49 relations: Complex polytope, Complex reflection group, Configuration (polytope), Convex polytope, Coxeter element, Coxeter group, Coxeter–Dynkin diagram, Cross-polytope, Cubic surface, Dodecagon, Dynkin diagram, E6 (mathematics), E6 polytope, Edmund Hess, Emanuel Lodewijk Elte, Geometry, Gosset–Elte figures, Harold Scott MacDonald Coxeter, Hessian polyhedron, Isosceles triangle, Ludwig Schläfli, Orthographic projection, Petrie polygon, Projection (linear algebra), Rectified 5-cell, Rectified 5-orthoplexes, Rectified 5-simplexes, Regular polytope, Schläfli graph, Schläfli symbol, Semiregular polytope, Simplex, Tetrahedron, Thorold Gosset, Triangle, Triangular prism, Uniform 5-polytope, Uniform 6-polytope, Uniform k 21 polytope, Uniform polytope, Vertex figure, 1 22 polytope, 2 22 honeycomb, 24-cell, 4 21 polytope, 5-cell, 5-demicube, 5-orthoplex, 5-simplex.

## Complex polytope

In geometry, a complex polytope is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one.

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## Complex reflection group

In mathematics, a complex reflection group is a finite group acting on a finite-dimensional complex vector space that is generated by complex reflections: non-trivial elements that fix a complex hyperplane pointwise.

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## 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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## Cross-polytope

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

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## Cubic surface

A cubic surface is a projective variety studied in algebraic geometry.

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## Dodecagon

In geometry, a dodecagon or 12-gon is any twelve-sided polygon.

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## Dynkin diagram

In the mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled (drawn as a double or triple line).

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## 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.

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## E6 polytope

In 6-dimensional geometry, there are 39 uniform polytopes with E6 symmetry.

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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 (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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## Hessian polyhedron

In geometry, the Hessian polyhedron is a regular complex polyhedron 333,, in \mathbb^3.

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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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## Ludwig Schläfli

Ludwig Schläfli (15 January 1814 – 20 March 1895) was a Swiss mathematician, specialising in geometry and complex analysis (at the time called function theory) who was one of the key figures in developing the notion of higher-dimensional spaces.

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## Orthographic projection

Orthographic projection (sometimes orthogonal projection), is a means of representing three-dimensional objects in two dimensions.

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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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## 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 5-orthoplexes

In five-dimensional geometry, a rectified 5-orthoplex is a convex uniform 5-polytope, being a rectification of the regular 5-orthoplex.

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## Rectified 5-simplexes

In five-dimensional geometry, a rectified 5-simplex is a convex uniform 5-polytope, being a rectification of the regular 5-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 graph

In the mathematical field of graph theory, the Schläfli graph, named after Ludwig Schläfli, is a 16-regular undirected graph with 27 vertices and 216 edges.

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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-uniform 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 (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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## 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 edges and three vertices.

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## 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.

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## Uniform 5-polytope

In geometry, a uniform 5-polytope is a five-dimensional uniform polytope.

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## Uniform 6-polytope

In six-dimensional geometry, a uniform polypeton (or uniform 6-polytope) is a six-dimensional uniform polytope.

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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

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 22 polytope

In 6-dimensional geometry, the 122 polytope is a uniform polytope, constructed from the E6 group.

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## 2 22 honeycomb

In geometry, the 222 honeycomb is a uniform tessellation of the six-dimensional Euclidean space.

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## 24-cell

In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol.

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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-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-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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## Redirects here:

Gosset 2 21 polytope, Rectified 2 21, Rectified 2 21 polytope, Truncated 2 21, Truncated 2 21 polytope.

## References

[1] https://en.wikipedia.org/wiki/2_21_polytope