In 8-dimensional geometry, the 4<sub>21</sub> is a semiregular uniform 8-polytope, constructed within the symmetry of the E<sub>8</sub> group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 8-ic semi-regular figure.
Its Coxeter symbol is 4<sub>21</sub>, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 4-node sequences, .
The rectified 4<sub>21</sub> is constructed by points at the mid-edges of the 4<sub>21</sub>. The birectified 4<sub>21</sub> is constructed by points at the triangle face centers of the 4<sub>21</sub>. The trirectified 4<sub>21</sub> is constructed by points at the tetrahedral centers of the 4<sub>21</sub>.
These polytopes are part of a family of 255 = 2<sup>8</sup> − 1 convex uniform 8-polytopes, made of uniform 7-polytope facets and vertex figures, defined by all permutations of one or more rings in this Coxeter-Dynkin diagram: .
The 4<sub>21</sub> polytope has 17,280 7-simplex and 2,160 7-orthoplex facets, and 240 vertices. Its vertex figure is the 3<sub>21</sub> polytope. As its vertices represent the root vectors of the simple Lie group E<sub>8</sub>, this polytope is sometimes referred to as the E<sub>8</sub> root polytope.
The vertices of this polytope can also 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.
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.
It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space.
The 240 vertices of the 4<sub>21</sub> polytope can be constructed in two sets: 112 () with coordinates obtained from by taking an arbitrary combination of signs and an arbitrary permutation of coordinates, and 128 roots (2<sup>7</sup>) with coordinates obtained from 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 are those whose coordinates sum to 4, namely the 28 obtained by permuting the coordinates of and the 28 obtained by permuting the coordinates of . These 56 points are the vertices of a 3<sub>21</sub> polytope in 7 dimensions.
Each vertex has 126 second nearest neighbors: for example, the nearest neighbors of the vertex are those whose coordinates sum to 0, namely the 56 obtained by permuting the coordinates of and the 70 obtained by permuting the coordinates of . These 126 points are the vertices of a 2<sub>31</sub> 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 vertices.
Another construction is by taking signed combination of 14 codewords of 8-bit Extended Hamming code(8,4) that give 14 ÃÂ 2<sup>4</sup> = 224 vertices and adding trivial signed axis for last 16 vertices. In this case, vertices are distance of from origin rather than .
Hamming 8-bit Code 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 â ñ ñ ñ ñ 0 0 0 0 2 1 1 0 0 1 1 0 0 â ñ ñ 0 0 ñ ñ 0 0 3 0 0 1 1 1 1 0 0 â 0 0 ñ ñ ñ ñ 0 0 4 1 0 1 0 1 0 1 0 â ñ 0 ñ 0 ñ 0 ñ 0 ñ2 0 0 0 0 0 0 0 5 0 1 0 1 1 0 1 0 â 0 ñ 0 ñ ñ 0 ñ 0 0 ñ2 0 0 0 0 0 0 6 0 1 1 0 0 1 1 0 â 0 ñ ñ 0 0 ñ ñ 0 0 0 ñ2 0 0 0 0 0 7 1 0 0 1 0 1 1 0 â ñ 0 0 ñ 0 ñ ñ 0 0 0 0 ñ2 0 0 0 0 8 0 1 1 0 1 0 0 1 â 0 ñ ñ 0 ñ 0 0 ñ 0 0 0 0 ñ2 0 0 0 9 1 0 0 1 1 0 0 1 â ñ 0 0 ñ ñ 0 0 ñ 0 0 0 0 0 ñ2 0 0 A 1 0 1 0 0 1 0 1 â ñ 0 ñ 0 0 ñ 0 ñ 0 0 0 0 0 0 ñ2 0 B 0 1 0 1 0 1 0 1 â 0 ñ 0 ñ 0 ñ 0 ñ 0 0 0 0 0 0 0 ñ2 C 1 1 0 0 0 0 1 1 â ñ ñ 0 0 0 0 ñ ñ D 0 0 1 1 0 0 1 1 â 0 0 ñ ñ 0 0 ñ ñ E 0 0 0 0 1 1 1 1 â 0 0 0 0 ñ ñ ñ ñ F 1 1 1 1 1 1 1 1 ( 224 vertices + 16 vertices )
Another decomposition gives the 240 points in 9-dimensions as an expanded 8-simplex, and two opposite birectified 8-simplexes, and .
This arises similarly to the relation of the A8 lattice and E8 lattice, sharing 8 mirrors of A8: .
