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7-cube
6-faces 14 4,34 5-faces 84 4,33 4-faces 280 4,3,3 Cells 560 4,3 Faces 672 4 Edges 448Vertices 128Vertex figure 6-simplex
6-simplex
Petrie polygon tetradecagonCoxeter group C7, [35,4]Dual 7-orthoplexProperties convexIn geometry, a 7-cube
7-cube
is a seven-dimensional hypercube with 128 vertices, 448 edges, 672 square faces, 560 cubic cells, 280 tesseract 4-faces, 84 penteract 5-faces, and 14 hexeract 6-faces. It can be named by its Schläfli symbol
Schläfli symbol
4,35 , being composed of 3 6-cubes around each 5-face. It can be called a hepteract, a portmanteau of tesseract (the 4-cube) and hepta for seven (dimensions) in Greek
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Orthogonal Projection
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P 2 = P. That is, whenever P is applied twice to any value, it gives the same result as if it were applied once (idempotent). It leaves its image unchanged.[1] Though abstract, this definition of "projection" formalizes and generalizes the idea of graphical projection
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Pascal's Triangle
In mathematics, Pascal's triangle
Pascal's triangle
is a triangular array of the binomial coefficients. In much of the Western world, it is named after French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in India,[1] Persia (Iran), China, Germany, and Italy.[2] The rows of Pascal's triangle
Pascal's triangle
are conventionally enumerated starting with row n = 0 at the top (the 0th row). The entries in each row are numbered from the left beginning with k = 0 and are usually staggered relative to the numbers in the adjacent rows. The triangle may be constructed in the following manner: In row 0 (the topmost row), there is a unique nonzero entry 1. Each entry of each subsequent row is constructed by adding the number above and to the left with the number above and to the right, treating blank entries as 0
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Portmanteau
A portmanteau (/pɔːrtˈmæntoʊ/ ( listen), /ˌpɔːrtmænˈtoʊ/[a][b]) or portmanteau word is a linguistic blend of words,[1] in which parts of multiple words or their phones (sounds) are combined into a new word,[1][2][3] as in smog, coined by blending smoke and fog,[2][4] or motel, from motor and hotel.[5] In linguistics, a portmanteau is defined as a single morph that represents two or more morphemes.[6][7][8][9] The definition overlaps with the grammatical term contraction, but contractions are formed from words that would otherwise appear together in sequence, such as do and not to make don't, whereas a portmanteau word is formed by combining two or more existing words that all relate to a singular concept. A portmanteau also differs from a compound, which does not involve the truncation of parts of the stems of the blended words
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Greek Language
Greek (Modern Greek: ελληνικά [eliniˈka], elliniká, "Greek", ελληνική γλώσσα [eliniˈci ˈɣlosa] ( listen), ellinikí glóssa, "Greek language") is an independent branch of the Indo-European family of languages, native to Greece
Greece
and other parts of the Eastern Mediterranean
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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 three-dimensional geometry a facet of a polyhedron is any polygon whose corners are vertices of the polyhedron, and is not a face.[1][2] 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.[3] 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
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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.[1] Such dual figures remain combinatorial or abstract polyhedra, but not all are also geometric polyhedra.[2] 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
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Alternation (geometry)
In geometry, an alternation or partial truncation, is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices.[1] Coxeter
Coxeter
labels an alternation by a prefixed by an h, standing for hemi or half. Because alternation reduce all polygon faces to half as many sides, it can only be applied for polytopes with all even-sided faces. An alternated square face becomes a digon, and being degenerate, is usually reduced to a single edge. More generally any vertex-uniform polyhedron or tiling with a vertex configuration consisting of all even-numbered elements can be alternated. For example, the alternation a vertex figure with 2a.2b.2c is a.3.b.3.c.3 where the three is the number of elements in this vertex figure. A special case is square faces whose order divide in half into degenerate digons
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Configuration (polytope)
In geometry, H. S. M. Coxeter
H. S. M. Coxeter
called a regular polytope a special kind of configuration. Other configurations in geometry are something different. These polytope configurations may more accurately called incidence matrices, where like elements are collected together in rows and columns. Regular polytopes will have one row and column per k-face element, while other polytopes will have one row and column for each k-face type by their symmetry classes. A polytope with no symmetry will have one row and column for every element, and the matrix will be filled with 0 if the elements are not connected, and 1 if they are connected. Elements of the same k will not be connected and will have a "*" table entry
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F-vector
Polyhedral combinatorics
Polyhedral combinatorics
is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes. Research in polyhedral combinatorics falls into two distinct areas. Mathematicians in this area study the combinatorics of polytopes; for instance, they seek inequalities that describe the relations between the numbers of vertices, edges, and faces of higher dimensions in arbitrary polytopes or in certain important subclasses of polytopes, and study other combinatorial properties of polytopes such as their connectivity and diameter (number of steps needed to reach any vertex from any other vertex)
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Cartesian Coordinates
A Cartesian coordinate system
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 (plural axes) of the system, and the point where they meet is its origin, at ordered pair (0, 0)
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Orthographic Projection
Orthographic projection
Orthographic projection
(sometimes orthogonal projection), is a means of representing three-dimensional objects in two dimensions. It is a form of parallel projection, in which all the projection lines are orthogonal to the projection plane,[1] 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,[1] but these are better known as multiview projections
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4-face
In solid geometry, a face is a flat (planar) surface that forms part of the boundary of a solid object;[1] 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).[2]Contents1 Polygonal face1.1 Number of polygonal faces of a polyhedron2 k-face2.1 Cell or 3-face 2.2 Facet or (n-1)-face 2.3 Ridge or (n-2)-face 2.4 Peak or (n-3)-face3 See also 4 References 5 External linksPolygonal face[edit] In elementary geometry, a face is a polygon on the boundary of a polyhedron.[2][3] 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 2-dimensional features of a 4-polytope
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Coxeter Plane
In mathematics, the Coxeter
Coxeter
number h is the order of a Coxeter
Coxeter
element of an irreducible Coxeter
Coxeter
group. It is named after H.S.M. Coxeter.[1]Contents1 Definitions 2 Group order 3 Coxeter
Coxeter
elements 4 Coxeter
Coxeter
plane 5 See also 6 Notes 7 ReferencesDefinitions[edit] Note that this article assumes a finite Coxeter
Coxeter
group. For infinite Coxeter
Coxeter
groups, there are multiple conjugacy classes of Coxeter elements, and they have infinite order. There are many different ways to define the Coxeter
Coxeter
number h of an irreducible root system. A Coxeter
Coxeter
element is a product of all simple reflections
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Dihedral Symmetry
In mathematics, a dihedral group is the group of symmetries of a regular polygon,[1][2] 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 differs in geometry and abstract algebra. In geometry, Dn or Dihn refers to the symmetries of the n-gon, a group of order 2n
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Harold Scott MacDonald Coxeter
Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, CC (February 9, 1907 – March 31, 2003)[2] was a British-born Canadian geometer. Coxeter is regarded as one of the greatest geometers of the 20th century. He was born in London
London
but spent most of his adult life in Canada. He was always called Donald, from his third name MacDonald.[3] He was most noted for his work on regular polytopes and higher-dimensional geometries
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