D6 polytope

Orthographic projections in the D6 Coxeter plane
6-demicube

6-orthoplex

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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.

## Contents

* 1 Graphs
* 2 References
* 3 Notes

## 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

|
|

[12/2] | D6
[10] | D5
[8] | D4
[6] | D3
[4] | A5
[6] | A3
[4]
1  |  | 80px | 80px | 80px | 80px | 80px | 80px |  =
6-demicube
Hemihexeract (hax)
2  |  | 80px | 80px | 80px | 80px | 80px | 80px |  =
cantic 6-cube
Truncated hemihexeract (thax)
3  |  | 80px | 80px | 80px | 80px | 80px |  |  =
runcic 6-cube
Small rhombated hemihexeract (sirhax)
4  |  | 80px | 80px | 80px | 80px | 80px |  |  =
steric 6-cube
Small prismated hemihexeract (sophax)
5  |  | 80px | 80px | 80px | 80px | 80px |  |  =
pentic 6-cube
Small cellated demihexeract (sochax)
6  |  | 80px | 80px | 80px | 80px | 80px |  |  =
runcicantic 6-cube
Great rhombated hemihexeract (girhax)
7  |  | 80px | 80px | 80px | 80px | 80px |  |  =
stericantic 6-cube
Prismatotruncated hemihexeract (pithax)
8  |  | 80px | 80px | 80px | 80px | 80px |  |  =
steriruncic 6-cube
Prismatorhombated hemihexeract (prohax)
9  |  | 80px | 80px | 80px | 80px | 80px |  |  =
Stericantic 6-cube
Cellitruncated hemihexeract (cathix)
10  |  | 80px | 80px | 80px | 80px | 80px |  |  =
Pentiruncic 6-cube
Cellirhombated hemihexeract (crohax)
11  |  | 80px | 80px | 80px | 80px | 80px |  |  =
Pentisteric 6-cube
Celliprismated hemihexeract (cophix)
12  |  | 80px | 80px | 80px | 80px | 80px |  |  =
Steriruncicantic 6-cube
Great prismated hemihexeract (gophax)
13  |  | 80px | 80px | 80px | 80px | 80px |  |  =
Pentiruncicantic 6-cube
Celligreatorhombated hemihexeract (cagrohax)
14  |  | 80px | 80px | 80px | 80px | 80px |  |  =
Pentistericantic 6-cube
Celliprismatotruncated hemihexeract (capthix)
15  |  | 80px | 80px | 80px | 80px | 80px |  |  =
Pentisteriruncic 6-cube
Celliprismatorhombated hemihexeract (caprohax)
16  |  | 80px | 80px | 80px | 80px | 80px |  |  =
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)". https://bendwavy.org/klitzing/dimensions/polypeta.htm.

## Notes

1. ↑ Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter

* v
* t
* e

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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