Gosset 2 21 polytope - AbsoluteAstronomy.com

orthogonal projections in E₆ Coxeter plane
2₂₁	Rectified 2₂₁
(1₂₂)	Birectified 2₂₁ (Rectified 1₂₂)

In 6-dimensional geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, the 2₂₁ polytope is a uniform 6-polytope, constructed within the symmetry of the E₆ group. It was discovered by Thorold Gosset

Thorold Gosset

Thorold Gosset was an English lawyer and an amateur mathematician. In mathematics, he is noted for discovering and classifying the semiregular polytopes in dimensions four and higher.According to H. S. M...

, published in his 1900 paper. He called it an 6-ic semi-regular figure.

Coxeter named it 2₂₁ by its bifurcating Coxeter-Dynkin diagram

Coxeter-Dynkin diagram

In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

, with a single ring on the end of one of the 2-node sequences. He also studied its connection with the 27 lines on the cubic surface

Cubic surface

A cubic surface is a projective variety studied in algebraic geometry. It is an algebraic surface in three-dimensional projective space defined by a single polynomial which is homogeneous of degree 3...

, which are naturally in correspondence with the vertices of 2₂₁.

The rectified 2₂₁ is constructed by points at the mid-edges of the 2₂₁. The birectified 2₂₁ is constructed by points at the triangle face centers of the 2₂₁, and is the same as the rectified 1₂₂.

These polytopes are a part of family of 39 convex uniform polytopes in 6-dimensions, made of uniform 5-polytope facets and vertex figure

Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...

s, defined by all permutations of rings in this Coxeter-Dynkin diagram

Coxeter-Dynkin diagram

In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

: .

2_21 polytope

2₂₁ polytope
Type	Uniform 6-polytope
Family	k₂₁ polytope
Schläfli symbol	{3,3,3^2,1}
Coxeter symbol	2₂₁
Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...
5-faces	99 total: 27 2₁₁ Pentacross 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 hypercells.... 72 {3⁴}
4-faces	648: 432 {3³} Pentachoron In geometry, the 5-cell is a four-dimensional object bounded by 5 tetrahedral cells. It is also known as the pentachoron, pentatope, or hyperpyramid... 216 {3³} Pentachoron In geometry, the 5-cell is a four-dimensional object bounded by 5 tetrahedral cells. It is also known as the pentachoron, pentatope, or hyperpyramid...
Cells	1080 {3,3} Tetrahedron In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
Faces	720 {3} Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
Edges	216
Vertices	27
Vertex figure Vertex figure In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...	1₂₁ (5-demicube)
Petrie polygon Petrie polygon In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon such that every consecutive sides belong to one of the facets...	Dodecagon Dodecagon In geometry, a dodecagon is any polygon with twelve sides and twelve angles.- Regular dodecagon :It usually refers to a regular dodecagon, having all sides of equal length and all angles equal to 150°...
Coxeter group 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 mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...	E₆, [3^2,2,1]
Properties	convex 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...

The 2₂₁ has 27 vertices, and 99 facets: 27 5-orthoplexes and 72 5-simplices. Its vertex figure

Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...

is a 5-demicube.

For visualization this 6-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 27 vertices within a 12-gonal regular polygon (called a Petrie polygon

Petrie polygon

In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon such that every consecutive sides belong to one of the facets...

). Its 216 edges are drawn between 2 rings of 12 vertices, and 3 vertices projected into the center. Higher elements (faces, cells, etc) can also be extracted and drawn on this projection.

Alternate names

E. L. Elte
E. L. Elte
Emanuel Lodewijk Elte was a Dutch mathematician. He is noted for discovering and classifying semiregular polytopes in dimensions four and higher....

named it V₂₇ (for its 27 vertices) in his 1912 listing of semiregular polytopes.
Icosihepta-heptacontidi-peton - 27-72 facetted polypeton (acronym jak) (Jonathan Bowers)

Coordinates

The 27 vertices can be expressed in 8-space as an edge-figure of the 4₂₁ polytope:

(-2,0,0,0,-2,0,0,0)(0,-2,0,0,-2,0,0,0)(0,0,-2,0,-2,0,0,0)(0,0,0,-2,-2,0,0,0)(0,0,0,0,-2,0,0,-2)(0,0,0,0,0,-2,-2,0)
( 2,0,0,0,-2,0,0,0)(0, 2,0,0,-2,0,0,0)(0,0, 2,0,-2,0,0,0)(0,0,0, 2,-2,0,0,0)(0,0,0,0,-2,0,0, 2)
(-1,-1,-1,-1,-1,-1,-1,-1)
(-1,-1,-1, 1,-1,-1,-1, 1) (-1,-1, 1,-1,-1,-1,-1, 1) (-1,-1, 1, 1,-1,-1,-1,-1) (-1, 1,-1,-1,-1,-1,-1, 1) (-1, 1,-1, 1,-1,-1,-1,-1) (-1, 1, 1,-1,-1,-1,-1,-1) (1,-1,-1,-1,-1,-1,-1, 1) (1,-1, 1,-1,-1,-1,-1,-1) (1,-1,-1, 1,-1,-1,-1,-1) (1, 1,-1,-1,-1,-1,-1,-1)
(-1, 1, 1, 1,-1,-1,-1, 1) (1,-1, 1, 1,-1,-1,-1, 1) (1, 1,-1, 1,-1,-1,-1, 1) (1, 1, 1,-1,-1,-1,-1, 1) (1, 1, 1, 1,-1,-1,-1,-1)

Construction

Its construction is based on the E6 group.

