Uniform k 21 polytope

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.

Thorold Gosset discovered this family as a part of his 1900 enumeration of the regular and semiregular polytopes, and so they are sometimes called Gosset's semiregular figures. Gosset named them by their dimension from 5 to 9, for example the 5-ic semiregular figure.

Family members

The sequence as identified by Gosset ends as an infinite tessellation (space-filling honeycomb) in 8-space, called the E8 lattice. (A final form was not discovered by Gosset and is called the E9 lattice: 621. It is a tessellation of hyperbolic 9-space constructed of (∞ 9-simplex and ∞ 9-orthoplex facets with all vertices at infinity.)

The family starts uniquely as 6-polytopes. The triangular prism and rectified 5-cell are included at the beginning for completeness. The demipenteract also exists in the demihypercube family.

They are also sometimes named by their symmetry group, like E6 polytope, although there are many uniform polytopes within the E6 symmetry.

The complete family of Gosset semiregular polytopes are:

  1. triangular prism: −121 (2 triangles and 3 square faces)
  2. rectified 5-cell: 021, Tetroctahedric (5 tetrahedra and 5 octahedra cells)
  3. demipenteract: 121, 5-ic semiregular figure (16 5-cell and 10 16-cell facets)
  4. 2 21 polytope: 221, 6-ic semiregular figure (72 5-simplex and 27 5-orthoplex facets)
  5. 3 21 polytope: 321, 7-ic semiregular figure (567 6-simplex and 126 6-orthoplex facets)
  6. 4 21 polytope: 421, 8-ic semiregular figure (17280 7-simplex and 2160 7-orthoplex facets)
  7. 5 21 honeycomb: 521, 9-ic semiregular check tessellates Euclidean 8-space (∞ 8-simplex and ∞ 8-orthoplex facets)
  8. 6 21 honeycomb: 621, tessellates hyperbolic 9-space (∞ 9-simplex and ∞ 9-orthoplex facets)

Each polytope is constructed from (n − 1)-simplex and (n − 1)-orthoplex facets.

The orthoplex faces are constructed from the Coxeter group Dn−1 and have a Schläfli symbol of {31,n−1,1} rather than the regular {3n−2,4}. This construction is an implication of two "facet types". Half the facets around each orthoplex ridge are attached to another orthoplex, and the others are attached to a simplex. In contrast, every simplex ridge is attached to an orthoplex.

Each has a vertex figure as the previous form. For example the rectified 5-cell has a vertex figure as a triangular prism.

Elements

Gosset semiregular figures
n-ic k21 Graph Name
Coxeter
diagram		 Facets Elements
(n − 1)-simplex
{3n−2}		 (n − 1)-orthoplex
{3n−4,1,1}		 Vertices Edges Faces Cells 4-faces 5-faces 6-faces 7-faces
3-ic −121 Triangular prism
 2 triangles

 3 squares

 6 9 5          
4-ic 021 Rectified 5-cell
 5 tetrahedron

 5 octahedron

 10 30 30 10        
5-ic 121 Demipenteract
 16 5-cell

 10 16-cell

 16 80 160 120 26      
6-ic 221 221 polytope
 72 5-simplexes

 27 5-orthoplexes

 27 216 720 1080 648 99    
7-ic 321 321 polytope
 576 6-simplexes

 126 6-orthoplexes

 56 756 4032 10080 12096 6048 702  
8-ic 421 421 polytope
 17280 7-simplexes

 2160 7-orthoplexes

 240 6720 60480 241920 483840 483840 207360 19440
9-ic 521 521 honeycomb
 ∞ 8-simplexes

 ∞ 8-orthoplexes

 ∞
10-ic 621 621 honeycomb
 ∞ 9-simplexes

 ∞ 9-orthoplexes

 ∞

See also

References

External links

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform 4-polytope 5-cell 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
Fundamental convex regular and uniform honeycombs in dimensions 2–10
Family {\tilde{A}}_{n-1} {\tilde{C}}_{n-1} {\tilde{B}}_{n-1} {\tilde{D}}_{n-1} {\tilde{G}}_2 / {\tilde{F}}_4 / {\tilde{E}}_{n-1}
Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
Uniform 5-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
Uniform 6-honeycomb {3[6]} δ6 hδ6 qδ6
Uniform 7-honeycomb {3[7]} δ7 hδ7 qδ7 222
Uniform 8-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
Uniform 9-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
Uniform n-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
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