Orthogonal projections in E_{6} Coxeter plane  

4_{21} 
1_{42} 
2_{41} 
Rectified 4_{21} 
Rectified 1_{42} 
Rectified 2_{41} 
Birectified 4_{21} 
Trirectified 4_{21} 
In 8dimensional geometry, the 4_{21} is a semiregular uniform 8polytope, constructed within the symmetry of the E_{8} group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 8ic semiregular figure.^{[1]}
Its Coxeter symbol is 4_{21}, describing its bifurcating CoxeterDynkin diagram, with a single ring on the end of the 4node sequences, .
The rectified 4_{21} is constructed by points at the midedges of the 4_{21}. The birectified 4_{21} is constructed by points at the triangle face centers of the 4_{21}. The trirectified 4_{21} is constructed by points at the tetrahedral centers of the 4_{21}.
These polytopes are part of a family of 255 = 2^{8} − 1 convex uniform 8polytopes, made of uniform 7polytope facets and vertex figures, defined by all permutations of one or more rings in this CoxeterDynkin diagram: .
4_{21}  

Type  Uniform 8polytope 
Family  k_{21} polytope 
Schläfli symbol  {3,3,3,3,3^{2,1}} 
Coxeter symbol  4_{21} 
Coxeter diagrams  = 
7faces  19440 total: 2160 4_{11} 17280 {3^{6}} 
6faces  207360: 138240 {3^{5}} 69120 {3^{5}} 
5faces  483840 {3^{4}} 
4faces  483840 {3^{3}} 
Cells  241920 {3,3} 
Faces  60480 {3} 
Edges  6720 
Vertices  240 
Vertex figure  3_{21} polytope 
Petrie polygon  30gon 
Coxeter group  E_{8}, [3^{4,2,1}], order 696729600 
Properties  convex 
The 4_{21} polytope has 17,280 7simplex and 2,160 7orthoplex facets, and 240 vertices. Its vertex figure is the 3_{21} polytope. As its vertices represent the root vectors of the simple Lie group E_{8}, this polytope is sometimes referred to as the E_{8} 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 8dimensional 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 8dimensional space.
The 240 vertices of the 4_{21} polytope can be constructed in two sets: 112 (2^{2}×^{8}C_{2}) with coordinates obtained from by taking an arbitrary combination of signs and an arbitrary permutation of coordinates, and 128 roots (2^{7}) 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_{21} 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_{31} 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 8bit Extended Hamming code(8,4) that give 14 × 2^{4} = 224 vertices and adding trivial signed axis for last 16 vertices. In this case, vertices are distance of from origin rather than .
Hamming 8bit 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 9dimensions as an expanded 8simplex, and two opposite birectified 8simplexes, and .
This arises similarly to the relation of the A8 lattice and E8 lattice, sharing 8 mirrors of A8:
.Name  4_{21} 
expanded 8simplex 
birectified 8simplex 
birectified 8simplex 

Vertices  240  72  84  84 
Image 
This polytope is the vertex figure for a uniform tessellation of 8dimensional space, represented by symbol 5_{21} and CoxeterDynkin diagram:
The facet information of this polytope can be extracted from its CoxeterDynkin diagram:
Removing the node on the short branch leaves the 7simplex:
Removing the node on the end of the 2length branch leaves the 7orthoplex in its alternated form (4_{11}):
Every 7simplex facet touches only 7orthoplex 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 7simplex has 7 6simplex facets, each incident to no other 6simplex, the 4_{21} polytope has 120,960 (7×17,280) 6simplex faces that are facets of 7simplexes. Since every 7orthoplex has 128 (2^{7}) 6simplex facets, half of which are not incident to 7simplexes, the 4_{21} polytope has 138,240 (2^{6}×2160) 6simplex faces that are not facets of 7simplexes. The 4_{21} polytope thus has two kinds of 6simplex faces, not interchanged by symmetries of this polytope. The total number of 6simplex faces is 259200 (120,960+138,240).
The vertex figure of a singlering polytope is obtained by removing the ringed node and ringing its neighbor(s). This makes the 3_{21} polytope.
Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.^{[4]}
E_{8}  kface  f_{k}  f_{0}  f_{1}  f_{2}  f_{3}  f_{4}  f_{5}  f_{6}  f_{7}  kfigure  notes  