The facet information can be extracted from its Coxeter-Dynkin diagram

Coxeter-Dynkin diagram

In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

, .

Removing the ring on the short branch leaves the 5-simplex, .

Removing the ring on the end of the 2-length branch leaves the 5-orthoplex in its alternated form: (2₁₁), .

Every simplex facet touches an 5-orthoplex facet, while alternate facets of the orthoplex touch either a simplex or another orthoplex.

The vertex figure

Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...

is determined by removing the ringed ring and ringing the neighboring ring. This makes 5-demicube (1₂₁ polytope), .

Images

Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow. The number of vertices by color are given in parentheses.

Coxeter plane orthographic projection
Orthographic projection
Orthographic projection is a means of representing a three-dimensional object in two dimensions. It is a form of parallel projection, where all the projection lines are orthogonal to the projection plane, resulting in every plane of the scene appearing in affine transformation on the viewing surface...

s
E6 [12]	D5 [8]	D4 / A2 [6]	B6 [12/2]
(1,3)	(1,3)	(3,9)	(1,3)
A5 [6]	A4 [5]	A3 / D3 [4]
(1,3)	(1,2)	(1,4,7)

Geometric folding

The 2₂₁ is related to the 24-cell by a geometric folding of the E6/F4 Coxeter-Dynkin diagram

Coxeter-Dynkin diagram

In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

s. This can be seen in the Coxeter plane projections. The 24 vertices of the 24-cell are projected in the same two rings as seen in the 2₂₁.

E₆	F₄
2₂₁	24-cell

This polytope can tessellate Euclidean 6-space, forming the 2₂₂ honeycomb with this Coxeter-Dynkin diagram: .

Rectified 2_21 polytope

Rectified 2₂₁ polytope
Type	Uniform 6-polytope
Schläfli symbol	t₁{3,3,3^2,1}
Coxeter symbol	t₁(2₂₁)
Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...
5-faces	126 total: 72 t₁{3⁴} 27 t₁{3³,4} 27 t₁{3,3^2,1}
4-faces	1350
Cells	4320
Faces	5040
Edges	2160
Vertices	216
Vertex figure Vertex figure In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...	rectified 5-cell Rectified 5-cell In four dimensional geometry, the rectified 5-cell is a uniform polychoron composed of 5 regular tetrahedral and 5 regular octahedral cells. Each edge has one tetrahedron and two octahedra. Each vertex has two tetrahedra and three octahedra. In total it has 30 triangle faces, 30 edges, and 10... prism
Coxeter group 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 mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...	E₆, [3^2,2,1]
Properties	convex 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...

The rectified 2₂₁ has 216 vertices, and 126 facets: 72 rectified 5-simplices, and 27 rectified 5-orthoplexes and 27 5-demicubes . Its vertex figure

Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...

is a rectified 5-cell

Rectified 5-cell

In four dimensional geometry, the rectified 5-cell is a uniform polychoron composed of 5 regular tetrahedral and 5 regular octahedral cells. Each edge has one tetrahedron and two octahedra. Each vertex has two tetrahedra and three octahedra. In total it has 30 triangle faces, 30 edges, and 10...

prism.

Alternate names

Rectified icosihepta-heptacontidi-peton as a rectified 27-72 facetted polypeton (acronym rojak) (Jonathan Bowers)

Construction

Its construction is based on the E₆ group and information can be extracted from the ringed Coxeter-Dynkin diagram

Coxeter-Dynkin diagram

In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

representing this polytope: .

Removing the ring on the short branch leaves the rectified 5-simplex, .

Removing the ring on the end of the other 2-length branch leaves the rectified 5-orthoplex in its alternated form: t₁(2₁₁), .

Removing the ring on the end of the same 2-length branch leaves the 5-demicube: (1₂₁), .

The vertex figure

Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...

is determined by removing the ringed ring and ringing the neighboring ring. This makes rectified 5-cell

Rectified 5-cell

prism, t₁{3,3,3}x{}, .

Images

Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.

Coxeter plane orthographic projection
Orthographic projection
Orthographic projection is a means of representing a three-dimensional object in two dimensions. It is a form of parallel projection, where all the projection lines are orthogonal to the projection plane, resulting in every plane of the scene appearing in affine transformation on the viewing surface...

s
E6 [12]	D5 [8]	D4 / A2 [6]

A5 [6]	A4 [5]	A3 / D3 [4]

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