4 21 polytope

421 polytope

The 4₂₁ is composed of 17,280 7-simplex and 2,160 7-orthoplex facets. Its vertex figure is the 3₂₁ polytope.

For visualization this 8-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 240 vertices within a regular triacontagon (called a Petrie polygon). Its 6720 edges are drawn between the 240 vertices. Specific higher elements (faces, cells, etc.) can also be extracted and drawn on this projection.

As its 240 vertices represent the root vectors of the simple Lie group E₈, the polytope is sometimes referred to as the E₈ polytope.

The vertices of this polytope can be obtained by taking the 240 integral octonions of norm 1. Because the octonions are a nonassociative normed division algebra, these 240 points have a multiplication operation making them not into a group but rather a loop, in fact a Moufang loop.

Alternate names

Coordinates

It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space.

The 240 vertices of the 4₂₁ polytope can be constructed in two sets: 112 (2²×⁸C₂) with coordinates obtained from ( ± 2 , ± 2 , 0 , 0 , 0 , 0 , 0 , 0 ) by taking an arbitrary combination of signs and an arbitrary permutation of coordinates, and 128 roots (2⁷) with coordinates obtained from ( ± 1 , ± 1 , ± 1 , ± 1 , ± 1 , ± 1 , ± 1 , ± 1 ) by taking an even number of minus signs (or, equivalently, requiring that the sum of all the eight coordinates be a multiple of 4).

Each vertex has 56 nearest neighbors; for example, the nearest neighbors of the vertex ( 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 ) are those whose coordinates sum to 4, namely the 28 obtained by permuting the coordinates of ( 2 , 2 , 0 , 0 , 0 , 0 , 0 , 0 ) and the 28 obtained by permuting the coordinates of ( 1 , 1 , 1 , 1 , 1 , 1 , − 1 , − 1 ) . These 56 points are the vertices of a 3₂₁ polytope in 7 dimensions.

Each vertex has 126 second nearest neighbors: for example, the nearest neighbors of the vertex ( 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 ) are those whose coordinates sum to 0, namely the 56 obtained by permuting the coordinates of ( 2 , − 2 , 0 , 0 , 0 , 0 , 0 , 0 ) and the 70 obtained by permuting the coordinates of ( 1 , 1 , 1 , 1 , − 1 , − 1 , − 1 , − 1 ) . These 126 points are the vertices of a 2₃₁ polytope in 7 dimensions.

Each vertex also has 56 third nearest neighbors, which are the negatives of its nearest neighbors, and one antipodal vertex, for a total of 1 + 56 + 126 + 56 + 1 = 240 vertices.

Tessellations

This polytope is the vertex figure for a uniform tessellation of 8-dimensional space, represented by symbol 5₂₁ and Coxeter-Dynkin diagram:

Construction and faces

The facet information of this polytope can be extracted from its Coxeter-Dynkin diagram:

Removing the node on the short branch leaves the 7-simplex:

Removing the node on the end of the 2-length branch leaves the 7-orthoplex in its alternated form (4₁₁):

Every 7-simplex facet touches only 7-orthoplex facets, while alternate facets of an orthoplex facet touch either a simplex or another orthoplex. There are 17,280 simplex facets and 2160 orthoplex facets.

Since every 7-simplex has 7 6-simplex facets, each incident to no other 6-simplex, the 4₂₁ polytope has 120,960 (7×17,280) 6-simplex faces that are facets of 7-simplexes. Since every 7-orthoplex has 128 (2⁷) 6-simplex facets, half of which are not incident to 7-simplexes, the 4₂₁ polytope has 138,240 (2⁶×2160) 6-simplex faces that are not facets of 7-simplexes. The 4₂₁ polytope thus has two kinds of 6-simplex faces, not interchanged by symmetries of this polytope. The total number of 6-simplex faces is 259200 (120,960+138,240).

The vertex figure of a single-ring polytope is obtained by removing the ringed node and ringing its neighbor(s). This makes the 3₂₁ polytope.

2D

These graphs represent orthographic projections in the E₈,E₇,E₆, and B₈,D₈,D₇,D₆,D₅,D₄,D₃,A₇,A₅ Coxeter planes. The vertex colors are by overlapping multiplicity in the projection: colored by increasing order of multiplicities as red, orange, yellow, green.

k21 family

The 4₂₁ polytope is last in a family called the k₂₁ polytopes. The first polytope in this family is the semiregular triangular prism which is constructed from three squares (2-orthoplexes) and two triangles (2-simplexes).

