Orthographic projections in the D6 Coxeter plane

6-demicube

6-orthoplex

In 6-dimensional geometry, there are 47 uniform polytopes with D6 symmetry, of which 16 are unique and 31 are shared with the B6 symmetry. There are two regular forms, the 6-orthoplex, and 6-demicube with 12 and 32 vertices respectively.

They can be visualized as symmetric orthographic projections in Coxeter planes of the D6 Coxeter group, and other subgroups.

Graphs

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Symmetric orthographic projections of these 16 polytopes can be made in the D6, D5, D4, D3, A5, A3, Coxeter planes. Ak has [k+1] symmetry, Dk has [2(k-1)] symmetry. B6 is also included although only half of its [12] symmetry exists in these polytopes.

These 16 polytopes are each shown in these 7 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.

# Coxeter plane graphs Coxeter diagram
Names
B6
[12/2]
D6
[10]
D5
[8]
D4
[6]
D3
[4]
A5
[6]
A3
[4]
1                         =            
6-demicube
Hemihexeract (hax)
2                         =            
cantic 6-cube
Truncated hemihexeract (thax)
3                         =            
runcic 6-cube
Small rhombated hemihexeract (sirhax)
4                         =            
steric 6-cube
Small prismated hemihexeract (sophax)
5                         =            
pentic 6-cube
Small cellated demihexeract (sochax)
6                         =            
runcicantic 6-cube
Great rhombated hemihexeract (girhax)
7                         =            
stericantic 6-cube
Prismatotruncated hemihexeract (pithax)
8                         =            
steriruncic 6-cube
Prismatorhombated hemihexeract (prohax)
9                         =            
Stericantic 6-cube
Cellitruncated hemihexeract (cathix)
10                         =            
Pentiruncic 6-cube
Cellirhombated hemihexeract (crohax)
11                         =            
Pentisteric 6-cube
Celliprismated hemihexeract (cophix)
12                         =            
Steriruncicantic 6-cube
Great prismated hemihexeract (gophax)
13                         =            
Pentiruncicantic 6-cube
Celligreatorhombated hemihexeract (cagrohax)
14                         =            
Pentistericantic 6-cube
Celliprismatotruncated hemihexeract (capthix)
15                         =            
Pentisteriruncic 6-cube
Celliprismatorhombated hemihexeract (caprohax)
16                         =            
Pentisteriruncicantic 6-cube
Great cellated hemihexeract (gochax)

References

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  • 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]
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
  • Klitzing, Richard. "6D uniform polytopes (polypeta)".

Notes

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Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 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