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7-simplex
In 7-dimensional geometry, a 7-simplex
7-simplex
is a self-dual regular 7-polytope. It has 8 vertices, 28 edges, 56 triangle faces, 70 tetrahedral cells, 56 5-cell
5-cell
5-faces, 28 5-simplex
5-simplex
6-faces, and 8 6-simplex
6-simplex
7-faces. Its dihedral angle is cos−1(1/7), or approximately 81.79°.Contents1 Alternate names 2 As a configuration 3 Coordinates 4 Images 5 Related polytopes 6 Notes 7 External linksAlternate names[edit] It can also be called an octaexon, or octa-7-tope, as an 8-facetted polytope in 7-dimensions. The name octaexon is derived from octa for eight facets in Greek and -ex for having six-dimensional facets, and -on. Jonathan Bowers gives an octaexon the acronym oca.[1] As a configuration[edit] The elements of the regular polytopes can be expressed in a configuration matrix
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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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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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Face (geometry)
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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Cell (mathematics)
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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Dihedral Angle
A dihedral angle is the angle between two intersecting planes. In chemistry it is the angle between planes through two sets of three atoms, having two atoms in common. In solid geometry it is defined as the union of a line and two half-planes that have this line as a common edge
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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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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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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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Vertex (geometry)
In geometry, a vertex (plural: vertices or vertexes) is a point where two or more curves, lines, or edges meet
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Cartesian Coordinate
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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Triakis Tetrahedron
In geometry, a triakis tetrahedron (or kistetrahedron[1]) is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated tetrahedron. It can be seen as a tetrahedron with triangular pyramids added to each face; that is, it is the Kleetope
Kleetope
of the tetrahedron. It is very similar to the net for the 5-cell, as the net for a tetrahedron is a triangle with other triangles added to each edge, the net for the 5-cell
5-cell
a tetrahedron with pyramids attached to each face. This interpretation is expressed in the name. The length of the shorter edges is 3/5 that of the longer edges[2]
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Amplituhedron
An amplituhedron is a geometric structure introduced in 2013 by Nima Arkani-Hamed and Jaroslav Trnka. It enables simplified calculation of particle interactions in some quantum field theories. In planar N = 4 supersymmetric Yang–Mills theory, also equivalent to the perturbative topological B model string theory in twistor space, an amplituhedron is defined as a mathematical space known as the positive Grassmannian.[1][2] Amplituhedron theory challenges the notion that space-time locality and unitarity are necessary components of a model of particle interactions. Instead, they are treated as properties that emerge from an underlying phenomenon.[3][4] The connection between the amplituhedron and scattering amplitudes is at present a conjecture that has passed many non-trivial checks, including an understanding of how locality and unitarity arise as consequences of positivity.[1] Research has been led by Nima Arkani-Hamed
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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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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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