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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. [1]

53 relations: Alicia Boole Stott, Coxeter group, Coxeter–Dynkin diagram, Cross-polytope, Demihypercube, E6 (mathematics), E8 lattice, E9 honeycomb, Edge (geometry), En (Lie algebra), Face (geometry), Facet (geometry), Geometry, Gosset–Elte figures, Graph (discrete mathematics), Harold Scott MacDonald Coxeter, John Horton Conway, Norman Johnson (mathematician), Octahedron, Polytope, Rectified 5-cell, Regular polytope, Schläfli symbol, Semiregular polytope, Simplex, Square, Tetrahedron, Thorold Gosset, Triangle, Triangular prism, Uniform 1 k2 polytope, Uniform 2 k1 polytope, Uniform polytope, Vertex (geometry), Vertex figure, 16-cell, 2 21 polytope, 3 21 polytope, 4 21 polytope, 5 21 honeycomb, 5-cell, 5-demicube, 5-orthoplex, 5-simplex, 6-orthoplex, 6-polytope, 6-simplex, 7-orthoplex, 7-simplex, 8-orthoplex, ... Expand index (3 more) »

## Alicia Boole Stott

Alicia Boole Stott (8 June 1860 – 17 December 1940) was an Irish-English mathematician.

## 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).

## 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).

## Cross-polytope

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

## Demihypercube

In geometry, demihypercubes (also called n-demicubes, n-hemicubes, and half measure polytopes) are a class of n-polytopes constructed from alternation of an n-hypercube, labeled as h&gamma;n for being half of the hypercube family, &gamma;n.

## E6 (mathematics)

In mathematics, E6 is the name of some closely related Lie groups, linear algebraic groups or their Lie algebras \mathfrak_6, all of which have dimension 78; the same notation E6 is used for the corresponding root lattice, which has rank 6.

## E8 lattice

In mathematics, the E8 lattice is a special lattice in R8.

## E9 honeycomb

In geometry, an E9 honeycomb is a tessellation of uniform polytopes in hyperbolic 9-dimensional space.

## Edge (geometry)

In geometry, an edge is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope.

## En (Lie algebra)

In mathematics, especially in Lie theory, En is the Kac–Moody algebra whose Dynkin diagram is a bifurcating graph with three branches of length 1,2, and k, with k.

## 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.

## 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.

## 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.

## 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.

## Graph (discrete mathematics)

In mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related".

## Harold Scott MacDonald Coxeter

Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, (February 9, 1907 &ndash; March 31, 2003) was a British-born Canadian geometer.

## John Horton Conway

John Horton Conway FRS (born 26 December 1937) is an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory.

## Norman Johnson (mathematician)

Norman Woodason Johnson (November 12, 1930 – July 13, 2017) was a mathematician, previously at Wheaton College, Norton, Massachusetts.

## Octahedron

In geometry, an octahedron (plural: octahedra) is a polyhedron with eight faces, twelve edges, and six vertices.

## Polytope

In elementary geometry, a polytope is a geometric object with "flat" sides.

## 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.

## 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.

## Schläfli symbol

In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations.

## Semiregular polytope

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

## Simplex

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

## Square

In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, or (100-gradian angles or right angles). It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted.

## 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.

## Thorold Gosset

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

## Triangle

A triangle is a polygon with three edges and three vertices.

## Triangular prism

In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides.

## Uniform 1 k2 polytope

In geometry, 1k2 polytope is a uniform polytope in n-dimensions (n.

## Uniform 2 k1 polytope

In geometry, 2k1 polytope is a uniform polytope in n dimensions (n.

## Uniform polytope

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

## Vertex (geometry)

In geometry, a vertex (plural: vertices or vertexes) is a point where two or more curves, lines, or edges meet.

## Vertex figure

In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.

## 16-cell

In four-dimensional geometry, a 16-cell is a regular convex 4-polytope.

## 2 21 polytope

In 6-dimensional geometry, the 221 polytope is a uniform 6-polytope, constructed within the symmetry of the E6 group.

## 3 21 polytope

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

## 4 21 polytope

In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group.

## 5 21 honeycomb

In geometry, the 521 honeycomb is a uniform tessellation of 8-dimensional Euclidean space.

## 5-cell

In geometry, the 5-cell is a four-dimensional object bounded by 5 tetrahedral cells.

## 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.

## 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.

## 5-simplex

In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope.

## 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.

## 6-polytope

In six-dimensional geometry, a six-dimensional polytope or 6-polytope is a polytope, bounded by 5-polytope facets.

## 6-simplex

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

## 7-orthoplex

In geometry, a 7-orthoplex, or 7-cross polytope, is a regular 7-polytope with 14 vertices, 84 edges, 280 triangle faces, 560 tetrahedron cells, 672 5-cells 4-faces, 448 5-faces, and 128 6-faces.

## 7-simplex

In 7-dimensional geometry, a 7-simplex is a self-dual regular 7-polytope.

## 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.

## 8-simplex

In geometry, an 8-simplex is a self-dual regular 8-polytope.

## 9-orthoplex

In geometry, a 9-orthoplex or 9-cross polytope, is a regular 9-polytope with 18 vertices, 144 edges, 672 triangle faces, 2016 tetrahedron cells, 4032 5-cells 4-faces, 5376 5-simplex 5-faces, 4608 6-simplex 6-faces, 2304 7-simplex 7-faces, and 512 8-simplex 8-faces.

## 9-simplex

In geometry, a 9-simplex is a self-dual regular 9-polytope.

## References

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