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48 relations: Configuration (polytope), Convex polytope, Coxeter element, Coxeter group, Coxeter–Dynkin diagram, E6 polytope, E8 (mathematics), E8 polytope, Edge (geometry), Emanuel Lodewijk Elte, Equilateral triangle, Face (geometry), Facet (geometry), Geometry, Gosset–Elte figures, Harold Scott MacDonald Coxeter, Hyperplane, Isosceles triangle, Petrie polygon, Projection (linear algebra), Pyramid (geometry), Rectification (geometry), Rectified 6-simplexes, Rectified 7-simplexes, Rectified 8-orthoplexes, Regular polygon, Schläfli symbol, Tetrahedron, Triangle, Uniform 2 k1 polytope, Uniform 5-polytope, Uniform 8-polytope, Uniform polytope, Vertex (geometry), Vertex figure, Wythoff construction, 1 42 polytope, 2 21 polytope, 2 31 polytope, 2 51 honeycomb, 4 21 polytope, 5-cell, 5-orthoplex, 5-simplex, 6-simplex, 7-demicube, 7-simplex, 8-orthoplex.
Configuration (polytope)
In geometry, H. S. M. Coxeter called a regular polytope a special kind of configuration.
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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.
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Coxeter element
In mathematics, the Coxeter number h is the order of a Coxeter element of an irreducible Coxeter group.
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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).
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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).
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E6 polytope
In 6-dimensional geometry, there are 39 uniform polytopes with E6 symmetry.
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E8 (mathematics)
In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8.
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E8 polytope
In 8-dimensional geometry, there are 255 uniform polytopes with E8 symmetry.
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Edge (geometry)
In geometry, an edge is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope.
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Emanuel Lodewijk Elte
Emanuel Lodewijk Elte (16 March 1881 in Amsterdam – 9 April 1943 in Sobibór) at joodsmonument.nl was a Dutch mathematician.
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Equilateral triangle
In geometry, an equilateral triangle is a triangle in which all three sides are equal.
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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; a three-dimensional solid bounded exclusively by flat faces is a polyhedron.
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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.
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Geometry
Geometry (from the 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.
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Gosset–Elte figures
In geometry, the Gosset–Elte figures, named by Coxeter after Thorold Gosset and E. L. Elte, are a group of uniform polytopes which are not regular, generated by a Wythoff construction with mirrors all related by order-2 and order-3 dihedral angles.
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Harold Scott MacDonald Coxeter
Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, (February 9, 1907 – March 31, 2003) was a British-born Canadian geometer.
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Hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space.
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Isosceles triangle
In geometry, an isosceles triangle is a triangle that has two sides of equal length.
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Petrie polygon
In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon in which every (n – 1) consecutive sides (but no n) belongs to one of the facets.
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Projection (linear algebra)
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that.
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Pyramid (geometry)
In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex.
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Rectification (geometry)
In Euclidean geometry, rectification or complete-truncation is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points.
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Rectified 6-simplexes
In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex.
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Rectified 7-simplexes
In seven-dimensional geometry, a rectified 7-simplex is a convex uniform 7-polytope, being a rectification of the regular 7-simplex.
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Rectified 8-orthoplexes
In eight-dimensional geometry, a rectified 8-orthoplex is a convex uniform 8-polytope, being a rectification of the regular 8-orthoplex.
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Regular polygon
In Euclidean geometry, a regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length).
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Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations.
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Tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners.
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Triangle
A triangle is a polygon with three edges and three vertices.
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Uniform 2 k1 polytope
In geometry, 2k1 polytope is a uniform polytope in n dimensions (n.
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Uniform 5-polytope
In geometry, a uniform 5-polytope is a five-dimensional uniform polytope.
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Uniform 8-polytope
In eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets.
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Uniform polytope
A uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets.
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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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Vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
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Wythoff construction
In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling.
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1 42 polytope
In 8-dimensional geometry, the 142 is a uniform 8-polytope, constructed within the symmetry of the E8 group.
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2 21 polytope
In 6-dimensional geometry, the 221 polytope is a uniform 6-polytope, constructed within the symmetry of the E6 group.
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2 31 polytope
In 7-dimensional geometry, 231 is a uniform polytope, constructed from the E7 group.
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2 51 honeycomb
In 8-dimensional geometry, the 251 honeycomb is a space-filling uniform tessellation.
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4 21 polytope
In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group.
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5-cell
In geometry, the 5-cell is a four-dimensional object bounded by 5 tetrahedral cells.
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5-orthoplex
In five-dimensional geometry, a 5-orthoplex, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-faces.
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5-simplex
In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope.
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6-simplex
In geometry, a 6-simplex is a self-dual regular 6-polytope.
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7-demicube
In geometry, a demihepteract or 7-demicube is a uniform 7-polytope, constructed from the 7-hypercube (hepteract) with alternated vertices removed.
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7-simplex
In 7-dimensional geometry, a 7-simplex is a self-dual regular 7-polytope.
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8-orthoplex
In geometry, an 8-orthoplex or 8-cross polytope is a regular 8-polytope with 16 vertices, 112 edges, 448 triangle faces, 1120 tetrahedron cells, 1792 5-cells 4-faces, 1792 5-faces, 1024 6-faces, and 256 7-faces.
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Gosset 2 41 polytope, Rectified 2 41 polytope.
References
[1] https://en.wikipedia.org/wiki/2_41_polytope