This polytope is the vertex figure for a uniform tessellation of 8-dimensional space, represented by symbol 5<sub>21</sub> and Coxeter-Dynkin diagram:
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 (4<sub>11</sub>):
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 4<sub>21</sub> polytope has 120,960 (7ÃÂ17,280) 6-simplex faces that are facets of 7-simplexes. Since every 7-orthoplex has 128 (2<sup>7</sup>) 6-simplex facets, half of which are not incident to 7-simplexes, the 4<sub>21</sub> polytope has 138,240 (2<sup>6</sup>ÃÂ2160) 6-simplex faces that are not facets of 7-simplexes. The 4<sub>21</sub> 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 3<sub>21</sub> polytope.
Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.
These graphs represent orthographic projections in the E<sub>8</sub>, E<sub>7</sub>, E<sub>6</sub>, and B<sub>8</sub>, D<sub>8</sub>, D<sub>7</sub>, D<sub>6</sub>, D<sub>5</sub>, D<sub>4</sub>, D<sub>3</sub>, A<sub>7</sub>, A<sub>5</sub> Coxeter planes. The vertex colors are by overlapping multiplicity in the projection: colored by increasing order of multiplicities as red, orange, yellow, green.
The 4<sub>21</sub> polytope is last in a family called the k<sub>21</sub> 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).
The 4<sub>21</sub> 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 4<sub>21</sub> polytope are projected into 4-space as two copies of the 120 vertices of the 600-cell, one copy smaller (scaled by the golden ratio) 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 4<sub>21</sub>. The other 4 rings of the 4<sub>21</sub> graph also match a smaller copy of the four rings of the 600-cell.
In 4-dimensional complex geometry, the regular complex polytope <sub>3</sub>{3}<sub>3</sub>{3}<sub>3</sub>{3}<sub>3</sub>, and Coxeter diagram exists with the same vertex arrangement as the 4<sub>21</sub> polytope. It is self-dual. Coxeter called it the Witting polytope, after Alexander Witting. Coxeter expresses its Shephard group symmetry by <sub>3</sub>[3]<sub>3</sub>[3]<sub>3</sub>[3]<sub>3</sub>.
The 4<sub>21</sub> 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.
The rectified 4<sub>21</sub> can be seen as a rectification of the 4<sub>21</sub> polytope, creating new vertices on the center of edges of the 4<sub>21</sub>.
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 4<sub>21</sub>. Vertices are positioned at the midpoint of all the edges of 4<sub>21</sub>, 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 3<sub>21</sub>:
The vertex figure is determined by removing the ringed node and adding a ring to the neighboring node. This makes a 2<sub>21</sub> prism.
The Cartesian coordinates of the 6720 vertices of the rectified 4<sub>21</sub> is given by all permutations of coordinates from three other uniform polytope:
These graphs represent orthographic projections in the E<sub>8</sub>, E<sub>7</sub>, E<sub>6</sub>, and B<sub>8</sub>, D<sub>8</sub>, D<sub>7</sub>, D<sub>6</sub>, D<sub>5</sub>, D<sub>4</sub>, D<sub>3</sub>, A<sub>7</sub>, A<sub>5</sub> Coxeter planes. The vertex colors are by overlapping multiplicity in the projection: colored by increasing order of multiplicities as red, orange, yellow, green.
The birectified 4<sub>21</sub> can be seen as a second rectification of the uniform 4<sub>21</sub> polytope. Vertices of this polytope are positioned at the centers of all the 60480 triangular faces of the 4<sub>21</sub>.
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 4<sub>21</sub>. Vertices are positioned at the center of all the triangle faces of 4<sub>21</sub>.
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 3<sub>21</sub>. 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.
These graphs represent orthographic projections in the E<sub>8</sub>, E<sub>7</sub>, E<sub>6</sub>, and B<sub>8</sub>, D<sub>8</sub>, D<sub>7</sub>, D<sub>6</sub>, D<sub>5</sub>, D<sub>4</sub>, D<sub>3</sub>, A<sub>7</sub>, A<sub>5</sub> 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.
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 4<sub>21</sub>. Vertices are positioned at the center of all the triangle faces of 4<sub>21</sub>.
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 3<sub>21</sub>:
The vertex figure is determined by removing the ringed node and ring the neighbor nodes. This makes a tetrahedron-rectified 5-cell duoprism.
These graphs represent orthographic projections in the E<sub>7</sub>, E<sub>6</sub>, B<sub>8</sub>, D<sub>8</sub>, D<sub>7</sub>, D<sub>6</sub>, D<sub>5</sub>, D<sub>4</sub>, D<sub>3</sub>, A<sub>7</sub>, and A<sub>5</sub> Coxeter planes. The vertex colors are by overlapping multiplicity in the projection: colored by increasing order of multiplicities as red, orange, yellow, green.
(E<sub>8</sub> and B<sub>8</sub> were too large to display)