Its Coxeter symbol is 421, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 4-node sequences, .
The rectified 421 is constructed by points at the mid-edges of the 421. The birectified 421 is constructed by points at the triangle face centers of the 421. The trirectified 421 is constructed by points at the tetrahedral centers of the 421.
These polytopes are part of a family of 255 = 28 - 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: .
| bgcolor=#e7dcc3 colspan=2 | 421 | |
|---|---|---|
| Type | Uniform 8-polytope | |
| Family | k21 polytope | |
| Schläfli symbol | ||
| Coxeter symbol | 421 | |
| Coxeter diagrams | = | |
| 7-faces | 19440 total: 2160 411 17280 | |
| 6-faces | 207360: 138240 69120 | |
| 5-faces | 483840 | |
| 4-faces | 483840 | |
| Cells | 241920 | |
| Faces | 60480 | |
| Edges | 6720 | |
| Vertices | 240 | |
| Vertex figure | 321 polytope | |
| Petrie polygon | 30-gon | |
| Coxeter group | E8, [3<sup>4,2,1</sup>], order 696729600 | |
| Properties | convex |
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 421 polytope can be constructed in two sets: 112 with coordinates obtained from
(\pm2,\pm2,0,0,0,0,0,0)
(\pm1,\pm1,\pm1,\pm1,\pm1,\pm1,\pm1,\pm1)
Each vertex has 56 nearest neighbors; for example, the nearest neighbors of the vertex
(1,1,1,1,1,1,1,1)
(2,2,0,0,0,0,0,0)
(1,1,1,1,1,1,-1,-1)
Each vertex has 126 second nearest neighbors: for example, the nearest neighbors of the vertex
(1,1,1,1,1,1,1,1)
(2,-2,0,0,0,0,0,0)
(1,1,1,1,-1,-1,-1,-1)
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
Another construction is by taking signed combination of 14 codewords of 8-bit Extended Hamming code(8,4) that give 14 × 24 = 224 vertices and adding trivial signed axis
(\pm2,0,0,0,0,0,0,0)
\sqrt{4}
\sqrt{8}
Hamming 8-bit Code 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 ⇒ ± ± ± ± 0 0 0 0 2 1 1 0 0 1 1 0 0 ⇒ ± ± 0 0 ± ± 0 0 3 0 0 1 1 1 1 0 0 ⇒ 0 0 ± ± ± ± 0 0 4 1 0 1 0 1 0 1 0 ⇒ ± 0 ± 0 ± 0 ± 0 ±2 0 0 0 0 0 0 0 5 0 1 0 1 1 0 1 0 ⇒ 0 ± 0 ± ± 0 ± 0 0 ±2 0 0 0 0 0 0 6 0 1 1 0 0 1 1 0 ⇒ 0 ± ± 0 0 ± ± 0 0 0 ±2 0 0 0 0 0 7 1 0 0 1 0 1 1 0 ⇒ ± 0 0 ± 0 ± ± 0 0 0 0 ±2 0 0 0 0 8 0 1 1 0 1 0 0 1 ⇒ 0 ± ± 0 ± 0 0 ± 0 0 0 0 ±2 0 0 0 9 1 0 0 1 1 0 0 1 ⇒ ± 0 0 ± ± 0 0 ± 0 0 0 0 0 ±2 0 0 A 1 0 1 0 0 1 0 1 ⇒ ± 0 ± 0 0 ± 0 ± 0 0 0 0 0 0 ±2 0 B 0 1 0 1 0 1 0 1 ⇒ 0 ± 0 ± 0 ± 0 ± 0 0 0 0 0 0 0 ±2 C 1 1 0 0 0 0 1 1 ⇒ ± ± 0 0 0 0 ± ± D 0 0 1 1 0 0 1 1 ⇒ 0 0 ± ± 0 0 ± ± E 0 0 0 0 1 1 1 1 ⇒ 0 0 0 0 ± ± ± ± 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 .
: 72 vertices
: 84 vertices
: 84 verticesThis 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 521 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 (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.
Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.[4]
| Configuration matrix | ||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| E8 | width=80 | k-face | fk | f0 | f1 | f2 | f3 | f4 | f5 | f6 | f7 | k-figure | notes | |||
| E7 | f0 | 240 | 56 | 756 | 4032 | 10080 | 12096 | 4032 | 2016 | 576 | 126 | E8/E7 = 192×10!/(72×8 | ) = 240 | |||
| A1E6 | f1 | 2 | 6720 | 27 | 216 | 720 | 1080 | 432 | 216 | 72 | 27 | E8/A1E6 = 192×10!/(2×72×6 | ) = 6720 | |||
| A2D5 | f2 | 3 | 3 | 60480 | 16 | 80 | 160 | 80 | 40 | 16 | 10 | E8/A2D5 = 192×10!/(6×24×5 | ) = 60480 | |||
| A3A4 | f3 | 4 | 6 | 4 | 241920 | 10 | 30 | 20 | 10 | 5 | 5 | E8/A3A4 = 192×10!/(4 | ×5!) = 241920 | |||
| A4A2A1 | f4 | 5 | 10 | 10 | 5 | 483840 | 6 | 6 | 3 | 2 | 3 | E8/A4A2A1 = 192×10!/(5 | ×3!×2) = 483840 | |||
| A5A1 | f5 | 6 | 15 | 20 | 15 | 6 | 483840 | 2 | 1 | 1 | 2 | E8/A5A1 = 192×10!/(6 | ×2) = 483840 | |||
| A6 | f6 | 7 | 21 | 35 | 35 | 21 | 7 | 138240 | 1 | 1 | E8/A6 = 192×(10!×7 | ) = 138240 | ||||
| A6A1 | 7 | 21 | 35 | 35 | 21 | 7 | 69120 | 0 | 2 | E8/A6A1 = 192×10!/(7 | ×2) = 69120 | |||||
| A7 | f7 | 8 | 28 | 56 | 70 | 56 | 28 | 8 | 0 | 17280 | E8/A7 = 192×10!/8 | = 17280 | ||||
| D7 | 14 | 84 | 280 | 560 | 672 | 448 | 64 | 64 | 2160 | E8/D7 = 192×10!/(26×7 | ) = 2160 | |||||
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.
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).
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 (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 421. The other 4 rings of the 421 graph also match a smaller copy of the four rings of the 600-cell.
In 4-dimensional complex geometry, the regular complex polytope 3333, 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.[6]
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.
| bgcolor=#e7dcc3 colspan=2 | Rectified 421 | |
|---|---|---|
| Type | Uniform 8-polytope | |
| Schläfli symbol | t1 | |
| Coxeter symbol | t1(421) | |
| Coxeter diagram | ||
| 7-faces | 19680 total: 240 321 17280 t1 2160 t1 | |
| 6-faces | 375840 | |
| 5-faces | 1935360 | |
| 4-faces | 3386880 | |
| Cells | 2661120 | |
| Faces | 1028160 | |
| Edges | 181440 | |
| Vertices | 6720 | |
| Vertex figure | 221 prism | |
| Coxeter group | E8, [3<sup>4,2,1</sup>] | |
| Properties | convex |
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.
The Cartesian coordinates of the 6720 vertices of the rectified 421 is given by all permutations of coordinates from three other uniform polytope:
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.
| bgcolor=#e7dcc3 colspan=2 | Birectified 421 polytope | |
|---|---|---|
| Type | Uniform 8-polytope | |
| Schläfli symbol | t2 | |
| Coxeter symbol | t2(421) | |
| Coxeter diagram | ||
| 7-faces | ||
| 6-faces | 382560 | |
| 5-faces | 2600640 | |
| 4-faces | 7741440 | |
| Cells | 9918720 | |
| Faces | 5806080 | |
| Edges | 1451520 | |
| Vertices | 60480 | |
| Vertex figure | 5-demicube-triangular duoprism | |
| Coxeter group | E8, [3<sup>4,2,1</sup>] | |
| Properties | convex |
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.
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.
| bgcolor=#e7dcc3 colspan=2 | Trirectified 421 polytope | |
|---|---|---|
| Type | Uniform 8-polytope | |
| Schläfli symbol | t3 | |
| Coxeter symbol | t3(421) | |
| Coxeter diagram | ||
| 7-faces | 19680 | |
| 6-faces | 382560 | |
| 5-faces | 2661120 | |
| 4-faces | 9313920 | |
| Cells | 16934400 | |
| Faces | 14515200 | |
| Edges | 4838400 | |
| Vertices | 241920 | |
| Vertex figure | tetrahedron-rectified 5-cell duoprism | |
| Coxeter group | E8, [3<sup>4,2,1</sup>] | |
| Properties | convex |
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.
These graphs represent orthographic projections in the E7, E6, B8, D8, D7, D6, D5, D4, D3, A7, and 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)