3 21 polytope


321

231

132

Rectified 321

birectified 321

Rectified 231

Rectified 132
Orthogonal projections in E6 Coxeter plane

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
TypeUniform 7-polytope
Familyk21 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 figure221 polytope
Petrie polygonoctadecagon
Coxeter groupE7, [33,2,1], order 2903040
Propertiesconvex

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 called a Gosset graph.

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

Alternate names

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 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]

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 = {\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

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 {\tilde {E}}_{7}=E7+ {\bar {T}}_{8}=E7++
Coxeter
diagram
Symmetry [3−1,3,1] [30,3,1] [[3<sup>1,3,1</sup>]] [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
TypeUniform 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 figure5-demicube prism
Petrie polygonoctadecagon
Coxeter groupE7, [33,2,1], order 2903040
Propertiesconvex

Alternate names

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
TypeUniform 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 figure5-cell-triangle duoprism
Petrie polygonoctadecagon
Coxeter groupE7, [33,2,1], order 2903040
Propertiesconvex

Alternate names

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]

See also

Notes

  1. 1 2 Gosset, 1900
  2. Elte, 1912
  3. Klitzing, (o3o3o3o *c3o3o3x - naq)
  4. Klitzing. (o3o3o3o *c3o3x3o - ranq)
  5. Klitzing, (o3o3o3o *c3x3o3o - branq)

References

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