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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
5-cell
facets. It has a dihedral angle of cos−1(1/5), or approximately 78.46°.Contents1 Alternate names 2 As a configuration 3 Regular hexateron cartesian coordinates 4 Projected images 5 Related uniform 5-polytopes 6 Other forms 7 Notes 8 References 9 External linksAlternate names[edit] It 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.[1] As a configuration[edit] The elements of the regular polytopes can be expressed in a configuration matrix
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Hosohedron
In geometry, an n-gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices. A regular n-gonal hosohedron has Schläfli symbol 2, n , with each spherical lune having internal angle 2π/n radians (360/n degrees).[1][2]Contents1 Hosohedra as regular polyhedra 2 Kaleidoscopic symmetry 3 Relationship with the Steinmetz solid 4 Derivative polyhedra 5 Apeirogonal hosohedron 6 Hosotopes 7 Etymology 8 See also 9 References 10 External linksHosohedra as regular polyhedra[edit] Further information: List_of_regular_polytopes_and_compounds § Spherical_2 For a regular polyhedron whose Schläfli symbol is m, n , the number of polygonal faces may be found by: N 2 = 4 n 2 m + 2 n − m n
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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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Schläfli Symbol
In geometry, the Schläfli symbol
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-cen
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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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Hyperplane
In geometry a hyperplane is a subspace whose dimension is one less than that of its ambient space. If a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyperplanes are the 1-dimensional lines. This notion can be used in any general space in which the concept of the dimension of a subspace is defined. In different settings, the objects which are hyperplanes may have different properties. For instance, a hyperplane of an n-dimensional affine space is a flat subset with dimension n − 1. By its nature, it separates the space into two half spaces
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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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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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Stereographic Projection
In geometry, the stereographic projection is a particular mapping (function) that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point: the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it preserves angles at which curves meet. It is neither isometric nor area-preserving: that is, it preserves neither distances nor the areas of figures. Intuitively, then, the stereographic projection is a way of picturing the sphere as the plane, with some inevitable compromises. Because the sphere and the plane appear in many areas of mathematics and its applications, so does the stereographic projection; it finds use in diverse fields including complex analysis, cartography, geology, and photography
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Schlegel Diagram
In geometry, a Schlegel diagram
Schlegel diagram
is a projection of a polytope from R d displaystyle R^ d into R d − 1 displaystyle R^ d-1 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^ d-1 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
Schlegel diagram
is a projection of a polyhedron into a plane figure and a projection of a 4-polytope
4-polytope
to 3-space, respectively
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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.[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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Four-dimensional Space
A four-dimensional space or 4D space is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible generalization of the observation that one only needs three numbers, called dimensions, to describe the sizes or locations of objects in the everyday world. For example, the volume of a rectangular box is found by measuring its length (often labeled x), width (y), and depth (z). The idea of adding a fourth dimension began with Joseph-Louis Lagrange in the mid-1700s and culminated in a precise formalization of the concept in 1854 by Bernhard Riemann. In 1880 Charles Howard Hinton popularized these insights in an essay titled What is the Fourth Dimension?, which explained the concept of a four-dimensional cube with a step-by-step generalization of the properties of lines, squares, and cubes
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Six-dimensional Space
Six-dimensional space
Six-dimensional space
is any space that has six dimensions, six degrees of freedom, and that needs six pieces of data, or coordinates, to specify a location in this space. There are an infinite number of these, but those of most interest are simpler ones that model some aspect of the environment. Of particular interest is six-dimensional Euclidean space, in which 6-polytopes and the 5-sphere are constructed. Six-dimensional elliptical space and hyperbolic spaces are also studied, with constant positive and negative curvature. Formally, six-dimensional Euclidean space, ℝ6, is generated by considering all real 6-tuples as 6-vectors in this space. As such it has the properties of all Euclidean spaces, so it is linear, has a metric and a full set of vector operations. In particular the dot product between two 6-vectors is readily defined, and can be used to calculate the metric
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Seven-dimensional Space
In mathematics, a sequence of n real numbers can be understood as a location in n-dimensional space. When n = 7, the set of all such locations is called 7-dimensional space. Often such a space is studied as a vector space, without any notion of distance. Seven-dimensional Euclidean space
Euclidean space
is seven-dimensional space equipped with a Euclidean metric, which is defined by the dot product.[disputed – discuss] More generally, the term may refer to a seven-dimensional vector space over any field, such as a seven-dimensional complex vector space, which has 14 real dimensions. It may also refer to a seven-dimensional manifold such as a 7-sphere, or a variety of other geometric constructions. Seven-dimensional spaces have a number of special properties, many of them related to the octonions
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