# 4 21 polytope

Orthogonal projections in E6 Coxeter plane

421

142

241

Rectified 421

Rectified 142

Rectified 241

Birectified 421

Trirectified 421

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. He called it an 8-ic semi-regular figure.[1]

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, and is the same as the rectified 142.

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: .

## 421 polytope

421
Type Uniform 8-polytope
Family k21 polytope
Schläfli symbol {3,3,3,3,32,1}
Coxeter symbol 421
Coxeter diagram
7-faces 19440 total:
2160 411
17280 {36}
6-faces 207360:
138240 {35}
69120 {35}
5-faces 483840 {34}
4-faces 483840 {33}
Cells 241920 {3,3}
Faces 60480 {3}
Edges 6720
Vertices 240
Vertex figure 321 polytope
Petrie polygon 30-gon
Coxeter group E8, [34,2,1]
Properties convex

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

• This polytope was discovered by Thorold Gosset, who described it in his 1900 paper as an 8-ic semi-regular figure.[1] It is the last finite semiregular figure in his enumeration, semiregular to him meaning that it contained only regular facets.
• E. L. Elte named it V240 (for its 240 vertices) in his 1912 listing of semiregular polytopes.[2]
• H.S.M. Coxeter called it 421 because its Coxeter-Dynkin diagram has three branches of length 4, 2, and 1, with a single node on the terminal node of the 4 branch.
• Dischiliahectohexaconta-myriaheptachiliadiacosioctaconta-zetton (Acronym Fy) - 2160-17280 facetted polyzetton (Jonathan Bowers)[3]

### 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 $(\pm 2,\pm 2,0,0,0,0,0,0)\,$ by taking an arbitrary combination of signs and an arbitrary permutation of coordinates, and 128 roots (27) with coordinates obtained from $(\pm 1,\pm 1,\pm 1,\pm 1,\pm 1,\pm 1,\pm 1,\pm 1)\,$ 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.

### Projections

 The 421 graph created as string art. E8 Coxeter plane projection

#### 3D

 Mathematical representation of the physical Zome model isomorphic (?) to E8. This is constructed from VisibLie_E8 pictured with all 3360 edges of length √2(√5-1) from two concentric 600-cells (at the golden ratio) with orthogonal projections to perspective 3-space The actual split real even E8 421 polytope projected into perspective 3-space pictured with all 6720 edges of length √2[4]

#### 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.

E8 / H4
[30]
[20] [24]

(Colors: 1)

(Colors: 1)

(Colors: 1)
E7
[18]
E6 / F4
[12]
[6]

(Colors: 1,3,6)

(Colors: 1,8,24)

(Colors: 1,2,3)
D3 / B2 / A3
[4]
D4 / B3 / A2 / G2
[6]
D5 / B4
[8]

(Colors: 1,12,32,60)

(Colors: 1,27,72)

(Colors: 1,8,24)
D6 / B5 / A4
[10]
D7 / B6
[12]
D8 / B7 / A6
[14]

(Colors: 1,5,10,20)

(Colors: 1,3,9,12)

(Colors: 1,2,3)
B8
[16/2]
A5
[6]
A7
[8]

(Colors: 1)

(Colors: 3,8,24,30)

(Colors: 1,2,4,8)

### 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 polytope can be projected into 3-space as a physical vertex-edge model. Pictured here as 2 concentric 600-cells (at the golden ratio) using Zome tools.[5] (Not all of the 3360 edges of length √2(√5-1) are represented.)

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.

E8/H4 Coxeter planes
E8 H4

421

600-cell
[20] symmetry planes

421

600-cell

### Related polytopes

Using a complex number coordinate system, it can also be constructed as a 4-dimensional regular complex polytope, named as: 3{3}3{3}3{3}3. Coxeter called it the Witting polytope, after Alexander Witting.[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.

k21 figures in n dimensional
Space Finite Euclidean Hyperbolic
En 3 4 5 6 7 8 9 10
Coxeter
group
E3=A2A1 E4=A4 E5=D5 E6 E7 E8 E9 = ${\tilde{E}}_{8}$ = E8+ E10 = ${\bar{T}}_8$ = E8++
Coxeter
diagram
Symmetry [3−1,2,1] [30,2,1] [31,2,1] [32,2,1] [33,2,1] [34,2,1] [35,2,1] [36,2,1]
Order 12 120 192 51,840 2,903,040 696,729,600
Graph - -
Name −121 021 121 221 321 421 521 621

