# 3 21 polytope

Last updated
 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]

## Contents

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-faces702 total:
126 311
576 {35}
5-faces6048:
4032 {34}
2016 {34}
4-faces12096 {33}
Cells10080 {3,3}
Faces4032 {3}
Edges756
Vertices56
Vertex figure 221 polytope
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]

E7k-facefkf0f1f2f3f4f5f6 k-figures notes
E6( )f0562721672010804322167227 221 E7/E6 = 72x8!/72x6! = 56
D5A1{ }f12756168016080401610 5-demicube E7/D5A1 = 72x8!/16/5!/2 = 756
A4A2 {3} f23340321030201055 rectified 5-cell E7/A4A2 = 72x8!/5!/2 = 4032
A3A2A1 {3,3} f34641008066323 triangular prism E7/A3A2A1 = 72x8!/4!/3!/2 = 10080
A4A1 {3,3,3} f4510105120962112 isosceles triangle E7/A4A1 = 72x8!/5!/2 = 12096
A5A1 {3,3,3,3} f5615201564032*11{ }E7/A5A1 = 72x8!/6!/2 = 4032
A561520156*201602E7/A5 = 72x8!/6! = 2016
A6 {3,3,3,3,3} f6721353521100576*( )E7/A6 = 72x8!/7! = 576
D6 {3,3,3,3,4} 12601602401923232*126E7/D6 = 72x8!/32/6! = 126

### Images

Coxeter plane projections
E7E6 / F4B7 / A6

[18]

[12]

[7x2]
A5D7 / B6D6 / B5

[6]

[12/2]

[10]
D5 / B4 / A4D4 / B3 / A2 / G2D3 / B2 / A3

[8]

[6]

[4]

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.

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
SpaceFiniteEuclideanHyperbolic
n 4 5 6 7 8 9
Coxeter
group
A3A1A5D6 E7 ${\displaystyle {\tilde {E}}_{7}}$=E7+${\displaystyle {\bar {T}}_{8}}$=E7++
Coxeter
diagram
Symmetry [3−1,3,1][30,3,1][[3<sup>1,3,1</sup>]]
= [4,3,3,3,3]
[32,3,1][33,3,1][34,3,1]
Order 4872046,0802,903,040
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-faces758
5-faces44352
4-faces70560
Cells48384
Faces11592
Edges12096
Vertices756
Vertex figure 5-demicube prism
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
E7E6 / F4B7 / A6

[18]

[12]

[7x2]
A5D7 / B6D6 / B5

[6]

[12/2]

[10]
D5 / B4 / A4D4 / B3 / A2 / G2D3 / 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-faces758
5-faces12348
4-faces68040
Cells161280
Faces161280
Edges60480
Vertices4032
Vertex figure 5-cell-triangle duoprism
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
E7E6 / F4B7 / A6

[18]

[12]

[7x2]
A5D7 / B6D6 / B5

[6]

[12/2]

[10]
D5 / B4 / A4D4 / B3 / A2 / G2D3 / B2 / A3

[8]

[6]

[4]

## Notes

1. 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)

## Related Research Articles

In six-dimensional geometry, a uniform polypeton is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes.

In geometry, the Gosset–Elte figures, named by Coxeter after Thorold Gosset and E. L. Elte, are a group of uniform polytopes which are not regular, generated by a Wythoff construction with mirrors all related by order-2 and order-3 dihedral angles. They can be seen as one-end-ringed Coxeter–Dynkin diagrams.

In geometry, a uniform k21 polytope is a polytope in k + 4 dimensions constructed from the En Coxeter group, and having only regular polytope facets. The family was named by their Coxeter symbol k21 by its bifurcating Coxeter–Dynkin diagram, with a single ring on the end of the k-node sequence.

In five-dimensional geometry, a rectified 5-simplex is a convex uniform 5-polytope, being a rectification of the regular 5-simplex.

In 7-dimensional geometry, 231 is a uniform polytope, constructed from the E7 group.

In 6-dimensional geometry, the 122 polytope is a uniform polytope, constructed from the E6 group. It was first published in E. L. Elte's 1912 listing of semiregular polytopes, named as V72 (for its 72 vertices).

In 7-dimensional geometry, 132 is a uniform polytope, constructed from the E7 group.

In 8-dimensional geometry, the 142 is a uniform 8-polytope, constructed within the symmetry of the E8 group.

In 8-dimensional geometry, the 241 is a uniform 8-polytope, constructed within the symmetry of the E8 group.

In 6-dimensional geometry, the 221 polytope is a uniform 6-polytope, constructed within the symmetry of the E6 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 6-ic semi-regular figure. It is also called the Schläfli polytope.

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.

In geometry, the 222 honeycomb is a uniform tessellation of the six-dimensional Euclidean space. It can be represented by the Schläfli symbol {3,3,32,2}. It is constructed from 221 facets and has a 122 vertex figure, with 54 221 polytopes around every vertex.

In 7-dimensional geometry, the 331 honeycomb is a uniform honeycomb, also given by Schläfli symbol {3,3,3,33,1} and is composed of 321 and 7-simplex facets, with 56 and 576 of them respectively around each vertex.

In 7-dimensional geometry, 133 is a uniform honeycomb, also given by Schläfli symbol {3,33,3}, and is composed of 132 facets.

In six-dimensional geometry, a pentellated 6-simplex is a convex uniform 6-polytope with 5th order truncations of the regular 6-simplex.

In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex.

In geometry, the 521 honeycomb is a uniform tessellation of 8-dimensional Euclidean space. The symbol 521 is from Coxeter, named for the length of the 3 branches of its Coxeter-Dynkin diagram.

In seven-dimensional geometry, a rectified 7-simplex is a convex uniform 7-polytope, being a rectification of the regular 7-simplex.

In eight-dimensional geometry, a rectified 8-simplex is a convex uniform 8-polytope, being a rectification of the regular 8-simplex.

In geometry, an E9 honeycomb is a tessellation of uniform polytopes in hyperbolic 9-dimensional space. , also (E10) is a paracompact hyperbolic group, so either facets or vertex figures will not be bounded.

## 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
• (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
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / / Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds