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3 21 polytope

## Summary

 Orthogonal projections in E7 Coxeter plane 321 231 132 Rectified 321 birectified 321 Rectified 231 Rectified 132

In 7-dimensional geometry, the 321 polytope is a uniform 7-polytope, constructed within the symmetry of the E7 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 7-ic semi-regular figure.[1]

Its Coxeter symbol is 321, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 3-node sequences.

The rectified 321 is constructed by points at the mid-edges of the 321. The birectified 321 is constructed by points at the triangle face centers of the 321. The trirectified 321 is constructed by points at the tetrahedral centers of the 321, and is the same as the rectified 132.

These polytopes are part of a family of 127 (27-1) convex uniform polytopes in 7-dimensions, made of uniform 6-polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .

## 321 polytope

321 polytope
Type Uniform 7-polytope
Family k21 polytope
Schläfli symbol {3,3,3,32,1}
Coxeter symbol 321
Coxeter diagram
6-faces 702 total:
126 311
576 {35}
5-faces 6048:
4032 {34}
2016 {34}
4-faces 12096 {33}
Cells 10080 {3,3}
Faces 4032 {3}
Edges 756
Vertices 56
Vertex figure 221 polytope
Petrie polygon octadecagon
Coxeter group E7, [33,2,1], order 2903040
Properties convex

In 7-dimensional geometry, the 321 is a uniform polytope. It has 56 vertices, and 702 facets: 126 311 and 576 6-simplexes.

For visualization this 7-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 56 vertices within an 18-gonal regular polygon (called a Petrie polygon). Its 756 edges are drawn between 3 rings of 18 vertices, and 2 vertices in the center. Specific higher elements (faces, cells, etc.) can also be extracted and drawn on this projection.

The 1-skeleton of the 321 polytope is the Gosset graph.

This polytope, along with the 7-simplex, can tessellate 7-dimensional space, represented by 331 and Coxeter-Dynkin diagram:          .

### Alternate names

• It is also called the Hess polytope for Edmund Hess who first discovered it.
• It was enumerated by Thorold Gosset in his 1900 paper. He called it an 7-ic semi-regular figure.[1]
• E. L. Elte named it V56 (for its 56 vertices) in his 1912 listing of semiregular polytopes.[2]
• H.S.M. Coxeter called it 321 due to its bifurcating Coxeter-Dynkin diagram, having 3 branches of length 3, 2, and 1, and having a single ring on the final node of the 3 branch.
• Hecatonicosihexa-pentacosiheptacontihexa-exon (Acronym Naq) - 126-576 facetted polyexon (Jonathan Bowers)[3]

### Coordinates

The 56 vertices can be most simply represented in 8-dimensional space, obtained by the 28 permutations of the coordinates and their opposite:

± (-3, -3, 1, 1, 1, 1, 1, 1)

### Construction

Its construction is based on the E7 group. Coxeter named it as 321 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 3-node sequence.

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

Removing the node on the short branch leaves the 6-simplex,            .

Removing the node on the end of the 2-length branch leaves the 6-orthoplex in its alternated form: 311,          .

Every simplex facet touches a 6-orthoplex facet, while alternate facets of the orthoplex touch either a simplex or another orthoplex.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 221 polytope,          .

Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.[4]

E7             k-face fk f0 f1 f2 f3 f4 f5 f6 k-figures notes
E6             ( ) f0 56 27 216 720 1080 432 216 72 27 221 E7/E6 = 72x8!/72x6! = 56
D5A1             { } f1 2 756 16 80 160 80 40 16 10 5-demicube E7/D5A1 = 72x8!/16/5!/2 = 756
A4A2             {3} f2 3 3 4032 10 30 20 10 5 5 rectified 5-cell E7/A4A2 = 72x8!/5!/2 = 4032
A3A2A1             {3,3} f3 4 6 4 10080 6 6 3 2 3 triangular prism E7/A3A2A1 = 72x8!/4!/3!/2 = 10080
A4A1             {3,3,3} f4 5 10 10 5 12096 2 1 1 2 isosceles triangle E7/A4A1 = 72x8!/5!/2 = 12096
A5A1             {3,3,3,3} f5 6 15 20 15 6 4032 * 1 1 { } E7/A5A1 = 72x8!/6!/2 = 4032
A5             6 15 20 15 6 * 2016 0 2 E7/A5 = 72x8!/6! = 2016
A6             {3,3,3,3,3} f6 7 21 35 35 21 10 0 576 * ( ) E7/A6 = 72x8!/7! = 576
D6             {3,3,3,3,4} 12 60 160 240 192 32 32 * 126 E7/D6 = 72x8!/32/6! = 126

### Images

Coxeter plane projections
E7 E6 / F4 B7 / A6

[18]

[12]

[7x2]
A5 D7 / B6 D6 / B5

[6]

[12/2]

[10]
D5 / B4 / A4 D4 / B3 / A2 / G2 D3 / B2 / A3

[8]

[6]

[4]

### Related polytopes

The 321 is fifth 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 = ${\displaystyle {\tilde {E}}_{8}}$  = E8+ E10 = ${\displaystyle {\bar {T}}_{8}}$  = 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 1,920 51,840 2,903,040 696,729,600
Graph             - -
Name −121 021 121 221 321 421 521 621

It is in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3k1 series. (A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron.)

