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6-simplex
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
Vertex figure
5-simplex
5-simplex
Circumradius 0.645497 Properties convex , isogonal self-dual In geometry , a 6-simplex
6-simplex
is a self-dual regular 6-polytope . It has 7 vertices , 21 edges , 35 triangle faces , 35 tetrahedral cells , 21 5-cell
5-cell
4-faces, and 7 5-simplex
5-simplex
5-faces. Its dihedral angle is cos−1(1/6), or approximately 80.41°. CONTENTS * 1 Alternate names * 2 Coordinates * 3 Images * 4 Related uniform 6-polytopes * 5 Notes * 6 References * 7 External links ALTERNATE NAMESIt can also be called a HEPTAPETON, or HEPTA-6-TOPE, as a 7-facetted polytope in 6-dimensions. The name heptapeton is derived from hepta for seven facets in Greek and -peta for having five-dimensional facets, and -on. Jonathan Bowers gives a heptapeton the acronym HOP
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Uniform Polypeton
In six-dimensional geometry , a UNIFORM POLYPETON (or UNIFORM 6-POLYTOPE ) is a six-dimensional uniform polytope . A uniform polypeton is vertex-transitive , and all facets are uniform 5-polytopes . The complete set of CONVEX UNIFORM POLYPETA has not been determined, but most can be made as Wythoff constructions from a small set of symmetry groups . These construction operations are represented by the permutations of rings of the Coxeter-Dynkin diagrams . Each combination of at least one ring on every connected group of nodes in the diagram produces a uniform 6-polytope. The simplest uniform polypeta are regular polytopes : the 6-simplex {3,3,3,3,3}, the 6-cube
6-cube
(hexeract) {4,3,3,3,3}, and the 6-orthoplex (hexacross) {3,3,3,3,4}
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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. CONTENTS * 1 Description * 2 Cases * 2.1 Symmetry groups * 2.2 Regular polygons (plane) * 2.3 Regular polyhedra (3 dimensions) * 2.4 Regular 4-polytopes (4 dimensions) * 2.5 Regular n-polytopes (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 p-sided 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
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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. Dynkin diagrams correspond to and are used to classify root systems and therefore semisimple Lie algebras
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Euler Characteristic
In mathematics , and more specifically in algebraic topology and polyhedral combinatorics , the EULER CHARACTERISTIC (or EULER NUMBER, or EULER–POINCARé CHARACTERISTIC) is a topological invariant , a number that describes a topological space 's shape or structure regardless of the way it is bent. It is commonly denoted by {displaystyle chi } (Greek lower-case letter chi ). The Euler characteristic
Euler characteristic
was originally defined for polyhedra and used to prove various theorems about them, including the classification of the Platonic solids . Leonhard Euler
Leonhard Euler
, for whom the concept is named, was responsible for much of this early work. In modern mathematics, the Euler characteristic
Euler characteristic
arises from homology and, more abstractly, homological algebra
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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 ). 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 infinite-dimensional Kac–Moody algebras . Standard references include (Humphreys 1992 ) and (Davis 2007 )
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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. 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 space-filling 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
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5-simplex
In five-dimensional geometry , a 5-simplex
5-simplex
is a self-dual regular 5-polytope . It has six vertices , 15 edges , 20 triangle faces , 15 tetrahedral cells , and 6 5-cell 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 5-polytopes * 5 Other forms * 6 Notes * 7 References * 8 External links ALTERNATE NAMESIt can also be called a HEXATERON, or HEXA-5-TOPE, as a 6-facetted polytope in 5-dimensions. The name hexateron is derived from hexa- for having six facets and teron (with ter- being a corruption of tetra- ) for having four-dimensional facets. By Jonathan Bowers, a hexateron is given the acronym HIX
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Convex Polytope
A CONVEX POLYTOPE is a special case of a polytope , having the additional property that it is also a convex set of points in the n-dimensional space Rn. Some authors use the terms "convex polytope" and "CONVEX POLYHEDRON" interchangeably, while others prefer to draw a distinction between the notions of a polyhedron and a polytope. In addition, some texts require a polytope to be a bounded set , while others (including this article) allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue. Yet other texts treat a convex n-polytope as a surface or (n-1)-manifold. Convex polytopes play an important role both in various branches of mathematics and in applied areas, most notably in linear programming . A comprehensive and influential book in the subject, called Convex Polytopes, was published in 1967 by Branko Grünbaum
Branko Grünbaum
. In 2003 the 2nd edition of the book was published, with significant additional material contributed by new writers. In Grünbaum's book, and in some other texts in discrete geometry , convex polytopes are often simply called "polytopes". Grünbaum points out that this is solely to avoid the endless repetition of the word "convex", and that the discussion should throughout be understood as applying only to the convex variety. A polytope is called full-dimensional if it is an n-dimensional object in Rn
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Isogonal Figure
In geometry , a polytope (a polygon , polyhedron or tiling, for example) is ISOGONAL or VERTEX-TRANSITIVE 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 _n_-dimensional isogonal figure exist on an (n-1)-sphere . The term ISOGONAL has long been used for polyhedra. VERTEX-TRANSITIVE is a synonym borrowed from modern ideas such as symmetry groups and graph theory . The pseudorhombicuboctahedron – which is _not_ isogonal – demonstrates that simply asserting that "all vertices look the same" is not as restrictive as the definition used here, which involves the group of isometries preserving the polyhedron or tiling
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Self-dual
