3 21 polytope

231	Rectified 321	Birectified 321	Rectified 132
132	231	Rectified 231
Orthogonal projections in E6 Coxeter plane

In 7-dimensional geometry, the 3₂₁ polytope is a uniform 6-polytope, constructed within the symmetry of the E₇ group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 6-ic semi-regular figure.^[1]

Coxeter named it 3₂₁ by 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

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

321 polytope
Type	Uniform 7-polytope
Family	k21 polytope
Schläfli symbol	{3,3,3,32,1}
Coxeter symbol	321
Coxeter-Dynkin 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]
Properties	convex

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

For visualization this 7-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 56 vertices within a 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 called a Gosset graph.

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 nodea_1 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 nodea_1 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 an 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, .

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]

Rectified 3₂₁ polytope

Rectified 321 polytope
Type	Uniform 7-polytope
Schläfli symbol	t1{3,3,3,32,1}
Coxeter symbol	t1(321)
Coxeter-Dynkin 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]
Properties	convex

Alternate names

• Rectified hecatonicosihexa-pentacosiheptacontihexa-exon as a rectified 126-576 facetted polyexon (acronym ranq) (Jonathan Bowers)^[4]

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	t1{3,3,3,32,1}
Coxeter symbol	t1(321)
Coxeter-Dynkin 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]
Properties	convex

Alternate names

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

See also

• List of E7 polytopes

Notes

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)
• Richard Klitzing, 7D, uniform polytopes (polyexa) o3o3o3o *c3o3o3x - naq, o3o3o3o *c3o3x3o - ranq, o3o3o3o *c3x3o3o - branq