAHEDROID, or TETRAHEDRAL PYRAMID . It is a 4-SIMPLEX , the simplest possible convex regular 4-polytope (four-dimensional analogue of a Platonic solid
Platonic solid
), and is analogous to the tetrahedron in three dimensions and the triangle in two dimensions. The pentachoron is a four dimensional pyramid with a tetrahedral base. The REGULAR 5-CELL is bounded by regular tetrahedra , and is one of the six regular convex 4-polytopes , represented by Schläfli symbol {3,3,3}
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Pentatope Number
A PENTATOPE NUMBER is a number in the fifth cell of any row of Pascal\'s triangle starting with the 5-term row 1 4 6 4 1 either from left to right or from right to left. The first few numbers of this kind are : 1 , 5 , 15 , 35 , 70 , 126 , 210, 330, 495 , 715, 1001 , 1365 (sequence A000332 in the OEIS ) A pentatope with side length 5 contains 70 3-spheres. Each layer represents one of the first five tetrahedral numbers . For example the bottom (green) layer has 35 spheres in total. Pentatope numbers belong in the class of figurate numbers , which can be represented as regular, discrete geometric patterns. The formula for the nth pentatopic number is: ( n + 3 4 ) = n ( n + 1 ) ( n + 2 ) ( n + 3 ) 24 = n 4 4 ! . {displaystyle {n+3 choose 4}={frac {n(n+1)(n+2)(n+3)}{24}}={n^{overline {4}} over 4!}.} Two of every three pentatope numbers are also pentagonal numbers
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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^{d-1}} 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^{d-1}} 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 4-polytope
4-polytope
to 3-space , respectively. As such, Schlegel diagrams are commonly used as a means of visualizing four-dimensional polytopes
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Convex Regular 4-polytope
In mathematics , a REGULAR 4-POLYTOPE is a regular four-dimensional polytope . They are the four-dimensional analogs of the regular polyhedra in three dimensions and the regular polygons in two dimensions. Regular 4-polytopes were first described by the Swiss mathematician Ludwig Schläfli
Ludwig Schläfli
in the mid-19th century, although the full set were not discovered until later. There are six convex and ten star regular 4-polytopes, giving a total of sixteen. CONTENTS * 1 History * 2 Construction * 3 Regular convex 4-polytopes * 3.1 Properties * 3.2 Visualization * 4 Regular star (Schläfli–Hess) 4-polytopes * 4.1 Names * 4.2 Symmetry * 4.3 Properties * 5 See also * 6 References * 6.1 Citations * 6.2 Bibliography * 7 External links HISTORYThe convex regular 4-polytopes were first described by the Swiss mathematician Ludwig Schläfli
Ludwig Schläfli
in the mid-19th century
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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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Coxeter 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 order-3. Each diagram represents a 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
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3-face
In solid geometry , a FACE is a flat (planar ) surface that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by flat faces is a polyhedron . In more technical treatments of the geometry of polyhedra and higher-dimensional 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 _k_-face * 2.1 Cell or 3-face
3-face
* 2.2 Facet or (_n_-1)-face * 2.3 Ridge or (_n_-2)-face * 2.4 Peak or (_n_-3)-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
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Tetrahedron
In geometry , a TETRAHEDRON (plural: TETRAHEDRA or TETRAHEDRONS), also known as a TRIANGULAR PYRAMID, is a polyhedron composed of four triangular faces , six straight edges , and four vertex corners . The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces. The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex , and may thus also be called a 3-SIMPLEX. The tetrahedron is one kind of pyramid , which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid". Like all convex polyhedra , a tetrahedron can be folded from a single sheet of paper. It has two such nets
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2-face
In solid geometry , a FACE is a flat (planar ) surface that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by flat faces is a polyhedron . In more technical treatments of the geometry of polyhedra and higher-dimensional 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 k-face * 2.1 Cell or 3-face * 2.2 Facet or (n-1)-face * 2.3 Ridge or (n-2)-face * 2.4 Peak or (n-3)-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
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Edge (geometry)
In geometry , an EDGE is a particular type of line segment joining two vertices in a polygon , polyhedron , or higher-dimensional 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
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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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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. For example Coxeter (e.g
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Petrie Polygon
In geometry , a PETRIE POLYGON for a regular polytope of _n_ dimensions is a skew polygon such that every (_n_-1) consecutive side (but no _n_) belongs to one of the facets . The PETRIE POLYGON of a regular polygon is the regular polygon itself; that of a regular polyhedron is a skew polygon such that every two consecutive side (but no three) belongs to one of the faces . For every regular polytope there exists an orthogonal projection onto a plane such that one Petrie polygon becomes a regular polygon with the remainder of the projection interior to it. The plane in question is the Coxeter plane of the symmetry group of the polygon, and the number of sides, _h,_ is Coxeter number of the Coxeter group . These polygons and projected graphs are useful in visualizing symmetric structure of the higher-dimensional regular polytopes
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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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Coxeter Group
In mathematics , a COXETER GROUP, named after H. S. M. Coxeter , is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors ). Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups ; the symmetry groups of regular polyhedra are an example. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced (Coxeter 1934 ) as abstractions of reflection groups , and finite Coxeter groups were classified in 1935 (Coxeter 1935 )
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Dual Polytope
In geometry , any polyhedron is associated with a second DUAL figure, where the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra , but not all are also geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron. Duality preserves the symmetries of a polyhedron. Therefore, for many classes of polyhedra defined by their symmetries, the duals also belong to a symmetric class. Thus, the regular polyhedra – the (convex) Platonic solids and (star) Kepler-Poinsot polyhedra – form dual pairs, where the regular tetrahedron is self-dual . The dual of an isogonal polyhedron, having equivalent vertices, is one which is isohedral, having equivalent faces. The dual of an isotoxal polyhedron (having equivalent edges) is also isotoxal
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