In mathematics , a DUALITY, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A. Such involutions sometimes have fixed points , so that the dual of A is A itself. For example, Desargues\' theorem is self-dual in this sense under the standard duality in projective geometry . In mathematical contexts, duality has numerous meanings although it is "a very pervasive and important concept in (modern) mathematics" and "an important general theme that has manifestations in almost every area of mathematics". Many mathematical dualities between objects of two types correspond to pairings , bilinear functions from an object of one type and another object of the second type to some family of scalars. For instance, linear algebra duality corresponds in this way to bilinear maps from pairs of vector spaces to scalars, the duality between distributions and the associated test functions corresponds to the pairing in which one integrates a distribution against a test function, and Poincaré duality corresponds similarly to intersection number , viewed as a pairing between submanifolds of a given manifold. From a category theory viewpoint, duality can also be seen as a functor , at least in the realm of vector spaces
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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. Islamic scientists preserved Greek ideas and expanded on them during the Middle Ages . By the early 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat . Since then, and into modern times, geometry has expanded into non- Euclidean geometry and manifolds , describing spaces that lie beyond the normal range of human experience. While geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry
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Simplex
In geometry , a SIMPLEX (plural: SIMPLEXES or SIMPLICES) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions . Specifically, a K-SIMPLEX is a k-dimensional polytope which is the convex hull of its k + 1 vertices . More formally, suppose the k + 1 points u 0 , , u k R k {displaystyle u_{0},dots ,u_{k}in mathbb {R} ^{k}} are affinely independent , which means u 1 u 0 , , u k u 0 {displaystyle u_{1}-u_{0},dots ,u_{k}-u_{0}} are linearly independent . Then, the simplex determined by them is the set of points C = { 0 u 0 + + k u k i = 0 k i = 1 and i 0 for all i } {displaystyle C=left{theta _{0}u_{0}+dots +theta _{k}u_{k}~{bigg }~sum _{i=0}^{k}theta _{i}=1{mbox{ and }}theta _{i}geq 0{mbox{ for all }}iright}} . For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron, and a 4-simplex is a 5-cell . A single point may be considered a 0-simplex, and a line segment may be considered a 1-simplex. A simplex may be defined as the smallest convex set containing the given vertices. A REGULAR SIMPLEX is a simplex that is also a regular polytope . A regular n-simplex may be constructed from a regular (n − 1)-simplex by connecting a new vertex to all original vertices by the common edge length
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Duality (mathematics)
In mathematics , a DUALITY, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A. Such involutions sometimes have fixed points , so that the dual of A is A itself. For example, Desargues\' theorem is self-dual in this sense under the standard duality in projective geometry . In mathematical contexts, duality has numerous meanings although it is "a very pervasive and important concept in (modern) mathematics" and "an important general theme that has manifestations in almost every area of mathematics". Many mathematical dualities between objects of two types correspond to pairings , bilinear functions from an object of one type and another object of the second type to some family of scalars. For instance, linear algebra duality corresponds in this way to bilinear maps from pairs of vector spaces to scalars, the duality between distributions and the associated test functions corresponds to the pairing in which one integrates a distribution against a test function, and Poincaré duality corresponds similarly to intersection number , viewed as a pairing between submanifolds of a given manifold. From a category theory viewpoint, duality can also be seen as a functor , at least in the realm of vector spaces
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Regular Polytope
In mathematics , a REGULAR POLYTOPE is a polytope whose symmetry group acts transitively on its flags , thus giving it the highest degree of symmetry. All its elements or j-faces (for all 0 ≤ j ≤ n, where n is the dimension of the polytope) — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of dimension ≤ n. Regular polytopes are the generalized analog in any number of dimensions of regular polygons (for example, the square or the regular pentagon) and regular polyhedra (for example, the cube ). The strong symmetry of the regular polytopes gives them an aesthetic quality that interests both non-mathematicians and mathematicians. Classically, a regular polytope in n dimensions may be defined as having regular facets and regular vertex figures . These two conditions are sufficient to ensure that all faces are alike and all vertices are alike. Note, however, that this definition does not work for abstract polytopes . A regular polytope can be represented by a Schläfli symbol
Schläfli symbol
of the form {a, b, c, ...., y, z}, with regular facets as {a, b, c, ..., y}, and regular vertex figures as {b, c, ..., y, z}
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6-polytope
In six-dimensional geometry , a SIX-DIMENSIONAL POLYTOPE or 6-POLYTOPE is a polytope , bounded by 5-polytope facets . CONTENTS * 1 Definition * 2 Characteristics * 3 Classification * 4 Regular 6-polytopes * 5 Uniform 6-polytopes * 6 References * 7 External links DEFINITIONA 6-polytope
6-polytope
is a closed six-dimensional figure with vertices , edges , faces , cells (3-faces), 4-faces, and 5-faces. A vertex is a point where six or more edges meet. An edge is a line segment where four or more faces meet, and a face is a polygon where three or more cells meet. A cell is a polyhedron . A 4-face is a polychoron , and a 5-face is a 5-polytope . Furthermore, the following requirements must be met: * Each 4-face must join exactly two 5-faces (facets). * Adjacent facets are not in the same five-dimensional hyperplane . * The figure is not a compound of other figures which meet the requirements.CHARACTERISTICSThe topology of any given 6-polytope
6-polytope
is defined by its Betti numbers and torsion coefficients . The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, and is zero for all 6-polytopes, whatever their underlying topology
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