3k1 dimensional figures
Space Finite Euclidean Hyperbolic
n 4 5 6 7 8 9
Coxeter
group
A3A1 A5 D6 E7 ${\displaystyle {\tilde {E}}_{7}}$ =E7+ ${\displaystyle {\bar {T}}_{8}}$ =E7++
Coxeter
diagram

Symmetry [3−1,3,1] [30,3,1] [[31,3,1]]
= [4,3,3,3,3]
[32,3,1] [33,3,1] [34,3,1]
Order 48 720 46,080 2,903,040
Graph       - -
Name 31,-1 310 311 321 331 341

## Rectified 321 polytope

Rectified 321 polytope
Type Uniform 7-polytope
Schläfli symbol t1{3,3,3,32,1}
Coxeter symbol t1(321)
Coxeter diagram
6-faces 758
5-faces 44352
4-faces 70560
Cells 48384
Faces 11592
Edges 12096
Vertices 756
Vertex figure 5-demicube prism
Petrie polygon octadecagon
Coxeter group E7, [33,2,1], order 2903040
Properties convex

### Alternate names

• Rectified hecatonicosihexa-pentacosiheptacontihexa-exon as a rectified 126-576 facetted polyexon (acronym ranq) (Jonathan Bowers)[5]

### Construction

Its construction is based on the E7 group. Coxeter named it as 321 by its bifurcating Coxeter-Dynkin diagram, with a single node on the end of the 3-node sequence.

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

Removing the node on the short branch leaves the 6-simplex,            .

Removing the node on the end of the 2-length branch leaves the rectified 6-orthoplex in its alternated form: t1311,          .

Removing the node on the end of the 3-length branch leaves the 221,          .

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 5-demicube prism,          .

### Images

Coxeter plane projections
E7 E6 / F4 B7 / A6

[18]

[12]

[7x2]
A5 D7 / B6 D6 / B5

[6]

[12/2]

[10]
D5 / B4 / A4 D4 / B3 / A2 / G2 D3 / B2 / A3

[8]

[6]

[4]

## Birectified 321 polytope

Birectified 321 polytope
Type Uniform 7-polytope
Schläfli symbol t2{3,3,3,32,1}
Coxeter symbol t2(321)
Coxeter diagram
6-faces 758
5-faces 12348
4-faces 68040
Cells 161280
Faces 161280
Edges 60480
Vertices 4032
Vertex figure 5-cell-triangle duoprism
Petrie polygon octadecagon
Coxeter group E7, [33,2,1], order 2903040
Properties convex

### Alternate names

• Birectified hecatonicosihexa-pentacosiheptacontihexa-exon as a birectified 126-576 facetted polyexon (acronym branq) (Jonathan Bowers)[6]

### Construction

Its construction is based on the E7 group. Coxeter named it as 321 by its bifurcating Coxeter-Dynkin diagram, with a single node on the end of the 3-node sequence.

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

Removing the node on the short branch leaves the birectified 6-simplex,            .

Removing the node on the end of the 2-length branch leaves the birectified 6-orthoplex in its alternated form: t2(311),          .

Removing the node on the end of the 3-length branch leaves the rectified 221 polytope in its alternated form: t1(221),          .

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes rectified 5-cell-triangle duoprism,          .

### Images

Coxeter plane projections
E7 E6 / F4 B7 / A6

[18]

[12]

[7x2]
A5 D7 / B6 D6 / B5

[6]

[12/2]

[10]
D5 / B4 / A4 D4 / B3 / A2 / G2 D3 / B2 / A3

[8]

[6]

[4]

## Notes

1. ^ a b Gosset, 1900
2. ^ Elte, 1912
3. ^ Klitzing, (o3o3o3o *c3o3o3x - naq)
4. ^ Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203
5. ^ Klitzing. (o3o3o3o *c3o3x3o - ranq)
6. ^ Klitzing, (o3o3o3o *c3x3o3o - branq)

## 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
• H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
• 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 p342 (figure 3.7c) by Peter mcMullen: (18-gonal node-edge graph of 321)
• Klitzing, Richard. "7D uniform polytopes (polyexa)". o3o3o3o *c3o3o3x - naq, o3o3o3o *c3o3x3o - ranq, o3o3o3o *c3x3o3o - branq