3 21 polytope

In 7-dimensional geometry, the 321 polytope is a uniform 7-polytope, constructed within the symmetry of the E7 group. ^[1]

48 relations: Configuration (polytope), Convex polytope, Coxeter element, Coxeter group, Coxeter–Dynkin diagram, Cross-polytope, E7 (mathematics), E7 polytope, Edmund Hess, Emanuel Lodewijk Elte, Geometry, Gosset graph, Gosset–Elte figures, Harold Scott MacDonald Coxeter, Honeycomb (geometry), Hosohedron, Isosceles triangle, N-skeleton, Octadecagon, Petrie polygon, Projection (linear algebra), Rectified 5-cell, Rectified 6-orthoplexes, Rectified 6-simplexes, Regular polytope, Schläfli symbol, Semiregular polytope, Simplex, Tetrahedron, Thorold Gosset, Triangle, Triangular prism, Uniform 6-polytope, Uniform 7-polytope, Uniform k 21 polytope, Uniform polytope, Vertex figure, 1 32 polytope, 2 21 polytope, 2 31 polytope, 3 21 polytope, 3 31 honeycomb, 5-cell, 5-demicube, 5-simplex, 6-orthoplex, 6-simplex, 7-simplex.
7-polytopes

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 contained in the n-dimensional Euclidean space \mathbb^n.

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Coxeter element

In mathematics, a Coxeter element is an element of an irreducible Coxeter group which is a product of all simple reflections.

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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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In geometry, a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing a Coxeter group or sometimes a uniform polytope or uniform tiling constructed from the group.

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Cross-polytope

In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in n-dimensional Euclidean space.

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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.

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E7 polytope

In 7-dimensional geometry, there are 127 uniform polytopes with E7 symmetry. 3 21 polytope and E7 polytope are 7-polytopes.

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Edmund Hess

Edmund Hess (17 February 1843 – 24 December 1903) was a German mathematician who discovered several regular polytopes.

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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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Geometry

Geometry is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures.

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Gosset graph

The Gosset graph, named after Thorold Gosset, is a specific regular graph (1-skeleton of the 7-dimensional 321 polytope) with 56 vertices and valency 27.

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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 (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician.

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Honeycomb (geometry)

In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps.

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Hosohedron

In spherical 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.

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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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N-skeleton

In mathematics, particularly in algebraic topology, the of a topological space presented as a simplicial complex (resp. CW complex) refers to the subspace that is the union of the simplices of (resp. cells of) of dimensions In other words, given an inductive definition of a complex, the is obtained by stopping at the.

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Octadecagon

In geometry, an octadecagon (or octakaidecagon) or 18-gon is an eighteen-sided polygon.

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Petrie polygon

In geometry, a Petrie polygon for a regular polytope of dimensions is a skew polygon in which every consecutive sides (but no) 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 (an endomorphism) such that P\circ P.

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Rectified 5-cell

In four-dimensional geometry, the rectified 5-cell is a uniform 4-polytope composed of 5 regular tetrahedral and 5 regular octahedral cells.

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Rectified 6-orthoplexes

In six-dimensional geometry, a rectified 6-orthoplex is a convex uniform 6-polytope, being a rectification of the regular 6-orthoplex.

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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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Regular polytope

In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry.

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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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Semiregular polytope

In geometry, by Thorold Gosset's definition a semiregular polytope is usually taken to be a polytope that is vertex-transitive and has all its facets being regular polytopes.

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Simplex

In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions.

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Tetrahedron

In geometry, a tetrahedron (tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertices.

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Thorold Gosset

John Herbert de Paz Thorold Gosset (16 October 1869 – December 1962) was an English lawyer and an amateur mathematician.

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Triangle

A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry.

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Triangular prism

In geometry, a triangular prism or trigonal prism is a prism with 2 triangular bases.

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Uniform 6-polytope

In six-dimensional geometry, a uniform 6-polytope is a six-dimensional uniform polytope.

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Uniform 7-polytope

In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. 3 21 polytope and Uniform 7-polytope are 7-polytopes.

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Uniform k 21 polytope

In geometry, a uniform k21 polytope is a polytope in k + 4 dimensions constructed from the ''E''''n'' Coxeter group, and having only regular polytope facets.

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Uniform polytope

In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets.

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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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1 32 polytope

In 7-dimensional geometry, 132 is a uniform polytope, constructed from the E7 group. 3 21 polytope and 1 32 polytope are 7-polytopes.

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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. 3 21 polytope and 2 31 polytope are 7-polytopes.

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3 21 polytope

In 7-dimensional geometry, the 321 polytope is a uniform 7-polytope, constructed within the symmetry of the E7 group. 3 21 polytope and 3 21 polytope are 7-polytopes.

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3 31 honeycomb

In 7-dimensional geometry, the 331 honeycomb is a uniform honeycomb, also given by Schläfli symbol and is composed of 321 and 7-simplex facets, with 56 and 576 of them respectively around each vertex.

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5-cell

In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol.

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5-demicube

In five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube (penteract) with alternated vertices removed.

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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-orthoplex

In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 vertices, 60 edges, 160 triangle faces, 240 tetrahedron cells, 192 5-cell 4-faces, and 64 5-faces.

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6-simplex

In geometry, a 6-simplex is a self-dual regular 6-polytope.

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7-simplex

In 7-dimensional geometry, a 7-simplex is a self-dual regular 7-polytope. 3 21 polytope and 7-simplex are 7-polytopes.

See 3 21 polytope and 7-simplex

References

[1] https://en.wikipedia.org/wiki/3_21_polytope

Also known as Birectified 3 21 polytope, Gosset 3 21 polytope, Hess polytope, Hessian polytope, Rectified 3 21 polytope.

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3 21 polytope

Table of Contents

See also

7-polytopes

References