E_{7}  ( )  f_{0}  240  56  756  4032  10080  12096  4032  2016  576  126  3_21 polytope  E_{8}/E_{7} = 192×10!/(72×8!) = 240  
A_{1}E_{6}  { }  f_{1}  2  6720  27  216  720  1080  432  216  72  27  2_21 polytope  E_{8}/A_{1}E_{6} = 192×10!/(2×72×6!) = 6720  
A_{2}D_{5}  {3}  f_{2}  3  3  60480  16  80  160  80  40  16  10  5demicube  E_{8}/A_{2}D_{5} = 192×10!/(6×2^{4}×5!) = 60480  
A_{3}A_{4}  {3,3}  f_{3}  4  6  4  241920  10  30  20  10  5  5  Rectified 5cell  E_{8}/A_{3}A_{4} = 192×10!/(4!×5!) = 241920  
A_{4}A_{2}A_{1}  {3,3,3}  f_{4}  5  10  10  5  483840  6  6  3  2  3  Triangular prism  E_{8}/A_{4}A_{2}A_{1} = 192×10!/(5!×3!×2) = 483840  
A_{5}A_{1}  {3,3,3,3}  f_{5}  6  15  20  15  6  483840  2  1  1  2  Isosceles triangle  E_{8}/A_{5}A_{1} = 192×10!/(6!×2) = 483840  
A_{6}  {3,3,3,3,3}  f_{6}  7  21  35  35  21  7  138240  *  1  1  { }  E_{8}/A_{6} = 192×(10!×7!) = 138240  
A_{6}A_{1}  7  21  35  35  21  7  *  69120  0  2  E_{8}/A_{6}A_{1} = 192×10!/(7!×2) = 69120  
A_{7}  {3,3,3,3,3,3}  f_{7}  8  28  56  70  56  28  8  0  17280  *  ( )  E_{8}/A_{7} = 192×10!/8! = 17280  
D_{7}  {3,3,3,3,3,4}  14  84  280  560  672  448  64  64  *  2160  E_{8}/D_{7} = 192×10!/(2^{6}×7!) = 2160 
The 4_{21} graph created as string art. 
E_{8} Coxeter plane projection 
Mathematical representation of the physical Zome model isomorphic (?) to E8. This is constructed from VisibLie_E8 pictured with all 3360 edges of length √2(√51) from two concentric 600cells (at the golden ratio) with orthogonal projections to perspective 3space 
The actual split real even E8 4_{21} polytope projected into perspective 3space pictured with all 6720 edges of length √2 ^{[5]} 
E8 rotated to H4+H4φ, projected to 3D, converted to STL, and printed in nylon plastic. Projection basis used:

These graphs represent orthographic projections in the E_{8}, E_{7}, E_{6}, and B_{8}, D_{8}, D_{7}, D_{6}, D_{5}, D_{4}, D_{3}, A_{7}, A_{5} Coxeter planes. The vertex colors are by overlapping multiplicity in the projection: colored by increasing order of multiplicities as red, orange, yellow, green.
Orthogonal projections  

E_{8} / H_{4} [30] 
[20]  [24] 
(Colors: 1) 
(Colors: 1) 
(Colors: 1) 
E_{7} [18] 
E_{6} / F_{4} [12] 
[6] 
(Colors: 1,3,6) 
(Colors: 1,8,24) 
(Colors: 1,2,3) 
D_{3} / B_{2} / A_{3} [4] 
D_{4} / B_{3} / A_{2} / G_{2} [6] 
D_{5} / B_{4} [8] 
(Colors: 1,12,32,60) 
(Colors: 1,27,72) 
(Colors: 1,8,24) 
D_{6} / B_{5} / A_{4} [10] 
D_{7} / B_{6} [12] 
D_{8} / B_{7} / A_{6} [14] 
(Colors: 1,5,10,20) 
(Colors: 1,3,9,12) 
(Colors: 1,2,3) 
B_{8} [16/2] 
A_{5} [6] 
A_{7} [8] 
(Colors: 1) 
(Colors: 3,8,24,30) 
(Colors: 1,2,4,8) 
The 4_{21} polytope is last in a family called the k_{21} polytopes. The first polytope in this family is the semiregular triangular prism which is constructed from three squares (2orthoplexes) and two triangles (2simplexes).
The 4_{21} is related to the 600cell by a geometric folding of the CoxeterDynkin diagrams. This can be seen in the E8/H4 Coxeter plane projections. The 240 vertices of the 4_{21} polytope are projected into 4space as two copies of the 120 vertices of the 600cell, 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 600cell are projected in the same four rings as seen in the 4_{21}. The other 4 rings of the 4_{21} graph also match a smaller copy of the four rings of the 600cell.
E8/H4 Coxeter plane foldings  

