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7-polytope
In seven-dimensional geometry , a 7-POLYTOPE is a polytope contained by 6-polytope
6-polytope
facets. Each 5-polytope
5-polytope
ridge being shared by exactly two 6-polytope
6-polytope
facets . A UNIFORM 7-POLYTOPE is one which is vertex-transitive , and constructed from uniform 6-polytope
6-polytope
facets. CONTENTS * 1 Regular 7-polytopes * 2 Characteristics * 3 Uniform 7-polytopes by fundamental Coxeter groups * 4 The A7 family * 5 The B7 family * 6 The D7 family * 7 The E7 family * 8 Regular and uniform honeycombs * 8.1 Regular and uniform hyperbolic honeycombs * 9 Notes on the Wythoff construction for the uniform 7-polytopes * 10 References * 11 External links REGULAR 7-POLYTOPESRegular 7-polytopes are represented by the Schläfli symbol {p,q,r,s,t,u} with U {p,q,r,s,t} 6-polytopes facets around each 4-face
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Betti Number
In algebraic topology , the BETTI NUMBERS are used to distinguish topological spaces based on the connectivity of n-dimensional simplicial complexes . For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicial complexes or CW complexes), the sequence of Betti numbers is 0 from some point onward (Betti numbers vanish above the dimension of a space), and they are all finite. The nth Betti number
Betti number
represents the rank of the nth homology group , denoted Hn, which tells us the maximum amount of cuts that must be made before separating a surface into two pieces or 0-cycles, 1-cycles, etc. These numbers are used today in fields such as simplicial homology , computer science , digital images , etc. The term "Betti numbers" was coined by Henri Poincaré
Henri Poincaré
after Enrico Betti
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Torsion Coefficient (topology)
In mathematics , HOMOLOGY is a general way of associating a sequence of algebraic objects such as abelian groups or modules to other mathematical objects such as topological spaces . Homology groups were originally defined in algebraic topology . Similar constructions are available in a wide variety of other contexts, such as abstract algebra , groups , Lie algebras , Galois theory , and algebraic geometry . The original motivation for defining homology groups was the observation that two shapes can be distinguished by examining their holes. For instance, a circle is not a disk because the circle has a hole through it while the disk is solid, and the ordinary sphere is not a circle because the sphere encloses a two-dimensional hole while the circle encloses a one-dimensional hole. However, because a hole is "not there", it is not immediately obvious how to define a hole or how to distinguish different kinds of holes
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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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Vertex-transitive
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
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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 three-dimensional 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 higher-dimensional 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 three-dimensional 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
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Ridge (geometry)
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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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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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
Geometry
arose independently in a number of early cultures as a practical way for dealing with lengths , areas , and volumes . Geometry
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
Euclid
, whose treatment, Euclid\'s Elements , set a standard for many centuries to follow. Geometry
Geometry
arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC
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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 order-3. 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
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E7 (mathematics)
In mathematics , E7 is the name of several closely related Lie groups , linear algebraic groups or their Lie algebras E7, all of which have dimension 133; the same notation E7 is used for the corresponding root lattice , which has rank 7. The designation E7 comes from the Cartan–Killing classification of the complex simple Lie algebras , which fall into four infinite series labeled An, Bn, Cn, Dn, and five exceptional cases labeled E6 , E7, E8 , F4 , and G2 . The E7 algebra is thus one of the five exceptional cases. The fundamental group of the (adjoint) complex form, compact real form, or any algebraic version of E7 is the cyclic group Z/2Z, and its outer automorphism group is the trivial group . The dimension of its fundamental representation is 56
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Coxeter Group
In mathematics , a COXETER GROUP, named after H. S. M. Coxeter
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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5-polytope
In five-dimensional geometry , a FIVE-DIMENSIONAL POLYTOPE or 5-POLYTOPE is a 5-dimensional polytope , bounded by (4-polytope) facets. Each polyhedral cell being shared by exactly two 4-polytope facets. CONTENTS * 1 Definition * 2 Characteristics * 3 Classification * 4 Regular 5-polytopes * 5 Uniform 5-polytopes * 6 Pyramids * 7 See also * 8 References * 9 External links DEFINITIONA 5-polytope
5-polytope
is a closed five-dimensional figure with vertices , edges , faces , and cells , and 4-faces . A vertex is a point where five 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 , and a 4-face is a 4-polytope
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Polytope
In elementary geometry , a POLYTOPE is a geometric object with "flat" sides. It is a generalisation in any number of dimensions, of the three-dimensional polyhedron . Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or N-POLYTOPE. Flat sides mean that the sides of a (k+1)-polytope consist of k-polytopes that may have (k-1)-polytopes in common. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. Some theories further generalize the idea to include such objects as unbounded apeirotopes and tessellations , decompositions or tilings of curved manifolds including spherical polyhedra , and set-theoretic abstract polytopes . Polytopes in more than three dimensions were first discovered by Ludwig Schläfli
Ludwig Schläfli

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6-polytope
In six-dimensional geometry , a SIX-DIMENSIONAL POLYTOPE or 6-POLYTOPE is a polytope , bounded by 5-polytope
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
5-polytope

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Uniform Polytope
A UNIFORM POLYTOPE of dimension three or higher is a vertex-transitive polytope bounded by uniform facets . The uniform polytopes in two dimensions are the regular polygons (the definition is different in 2 dimensions to exclude vertex-transitive even-sided polygons that alternate two different lengths of edges). This is a generalization of the older category of SEMIREGULAR polytopes , but also includes the regular polytopes . Further, star regular faces and vertex figures (star polygons ) are allowed, which greatly expand the possible solutions. A strict definition requires uniform polytopes to be finite, while a more expansive definition allows uniform honeycombs (2-dimensional tilings and higher dimensional honeycombs ) of Euclidean and hyperbolic space to be considered polytopes as well
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