## Rectified 4_21 polytope

Rectified 421
Type Uniform 8-polytope
Schläfli symbol t1{3,3,3,3,32,1}
Coxeter symbol t1(421)
Coxeter diagram
7-faces 19680 total:

240 321
17280 t1{36}
2160 t1{35,4}

6-faces 375840
5-faces 1935360
4-faces 3386880
Cells 2661120
Faces 1028160
Edges 181440
Vertices 6720
Vertex figure 221 prism
Coxeter group E8, [34,2,1]
Properties convex

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

• Rectified dischiliahectohexaconta-myriaheptachiliadiacosioctaconta-zetton for rectified 2160-17280 polyzetton (Acronym riffy) (Jonathan Bowers)[7]

### 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.

### Projections

#### 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.

E8 / H4
[30]
[20] [24]
E7
[18]
E6 / F4
[12]
[6]
D3 / B2 / A3
[4]
D4 / B3 / A2 / G2
[6]
D5 / B4
[8]
D6 / B5 / A4
[10]
D7 / B6
[12]
D8 / B7 / A6
[14]
B8
[16/2]
A5
[6]
A7
[8]

## Birectified 4_21 polytope

Birectified 421 polytope
Type Uniform 8-polytope
Schläfli symbol t2{3,3,3,3,32,1}
Coxeter symbol t2(421)
Coxeter diagram
7-faces 19680 total:

17280 t2{36}
2160 t2{35,4}
240 t1(321)

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, [34,2,1]
Properties convex

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

• Birectified dischiliahectohexaconta-myriaheptachiliadiacosioctaconta-zetton for birectified 2160-17280 polyzetton (acronym borfy) (Jonathan Bowers)[8]

### 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.

### Projections

#### 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.

E8 / H4
[30]
[20] [24]
E7
[18]
E6 / F4
[12]
[6]
D3 / B2 / A3
[4]
D4 / B3 / A2 / G2
[6]
D5 / B4
[8]
D6 / B5 / A4
[10]
D7 / B6
[12]
D8 / B7 / A6
[14]
B8
[16/2]
A5
[6]
A7
[8]

## Trirectified 4_21 polytope

Trirectified 421 polytope
Type Uniform 8-polytope
Schläfli symbol t3{3,3,3,3,32,1}
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, [34,2,1]
Properties convex

### Alternative names

• Trirectified dischiliahectohexaconta-myriaheptachiliadiacosioctaconta-zetton for trirectified 2160-17280 polyzetton (acronym torfy) (Jonathan Bowers)[9]

### 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.

### Projections

#### 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)

E7
[18]
E6 / F4
[12]
D4 - E6
[6]
D3 / B2 / A3
[4]
D4 / B3 / A2 / G2
[6]
D5 / B4
[8]
D6 / B5 / A4
[10]
D7 / B6
[12]
D8 / B7 / A6
[14]
A5
[6]
A7
[8]

## Notes

1. ^ a b Gosset, 1900
2. ^ Elte, 1912
3. ^ Klitzing, (o3o3o3o *c3o3o3o3x - fy)
4. ^ e8Flyer.nb
5. ^ David Richter: Gosset's Figure in 8 Dimensions, A Zome Model
6. ^ Coxeter Regular Convex Polytopes, 12.5 The Witting polytope
7. ^ Klitzing, (o3o3o3o *c3o3o3x3o - riffy)
8. ^ Klitzing, (o3o3o3o *c3o3x3o3o - borfy)
9. ^ Klitzing, (o3o3o3o *c3x3o3o3o - torfy)

## References

• T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
• Elte, E. L. (1912), The Semiregular Polytopes of the Hyperspaces, Groningen: University of Groningen
• Coxeter, H. S. M., Regular Complex Polytopes, Cambridge University Press, (1974).
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] See p347 (figure 3.8c) by Peter McMullen: (30-gonal node-edge graph of 421)
• Richard Klitzing, 8D, uniform polytopes (polyzetta) o3o3o3o *c3o3o3o3x - fy, o3o3o3o *c3o3o3x3o - riffy, o3o3o3o *c3o3x3o3o - borfy, o3o3o3o *c3x3o3o3o - torfy