Gosset 2 41 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 2₄₁ 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 2₄₁ 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 2-node sequences.

The rectified 2₄₁ is constructed by points at the mid-edges of the 2₄₁. The birectified 2₄₁ is constructed by points at the triangle face centers of the 2₄₁, and is the same as the rectified 1₄₂.

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...

: .

2_41 polytope

2₄₁ polytope
Type	Uniform 8-polytope 8-polytope In eight-dimensional geometry, a polyzetton is a polytope contained by 7-polytope facets. Each 6-polytope ridge being shared by exactly two 7-polytope facets....
Family	2_k1 polytope Uniform 2 k1 polytope In geometry, 2k1 polytope is a uniform polytope in n dimensions constructed from the En Coxeter group. The family was named by Coxeter as 2k1 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequence...
Schläfli symbol	{3,3,3^4,1}
Coxeter symbol	2₄₁
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	17520: 240 2₃₁ Gosset 2 31 polytope In 7-dimensional geometry, 231 is a uniform polytope, constructed from the E7 group.Coxeter named it 231 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 2-node sequences.... 17280 {3⁶}
6-faces	144960: 6720 2₂₁ 138240 {3⁵}
5-faces	544320: 60480 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.... 483840 {3⁴}
4-faces	1209600: 241920 {2₀₁ 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... 967680 {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	1209600 {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	483840 {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	69120
Vertices	2160
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:...	1₄₁
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 2₄₁ is composed of 17,520 facets (240 2₃₁

Gosset 2 31 polytope

In 7-dimensional geometry, 231 is a uniform polytope, constructed from the E7 group.Coxeter named it 231 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 2-node sequences....

polytopes, 17,280 7-simplices), 144,960 6-faces (6,720 2₂₁ polytopes, 138,240 6-simplices), 544,320 5-faces (60,480 2₁₁

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....

, 483,840 5-simplices, 1,209,600 4-faces (4-simplices

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...

), 1,209,600 cells (tetrahedra

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...

), 483,840 faces

Face (geometry)

In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube...

(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 ....

s), 69,120 edges

Edge (geometry)

In geometry, an edge is a one-dimensional line segment joining two adjacent zero-dimensional vertices in a polygon. Thus applied, an edge is a connector for a one-dimensional line segment and two zero-dimensional objects....

, and 2160 vertices

Vertex (geometry)

In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...

. 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 7-demicube.

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 V₂₁₆₀ (for its 2160 vertices) in his 1912 listing of semiregular polytopes.
It is named 2₄₁ by Coxeter for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequence.
Diacositetracont-myriaheptachiliadiacosioctaconta-zetton (Acronym Bay) - 240-17280 facetted polyzetton (Jonathan Bowers)

Coordinates

The 2160 vertices can be defined as follows:

16 permutations of (±4,0,0,0,0,0,0,0)

1120 permutations of (±2,±2,±2,±2,0,0,0,0)

1024 permutations of (±3,±1,±1,±1,±1,±1,±1,±1) with an even number of minus-signs

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 short branch leaves the 7-simplex: . There are 17280 of these facets

Removing the node on the end of the 4-length branch leaves the 2₃₁

Gosset 2 31 polytope

, . There are 240 of these facets. They are centered at the positions of the 240 vertices in the 4₂₁

Gosset 4 21 polytope

In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group. It was discovered by Thorold Gosset, published in his 1900 paper...

polytope.

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 7-demicube, 1₄₁, .

Images

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...

projections can be 12, 18, or 30-sided based on the E6, E7, and E8 symmetries. The 2160 vertices are all displayed, but lower symmetry forms have projected positions overlapping, shown as different colored vertices. For comparison, a B6 coxeter group is also shown.

E8 [30]	[20]	[24]
(1)
E7 [18]	E6 [12]	[6]
	(1,8,24,32)

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

D6 / B5 / A4 [10]	D7 / B6 [12]	D8 / B7 / A6 [14]
	(1,3,9,12,18,21,36)
B8 [16/2]	A5 [6]	A7 [8]

Rectified 2_41 polytope

Rectified 2₄₁ polytope
Type	Uniform 8-polytope 8-polytope In eight-dimensional geometry, a polyzetton is a polytope contained by 7-polytope facets. Each 6-polytope ridge being shared by exactly two 7-polytope facets....
Schläfli symbol	t₁{3,3,3^4,1}
Coxeter symbol	t₁(2₄₁)
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 total: 240 t₁(2₂₁) 17280 t₁{3⁶} Rectified 7-simplex In seven-dimensional geometry, a rectified 7-simplex is a convex uniform 7-polytope, being a rectification of the regular 7-simplex.There are four unique degrees of rectifications, including the zeroth, the 7-simplex itself. Vertices of the rectified 7-simplex are located at the edge-centers of the... 2160 1₄₁
6-faces	313440
5-faces	1693440
4-faces	4717440
Cells	7257600
Faces	5322240
Edges	19680
Vertices	69120
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 6-simplex Rectified 6-simplex In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex.There are three unique degrees of rectifications, including the zeroth, the 6-simplex itself. Vertices of the rectified 6-simplex are located at the edge-centers of the... prism
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 rectified 2₄₁ is 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 2₄₁ polytope, with vertices positioned at the mid-edges of the 2₄₁.

Alternate names

Rectified Diacositetracont-myriaheptachiliadiacosioctaconta-zetton for rectifed 240-17280 facetted polyzetton (acronym robay) (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, defined by root vectors of the E₈

E8 (mathematics)

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...

.

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 short branch leaves the rectified 7-simplex

Rectified 7-simplex

In seven-dimensional geometry, a rectified 7-simplex is a convex uniform 7-polytope, being a rectification of the regular 7-simplex.There are four unique degrees of rectifications, including the zeroth, the 7-simplex itself. Vertices of the rectified 7-simplex are located at the edge-centers of the...

: .

Removing the node on the end of the 4-length branch leaves the rectified 2₃₁, .

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

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 rectified 6-simplex

Rectified 6-simplex

In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex.There are three unique degrees of rectifications, including the zeroth, the 6-simplex itself. Vertices of the rectified 6-simplex are located at the edge-centers of the...

prism, .

Images

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...

E8 [30]	[20]	[24]
(1)
E7 [18]	E6 [12]	[6]
	(1,8,24,32)

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

D6 / B5 / A4 [10]	D7 / B6 [12]	D8 / B7 / A6 [14]
	(1,3,9,12,18,21,36)
B8 [16/2]	A5 [6]	A7 [8]

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