E_{8}  H_{4} 
4_{21} 
600cell 
[20] symmetry planes  
4_{21} 
600cell 
In 4dimensional complex geometry, the regular complex polytope _{3}{3}_{3}{3}_{3}{3}_{3}, and Coxeter diagram exists with the same vertex arrangement as the 4_{21} polytope. It is selfdual. Coxeter called it the Witting polytope, after Alexander Witting. Coxeter expresses its Shephard group symmetry by _{3}[3]_{3}[3]_{3}[3]_{3}.^{[7]}
The 4_{21} 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.
k_{21} figures in n dimensional  

Space  Finite  Euclidean  Hyperbolic  
E_{n}  3  4  5  6  7  8  9  10  
Coxeter group 
E_{3}=A_{2}A_{1}  E_{4}=A_{4}  E_{5}=D_{5}  E_{6}  E_{7}  E_{8}  E_{9} = = E_{8}^{+}  E_{10} = = E_{8}^{++}  
Coxeter diagram 

Symmetry  [3^{−1,2,1}]  [3^{0,2,1}]  [3^{1,2,1}]  [3^{2,2,1}]  [3^{3,2,1}]  [3^{4,2,1}]  [3^{5,2,1}]  [3^{6,2,1}]  
Order  12  120  1,920  51,840  2,903,040  696,729,600  ∞  
Graph      
Name  −1_{21}  0_{21}  1_{21}  2_{21}  3_{21}  4_{21}  5_{21}  6_{21} 
Rectified 4_{21}  

Type  Uniform 8polytope 
Schläfli symbol  t_{1}{3,3,3,3,3^{2,1}} 
Coxeter symbol  t_{1}(4_{21}) 
Coxeter diagram  
7faces  19680 total: 240 3_{21} 
6faces  375840 
5faces  1935360 
4faces  3386880 
Cells  2661120 
Faces  1028160 
Edges  181440 
Vertices  6720 
Vertex figure  2_{21} prism 
Coxeter group  E_{8}, [3^{4,2,1}] 
Properties  convex 
The rectified 4_{21} can be seen as a rectification of the 4_{21} polytope, creating new vertices on the center of edges of the 4_{21}.
It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8dimensional space. It is named for being a rectification of the 4_{21}. Vertices are positioned at the midpoint of all the edges of 4_{21}, and new edges connecting them.
The facet information can be extracted from its CoxeterDynkin diagram.
Removing the node on the short branch leaves the rectified 7simplex:
Removing the node on the end of the 2length branch leaves the rectified 7orthoplex in its alternated form:
Removing the node on the end of the 4length branch leaves the 3_{21}:
The vertex figure is determined by removing the ringed node and adding a ring to the neighboring node. This makes a 2_{21} prism.
The Cartesian coordinates of the 6720 vertices of the rectified 4_{21} is given by all permutations of coordinates from three other uniform polytope:
Name  Rectified 4_{21} 
birectified 8cube = 
hexic 8cube = 
cantellated 8orthoplex = 

Vertices  6720  1792  3584  1344 
Image 
These graphs represent orthographic projections in the E_{8}, E_{7}, E_{6}, and B_{8}, D_{8}, D_{7}, D_{6}, D_{5}, D_{4}, D_{3}, A_{7}, A_{5} Coxeter planes. The vertex colors are by overlapping multiplicity in the projection: colored by increasing order of multiplicities as red, orange, yellow, green.
Orthogonal projections  

E_{8} / H_{4} [30] 
[20]  [24] 
E_{7} [18] 
E_{6} / F_{4} [12] 
[6] 
D_{3} / B_{2} / A_{3} [4] 
D_{4} / B_{3} / A_{2} / G_{2} [6] 
D_{5} / B_{4} [8] 
D_{6} / B_{5} / A_{4} [10] 
D_{7} / B_{6} [12] 
D_{8} / B_{7} / A_{6} [14] 
B_{8} [16/2] 
A_{5} [6] 
A_{7} [8] 
Birectified 4_{21} polytope  

