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5cell In geometry , the 5CELL is a fourdimensional object bounded by 5 tetrahedral cells . It is also known as a C5, PENTACHORON, PENTATOPE, PENTAHEDROID, or TETRAHEDRAL PYRAMID . It is a 4SIMPLEX , the simplest possible convex regular 4polytope (fourdimensional 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 5CELL is bounded by regular tetrahedra , and is one of the six regular convex 4polytopes , represented by Schläfli symbol {3,3,3}. CONTENTS * 1 Alternative names * 2 Geometry Geometry * 2.1 Construction * 2.2 Boerdijk–Coxeter helix * 2.3 Projections * 3 Irregular 5cell 5cell * 4 Compound * 5 Related polytopes and honeycomb * 6 References * 7 External links ALTERNATIVE NAMES * Pentachoron * 4simplex * Pentatope * Pentahedroid (Henry Parker Manning) * Pen (Jonathan Bowers: for pentachoron) * Hyperpyramid, tetrahedral pyramidGEOMETRYThe 5cell 5cell is selfdual , and its vertex figure is a tetrahedron. Its maximal intersection with 3dimensional space is the triangular prism . Its dihedral angle is cos−1(1/4), or approximately 75.52° [...More...]  "5cell" on: Wikipedia Yahoo 

Pentatope Number A PENTATOPE NUMBER is a number in the fifth cell of any row of Pascal\'s triangle starting with the 5term 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 3spheres. 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 . To be precise, the (3k − 2)th pentatope number is always the ((3k2 − k)/2)th pentagonal number and the (3k − 1)th pentatope number is always the ((3k2 + k)/2)th pentagonal number. The 3kth pentatope number is the generalized pentagonal number obtained by taking the negative index −(3k2 + k)/2 in the formula for pentagonal numbers. (These expressions always give integers). The infinite sum of the reciprocals of all pentatopal numbers is 4 3 {displaystyle 4 over 3} . This can be derived using telescoping series [...More...]  "Pentatope Number" 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. The diagram is 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 is a projection of a polyhedron into a plane figure and a projection of a 4polytope 4polytope to 3space , respectively. As such, Schlegel diagrams are commonly used as a means of visualizing fourdimensional polytopes. CONTENTS * 1 Construction * 2 Examples * 3 See also * 4 References * 5 Further reading * 6 External links CONSTRUCTIONThe most elementary Schlegel diagram, that of a polyhedron, was described by Duncan Sommerville as follows: A very useful method of representing a convex polyhedron is by plane projection. If it is projected from any external point, since each ray cuts it twice, it will be represented by a polygonal area divided twice over into polygons. It is always possible by suitable choice of the centre of projection to make the projection of one face completely contain the projections of all the other faces [...More...]  "Schlegel Diagram" on: Wikipedia Yahoo 

Convex Regular 4polytope In mathematics , a REGULAR 4POLYTOPE is a regular fourdimensional polytope . They are the fourdimensional analogs of the regular polyhedra in three dimensions and the regular polygons in two dimensions. Regular 4polytopes were first described by the Swiss mathematician Ludwig Schläfli Ludwig Schläfli in the mid19th century, although the full set were not discovered until later. There are six convex and ten star regular 4polytopes, giving a total of sixteen. CONTENTS * 1 History * 2 Construction * 3 Regular convex 4polytopes * 3.1 Properties * 3.2 Visualization * 4 Regular star (Schläfli–Hess) 4polytopes * 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 4polytopes were first described by the Swiss mathematician Ludwig Schläfli Ludwig Schläfli in the mid19th century. He discovered that there are precisely six such figures. Schläfli also found four of the regular star 4polytopes: the grand 120cell , great stellated 120cell , grand 600cell , and great grand stellated 120cell ). He skipped the remaining six because he would not allow forms that failed the Euler characteristic on cells or vertex figures (for zerohole tori: F − E + V = 2). That excludes cells and vertex figures as {5,5/2} and {5/2,5} [...More...]  "Convex Regular 4polytope" on: Wikipedia Yahoo 

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 19thcentury Swiss mathematician Ludwig Schläfli , who made important contributions in geometry and other areas. CONTENTS * 1 Description * 2 Cases * 2.1 Symmetry groups * 2.2 Regular polygons (plane) * 2.3 Regular polyhedra (3 dimensions) * 2.4 Regular 4polytopes (4 dimensions) * 2.5 Regular npolytopes (higher dimensions) * 2.6 Dual polytopes * 2.7 Prismatic polytopes * 3 Extension of Schläfli symbols * 3.1 Polygons and circle tilings * 3.2 Polyhedra and tilings * 3.2.1 Alternations, quarters and snubs * 3.2.2 Altered and holosnubbed * 3.3 Polychora and honeycombs * 3.3.1 Alternations, quarters and snubs * 3.3.2 Bifurcating families * 4 See also * 5 References * 6 Sources * 7 External links DESCRIPTIONThe Schläfli symbol Schläfli symbol is a recursive description, starting with {p} for a psided regular polygon that is convex . For example, {3} is an equilateral triangle , {4} is a square , {5} a convex regular pentagon and so on. Regular star polygons are not convex, and their Schläfli symbols {p/q} contain irreducible fractions p/q, where p is the number of vertices. For example, {5/2} is a pentagram [...More...]  "Schläfli Symbol" on: Wikipedia Yahoo 

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 order3. 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. Dynkin diagrams correspond to and are used to classify root systems and therefore semisimple Lie algebras [...More...]  "Coxeter Diagram" on: Wikipedia Yahoo 

