Gosset 1 42 polytope - AbsoluteAstronomy.com

Orthogonal projections in E₆ Coxeter plane
4₂₁	1₄₂	2₄₁
Rectified 4₂₁	Rectified 1₄₂	Rectified 2₄₁
Birectified 4₂₁	Trirectified 4₂₁

In 8-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 1₄₂ is a uniform 8-polytope, constructed within the symmetry of the E₈

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...

group.

Coxeter named it 1₄₂ 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 the 1-node sequences.

The rectified 1₂₁ is constructed by points at the mid-edges of the 1₄₂ and is the same as the birectified 2₄₁, and the quadrirectified 4₂₁.

These polytopes are part of a family of 255 (2⁸ − 1) convex uniform polytope

Uniform polytope

A uniform polytope is a vertex-transitive polytope made from uniform polytope facets of a lower dimension. Uniform polytopes of 2 dimensions are the regular polygons....

s in 8-dimensions, made of uniform polytope

Uniform polytope

A uniform polytope is a vertex-transitive polytope made from uniform polytope facets of a lower dimension. Uniform polytopes of 2 dimensions are the regular polygons....

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...

: .

1_42 polytope

1₄₂
Type	Uniform 8-polytope
Family	1_k2 polytope Uniform 1 k2 polytope In geometry, 1k2 polytope is a uniform polytope in n-dimensions constructed from the En Coxeter group. The family was named by Coxeter as 1k2 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequence...
Schläfli symbol	{3,3^4,2}
Coxeter symbol	1₄₂
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...
7-faces	2400: 240 1₃₂ 2160 1₄₁
6-faces	106080: 6720 1₂₂ 30240 1₃₁ 69120 {3⁵}
5-faces	725760: 60480 1₁₂ 181440 1₂₁ 483840 {3⁴}
4-faces	2298240: 241920 1₀₂ 604800 1₁₁ 16-cell In four dimensional geometry, a 16-cell or hexadecachoron is a regular convex 4-polytope. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century.... 1451520 {3³}
Cells	3628800: 1209600 1₀₁ 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... 2419200 {3²} 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	2419200 {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	483840
Vertices	17280
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:...	t₂{3⁶}
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...	30-gon Regular polygon A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...
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...	E₈ 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... , [3^4,2,1]
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 1₄₂ is composed of 2400 facets: 240 1₃₂

Gosset 1 32 polytope

In 7-dimensional geometry, 132 is a uniform polytope, constructed from the E7 group.Coxeter named it 132 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 1-node sequences....

polytopes, and 2160 7-demicubes (1₄₁). 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 birectified 7-simplex.

This polytope, along with the demiocteract

Demiocteract

In geometry, a demiocteract or 8-demicube is a uniform 8-polytope, constructed from the 8-hypercube, octeract, with alternated vertices deleted...

, can tessellate

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. It is an example of the more general mathematical tiling or tessellation in any number of dimensions....

8-dimensional space, represented by the symbol 1₅₂, and Coxeter-Dynkin diagram: .

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....

(1912) excluded this polytope from his listing of semiregular polytopes, because it has more than two types of 6-faces, but under his naming scheme it would be called V₁₇₂₈₀ for its 17280 vertices.
Coxeter named it 1₄₂ for 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 the 1-node branch.
Diacositetracont-dischiliahectohexaconta-zetton (Acronym bif) - 240-2160 facetted polyzetton (Jonathan Bowers)

Coordinates

The 17280 vertices can be defined as sign and location permutations of:

All sign combinations (32): (280×32=8960 vertices)

(4, 2, 2, 2, 2, 0, 0, 0)

Half of the sign combinations (128): ((1+8+56)×128=8320 vertices)

(2, 2, 2, 2, 2, 2, 2, 2)

(5, 1, 1, 1, 1, 1, 1, 1)

(3, 3, 3, 1, 1, 1, 1, 1)

The edge length is 2√2 in this coordinate set, and the polytope radius is 4√2.

Construction

It is created by a Wythoff construction

Wythoff construction

In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction.- Construction process :...

upon a set of 8 hyperplane

Hyperplane

A hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.A hyperplane of an n-dimensional space is a flat subset with dimension n − 1...

mirrors in 8-dimensional space.

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 node on the end of the 2-length branch leaves the 7-demicube, 1₄₁, .

Removing the node on the end of the 4-length branch leaves the 1₃₂, .

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 node and ringing the neighboring node. This makes the birectified 7-simplex, 0₄₂, .

Projections

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 are shown for the sub-symmetries of E₈: E₇, E₆, B₈, B₇, B₆, B₅, B₄, B₃, B₂, A₇, and A₅ Coxeter planes, as well as two more symmetry planes of order 20 and 24. Vertices are shown as circles, colored by their order of overlap in each projective plane.

E8 [30]	E7 [18]	E6 [12]
(1)	(1,3,6)	(8,16,24,32,48,64,96)
[20]	[24]	[6]
		(1,2,3,4,5,6,7,8,10,11,12,14,16,18,19,20)

D3 / B2 / A3 [4]	D4 / B3 / A2 [6]	D5 / B4 [8]
(32,160,192,240,480,512,832,960)	(72,216,432,720,864,1080)	(8,16,24,32,48,64,96)
D6 / B5 / A4 [10]	D7 / B6 [12]	D8 / B7 / A6 [14]

B8 [16/2]	A5 [6]	A7 [8]

Rectified 1_42 polytope

Rectified 1₄₂
Type	Uniform 8-polytope
Schläfli symbol	t₁{3,3^4,2}
Coxeter symbol	t₁(1₄₂)
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...
7-faces	19680
6-faces	382560
5-faces	2661120
4-faces	9072000
Cells	16934400
Faces	16934400
Edges	7257600
Vertices	483840
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:...	{3,3,3}x{3}x{}
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...	E₈ 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... , [3^4,2,1]
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 1₄₂ is named from being a rectification

Rectification (geometry)

In Euclidean geometry, rectification is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points...

of the 1₄₂ polytope, with vertices positioned at the mid-edges of the 1₄₂.

Alternate names

Birectified 2₄₁ polytope
Quadrirectified 4₂₁ polytope
Rectified diacositetracont-dischiliahectohexaconta-zetton as a rectified 240-2160 facetted polyzetton (Acronym buffy) (Jonathan Bowers)

Construction

It is created by a Wythoff construction

Wythoff construction

upon a set of 8 hyperplane

Hyperplane

A hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.A hyperplane of an n-dimensional space is a flat subset with dimension n − 1...

mirrors in 8-dimensional space.

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 node on the end of the 1-length branch leaves the birectified 7-simplex,

Removing the node on the end of the 2-length branch leaves the 7-demicube, 1₄₁, .

Removing the node on the end of the 3-length branch leaves the 1₃₂, .

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 node and ringing the neighboring node. This makes the 5-cell-triangle

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 ....

duoprism prism, .

Projections

Orthographic projection

s are shown for the sub-symmetries of B₆, B₅, B₄, B₃, B₂, A₇, and A₅ Coxeter planes. Vertices are shown as circles, colored by their order of overlap in each projective plane.

(Planes for E₈: E₇, E₆, B₈, B₇, [20], [24] are not shown for being too large to display.)

D3 / B2 / A3 [4]	D4 / B3 / A2 [6]	D5 / B4 [8]

D6 / B5 / A4 [10]	D7 / B6 [12]	[6]

A5 [6]	A7 [8]

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