6-orthoplex

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6-orthoplex
Hexacross

Orthogonal projection
inside Petrie polygon
Type Regular 6-polytope
Family orthoplex
Schläfli symbols {3,3,3,3,4}
{3,3,3,31,1}
Coxeter-Dynkin diagrams
=
5-faces 64 {34}
4-faces 192 {33}
Cells 240 {3,3}
Faces 160 {3}
Edges 60
Vertices 12
Vertex figure 5-orthoplex
Petrie polygon dodecagon
Coxeter groups B6, [4,34]
D6, [33,1,1]
Dual 6-cube
Properties convex, Hanner polytope

In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 vertices, 60 edges, 160 triangle faces, 240 tetrahedron cells, 192 5-cell 4-faces, and 64 5-faces.

It has two constructed forms, the first being regular with Schläfli symbol {34,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,3,31,1} or Coxeter symbol 311.

It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 6-hypercube, or hexeract.

Alternate names

  • Hexacross, derived from combining the family name cross polytope with hex for six (dimensions) in Greek.
  • Hexacontitetrapeton as a 64-facetted 6-polytope.

As a configuration

This configuration matrix represents the 6-orthoplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole 6-orthoplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element.[1][2]

[12104080803226082432163316061284642404451010519226152015664]

Construction

There are three Coxeter groups associated with the 6-orthoplex, one regular, dual of the hexeract with the C6 or [4,3,3,3,3] Coxeter group, and a half symmetry with two copies of 5-simplex facets, alternating, with the D6 or [33,1,1] Coxeter group. A lowest symmetry construction is based on a dual of a 6-orthotope, called a 6-fusil.

Name Coxeter Schläfli Symmetry Order
Regular 6-orthoplex {3,3,3,3,4} [4,3,3,3,3] 46080
Quasiregular 6-orthoplex {3,3,3,31,1} [3,3,3,31,1] 23040
6-fusil {3,3,3,4}+{} [4,3,3,3,3] 7680
{3,3,4}+{4} [4,3,3,2,4] 3072
2{3,4} [4,3,2,4,3] 2304
{3,3,4}+2{} [4,3,3,2,2] 1536
{3,4}+{4}+{} [4,3,2,4,2] 768
3{4} [4,2,4,2,4] 512
{3,4}+3{} [4,3,2,2,2] 384
2{4}+2{} [4,2,4,2,2] 256
{4}+4{} [4,2,2,2,2] 128
6{} [2,2,2,2,2] 64

Cartesian coordinates

Cartesian coordinates for the vertices of a 6-orthoplex, centered at the origin are

(±1,0,0,0,0,0), (0,±1,0,0,0,0), (0,0,±1,0,0,0), (0,0,0,±1,0,0), (0,0,0,0,±1,0), (0,0,0,0,0,±1)

Every vertex pair is connected by an edge, except opposites.

Images

orthographic projections
Coxeter plane B6 B5 B4
Graph
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

The 6-orthoplex can be projected down to 3-dimensions into the vertices of a regular icosahedron.[3]

2D 3D

Icosahedron
{3,5} =
H3 Coxeter plane
6-orthoplex
{3,3,3,31,1} =
D6 Coxeter plane
Icosahedron
6-orthoplex
This construction can be geometrically seen as the 12 vertices of the 6-orthoplex projected to 3 dimensions as the vertices of a regular icosahedron. This represents a geometric folding of the D6 to H3 Coxeter groups: : to . On the left, seen by these 2D Coxeter plane orthogonal projections, the two overlapping central vertices define the third axis in this mapping. Every pair of vertices of the 6-orthoplex are connected, except opposite ones: 30 edges are shared with the icosahedron, while 30 more edges from the 6-orthoplex project to the interior of the icosahedron.

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.)

This polytope is one of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.


References

  • H.S.M. Coxeter:
    • 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 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
      • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
      • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
  • Norman Johnson Uniform Polytopes, Manuscript (1991)
    • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. 1966
  • Klitzing, Richard. "6D uniform polytopes (polypeta) x3o3o3o3o4o - gee". https://bendwavy.org/klitzing/dimensions/polypeta.htm. 
Specific
  1. ↑ Coxeter, Regular Polytopes, sec 1.8 Configurations
  2. ↑ Coxeter, Complex Regular Polytopes, p.117
  3. ↑ Quasicrystals and Geometry, Marjorie Senechal, 1996, Cambridge University Press, p64. 2.7.1 The I6 crystal
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



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