3face 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 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. Sometimes "face" is also used to refer to the 2dimensional features of a 4polytope 4polytope . With this meaning, the 4dimensional tesseract has 24 square faces, each sharing two of 8 cubic cells. Regular examples by Schläfli symbol POLYHEDRON STAR POLYHEDRON EUCLIDEAN TILING HYPERBOLIC TILING 4POLYTOPE {4,3} {5/2,5} {4,4} {4,5} {4,3,3} The cube has 3 square faces per vertex [...More...]  "3face" on: Wikipedia Yahoo 

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 threedimensional case of the more general concept of a Euclidean simplex , and may thus also be called a 3SIMPLEX. 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 . For any tetrahedron there exists a sphere (called the circumsphere ) on which all four vertices lie, and another sphere (the insphere ) tangent to the tetrahedron's faces [...More...]  "Tetrahedron" on: Wikipedia Yahoo 

2face 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 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. Sometimes "face" is also used to refer to the 2dimensional features of a 4polytope . With this meaning, the 4dimensional tesseract has 24 square faces, each sharing two of 8 cubic cells. Regular examples by Schläfli symbol Schläfli symbol POLYHEDRON STAR POLYHEDRON EUCLIDEAN TILING HYPERBOLIC TILING 4POLYTOPE {4,3} {5/2,5} {4,4} {4,5} {4,3,3} The cube has 3 square faces per vertex [...More...]  "2face" 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. However, any polyhedron can be represented by its skeleton or edgeskeleton, a graph whose vertices are the geometric vertices of the polyhedron and whose edges correspond to the geometric edges. Conversely, the graphs that are skeletons of threedimensional polyhedra can be characterized by Steinitz\'s theorem as being exactly the 3vertexconnected planar graphs . NUMBER OF EDGES IN A POLYHEDRONAny convex polyhedron 's surface has Euler characteristic V E + F = 2 , {displaystyle VE+F=2,} where V is the number of vertices , E is the number of edges, and F is the number of faces [...More...]  "Edge (geometry)" 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. OF A POLYTOPEA vertex is a corner point of a polygon , polyhedron , or other higherdimensional polytope , formed by the intersection of edges , faces or facets of the object. In a polygon, a vertex is called "convex " if the internal angle of the polygon, that is, the angle formed by the two edges at the vertex, with the polygon inside the angle, is less than π radians ( 180°, two right angles ) ; otherwise, it is called "concave" or "reflex" [...More...]  "Vertex (geometry)" 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. For example Coxeter (e.g. 1948, 1954) varies his definition as convenient for the current area of discussion. Most of the following definitions of a vertex figure apply equally well to infinite tilings , or spacefilling tessellation with polytope cells . AS A FLAT SLICEMake a slice through the corner of the polyhedron, cutting through all the edges connected to the vertex. The cut surface is the vertex figure. This is perhaps the most common approach, and the most easily understood [...More...]  "Vertex Figure" on: Wikipedia Yahoo 

Petrie Polygon In geometry , a PETRIE POLYGON for a regular polytope of n dimensions is a skew polygon such that every (n1) 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 Petrie polygon becomes a regular polygon with the remainder of the projection interior to it. The plane in question is the Coxeter plane Coxeter plane of the symmetry group of the polygon, and the number of sides, h, is Coxeter number of the Coxeter group Coxeter group . These polygons and projected graphs are useful in visualizing symmetric structure of the higherdimensional regular polytopes. CONTENTS * 1 History * 2 The Petrie polygons of the regular polyhedra * 3 The Petrie polygon Petrie polygon of regular polychora (4polytopes) * 4 The Petrie polygon Petrie polygon projections of regular and uniform polytopes * 5 Notes * 6 References * 7 External links HISTORY The Petrie polygon Petrie polygon for a cube is a skew hexagon passing through 6 of 8 vertices [...More...]  "Petrie Polygon" on: Wikipedia Yahoo 

Pentagon In geometry , a PENTAGON (from the Greek πέντε pente and γωνία gonia, meaning five and angle ) is any fivesided polygon or 5gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simple or selfintersecting . A selfintersecting regular pentagon (or star pentagon) is called a pentagram . CONTENTS* 1 Regular pentagons * 1.1 Derivation of the area formula * 1.2 Inradius * 1.3 Chords from the circumscribed circle to the vertices * 1.4 Construction of a regular pentagon * 1.4.1 Richmond\'s method * 1.4.2 Carlyle circles * 1.4.3 Using trigonometry and the Pythagorean Theorem * 1.4.3.1 The construction * 1.4.4 † Proof that cos 36° = 1 + 5 4 {displaystyle {tfrac {1+{sqrt {5}}}{4}}} * 1.4.5 Side length is given * 1.4.5.1 The golden ratio * 1.5 Euclid\'s method * 1.5.1 Simply using a protractor (not a classical construction) * 1.6 Physical methods * 1.7 Symmetry * 2 Equilateral pentagons * 3 Cyclic pentagons * 4 General convex pentagons * 5 Graphs * 6 Examples of pentagons * 6.1 Plants * 6.2 Animals * 6.3 Artificial * 7 Pentagons in tiling * 8 Pentagons in polyhedra * 9 See also * 10 Inline notes and references * 11 External links REGULAR PENTAGONSA regular pentagon has Schläfli symbol Schläfli symbol {5} and interior angles are 108° [...More...]  "Pentagon" on: Wikipedia Yahoo 

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 ). Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the symmetry groups of regular polytopes , and the Weyl groups of simple Lie algebras . Examples of infinite Coxeter groups include the triangle groups corresponding to regular tessellations of the Euclidean plane and the hyperbolic plane , and the Weyl groups of infinitedimensional Kac–Moody algebras . Standard references include (Humphreys 1992 ) and (Davis 2007 ) [...More...]  "Coxeter Group" on: Wikipedia Yahoo 

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 