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

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

  • 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

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
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