Geometric folding

The 4₂₁ is related to the 600-cell by a geometric folding of the Coxeter-Dynkin diagrams. This can be seen in the E8/H4 Coxeter plane projections. The 240 vertices of the 4₂₁ polytope are projected into 4-space as two copies of the 120 vertices of the 600-cell, one copy smaller than the other with the same orientation. Seen as a 2D orthographic projection in the E8/H4 Coxeter plane, the 120 vertices of the 600-cell are projected in the same four rings as seen in the 4₂₁. The other 4 rings of the 4₂₁ graph also match a smaller copy of the four rings of the 600-cell.

Related polytopes

In 4-dimensional complex geometry, the regular complex polytope 3{3}3{3}3{3}3, and Coxeter diagram exists with the same vertex arrangement as the 4₂₁ polytope. It is self-dual. Coxeter called it the Witting polytope, after Alexander Witting. Coxeter expresses its Shephard group symmetry by 3[3]3[3]3[3]3.

The 4₂₁ is sixth in a dimensional series of semiregular polytopes. Each progressive uniform polytope is constructed vertex figure of the previous polytope. Thorold Gosset identified this series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes.

Rectified 4_21 polytope

The rectified 4₂₁ can be seen as a rectification of the 4₂₁ polytope, creating new vertices on the center of edges of the 4₂₁.

Alternative names

Construction

It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space. It is named for being a rectification of the 4₂₁. Vertices are positioned at the midpoint of all the edges of 4₂₁, and new edges connecting them.

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

Removing the node on the short branch leaves the rectified 7-simplex:

Removing the node on the end of the 2-length branch leaves the rectified 7-orthoplex in its alternated form:

Removing the node on the end of the 4-length branch leaves the 3₂₁:

The vertex figure is determined by removing the ringed node and adding a ring to the neighboring node. This makes a 2₂₁ prism.

2D

These graphs represent orthographic projections in the E₈,E₇,E₆, and B₈,D₈,D₇,D₆,D₅,D₄,D₃,A₇,A₅ Coxeter planes. The vertex colors are by overlapping multiplicity in the projection: colored by increasing order of multiplicities as red, orange, yellow, green.

Birectified 4_21 polytope

The birectified 4₂₁can be seen as a second rectification of the uniform 4₂₁ polytope. Vertices of this polytope are positioned at the centers of all the 60480 triangular faces of the 4₂₁.

Alternative names

Construction

It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space. It is named for being a birectification of the 4₂₁. Vertices are positioned at the center of all the triangle faces of 4₂₁.

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

Removing the node on the short branch leaves the birectified 7-simplex. There are 17280 of these facets.

Removing the node on the end of the 2-length branch leaves the birectified 7-orthoplex in its alternated form. There are 2160 of these facets.

Removing the node on the end of the 4-length branch leaves the rectified 3₂₁. There are 240 of these facets.

The vertex figure is determined by removing the ringed node and adding rings to the neighboring nodes. This makes a 5-demicube-triangular duoprism.

2D

These graphs represent orthographic projections in the E₈,E₇,E₆, and B₈,D₈,D₇,D₆,D₅,D₄,D₃,A₇,A₅ Coxeter planes. Edges are not drawn. The vertex colors are by overlapping multiplicity in the projection: colored by increasing order of multiplicities as red, orange, yellow, green, etc.

Alternative names

Construction

It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space. It is named for being a birectification of the 4₂₁. Vertices are positioned at the center of all the triangle faces of 4₂₁.

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

Removing the node on the short branch leaves the trirectified 7-simplex:

Removing the node on the end of the 2-length branch leaves the trirectified 7-orthoplex in its alternated form:

Removing the node on the end of the 4-length branch leaves the birectified 3₂₁:

The vertex figure is determined by removing the ringed node and ring the neighbor nodes. This makes a tetrahedron-rectified 5-cell duoprism.

2D

These graphs represent orthographic projections in the E₇,E₆, and B₈,D₈,D₇,D₆,D₅,D₄,D₃,A₇,A₅ Coxeter planes. The vertex colors are by overlapping multiplicity in the projection: colored by increasing order of multiplicities as red, orange, yellow, green.

(E₈ and B₈ were too large to display)