Type  Uniform 8polytope 
Schläfli symbol  t_{2}{3,3,3,3,3^{2,1}} 
Coxeter symbol  t_{2}(4_{21}) 
Coxeter diagram  
7faces  19680 total: 17280 t_{2}{3^{6}} 2160 t_{2}{3^{5},4} 240 t_{1}(3_{21}) 
6faces  382560 
5faces  2600640 
4faces  7741440 
Cells  9918720 
Faces  5806080 
Edges  1451520 
Vertices  60480 
Vertex figure  5demicubetriangular duoprism 
Coxeter group  E_{8}, [3^{4,2,1}] 
Properties  convex 
The birectified 4_{21} can be seen as a second rectification of the uniform 4_{21} polytope. Vertices of this polytope are positioned at the centers of all the 60480 triangular faces of the 4_{21}.
It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8dimensional space. It is named for being a birectification of the 4_{21}. Vertices are positioned at the center of all the triangle faces of 4_{21}.
The facet information can be extracted from its CoxeterDynkin diagram.
Removing the node on the short branch leaves the birectified 7simplex. There are 17280 of these facets.
Removing the node on the end of the 2length branch leaves the birectified 7orthoplex in its alternated form. There are 2160 of these facets.
Removing the node on the end of the 4length branch leaves the rectified 3_{21}. 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 5demicubetriangular duoprism.
These graphs represent orthographic projections in the E_{8}, E_{7}, E_{6}, and B_{8}, D_{8}, D_{7}, D_{6}, D_{5}, D_{4}, D_{3}, A_{7}, A_{5} 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.
Orthogonal projections  

E_{8} / H_{4} [30] 
[20]  [24] 
E_{7} [18] 
E_{6} / F_{4} [12] 
[6] 
D_{3} / B_{2} / A_{3} [4] 
D_{4} / B_{3} / A_{2} / G_{2} [6] 
D_{5} / B_{4} [8] 
D_{6} / B_{5} / A_{4} [10] 
D_{7} / B_{6} [12] 
D_{8} / B_{7} / A_{6} [14] 
B_{8} [16/2] 
A_{5} [6] 
A_{7} [8] 
Trirectified 4_{21} polytope  

Type  Uniform 8polytope 
Schläfli symbol  t_{3}{3,3,3,3,3^{2,1}} 
Coxeter symbol  t_{3}(4_{21}) 
Coxeter diagram  
7faces  19680 
6faces  382560 
5faces  2661120 
4faces  9313920 
Cells  16934400 
Faces  14515200 
Edges  4838400 
Vertices  241920 
Vertex figure  tetrahedronrectified 5cell duoprism 
Coxeter group  E_{8}, [3^{4,2,1}] 
Properties  convex 
It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8dimensional space. It is named for being a birectification of the 4_{21}. Vertices are positioned at the center of all the triangle faces of 4_{21}.
The facet information can be extracted from its CoxeterDynkin diagram.
Removing the node on the short branch leaves the trirectified 7simplex:
Removing the node on the end of the 2length branch leaves the trirectified 7orthoplex in its alternated form:
Removing the node on the end of the 4length branch leaves the birectified 3_{21}:
The vertex figure is determined by removing the ringed node and ring the neighbor nodes. This makes a tetrahedronrectified 5cell duoprism.
These graphs represent orthographic projections in the E_{7}, E_{6}, B_{8}, D_{8}, D_{7}, D_{6}, D_{5}, D_{4}, D_{3}, A_{7}, and A_{5} Coxeter planes. The vertex colors are by overlapping multiplicity in the projection: colored by increasing order of multiplicities as red, orange, yellow, green.
(E_{8} and B_{8} were too large to display)
Orthogonal projections  

E_{7} [18] 
E_{6} / F_{4} [12] 
D_{4}  E_{6} [6] 
D_{3} / B_{2} / A_{3} [4] 
D_{4} / B_{3} / A_{2} / G_{2} [6] 
D_{5} / B_{4} [8] 
D_{6} / B_{5} / A_{4} [10] 
D_{7} / B_{6} [12] 
D_{8} / B_{7} / A_{6} [14] 
A_{5} [6] 
A_{7} [8]  