D7 polytope

In 7-dimensional geometry, there are 95 uniform polytopes with D7 symmetry; 32 are unique, and 63 are shared with the B7 symmetry. There are two regular forms, the 7-orthoplex, and 7-demicube with 14 and 64 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 32 polytopes can be made in the D7, D6, D5, D4, D3, A5, A3, Coxeter planes. Ak has [k+1] symmetry, Dk has [2(k-1)] symmetry. B7 is also included although only half of its [14] symmetry exists in these polytopes.

These 32 polytopes are each shown in these 8 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
B7
[14/2]
D7
[12]
D6
[10]
D5
[8]
D4
[6]
D3
[4]
A5
[6]
A3
[4]
1 =
7-demicube
Demihepteract (Hesa)
2 =
Cantic 7-cube
Truncated demihepteract (Thesa)
3 =
Runcic 7-cube
Small rhombated demihepteract (Sirhesa)
4 =
Steric 7-cube
Small prismated demihepteract (Sphosa)
5 =
Pentic 7-cube
Small cellated demihepteract (Sochesa)
6 =
Hexic 7-cube
Small terated demihepteract (Suthesa)
7 =
Runcicantic 7-cube
Great rhombated demihepteract (Girhesa)
8 =
Stericantic 7-cube
Prismatotruncated demihepteract (Pothesa)
9 =
Steriruncic 7-cube
Prismatorhomated demihepteract (Prohesa)
10 =
Penticantic 7-cube
Cellitruncated demihepteract (Cothesa)
11 =
Pentiruncic 7-cube
Cellirhombated demihepteract (Crohesa)
12 =
Pentisteric 7-cube
Celliprismated demihepteract (Caphesa)
13 =
Hexicantic 7-cube
Teritruncated demihepteract (Tuthesa)
14 =
Hexiruncic 7-cube
Terirhombated demihepteract (Turhesa)
15 =
Hexisteric 7-cube
Teriprismated demihepteract (Tuphesa)
16 =
Hexipentic 7-cube
Tericellated demihepteract (Tuchesa)
17 =
Steriruncicantic 7-cube
Great prismated demihepteract (Gephosa)
18 =
Pentiruncicantic 7-cube
Celligreatorhombated demihepteract (Cagrohesa)
19 =
Pentistericantic 7-cube
Celliprismatotruncated demihepteract (Capthesa)
20 =
Pentisteriruncic 7-cube
Celliprismatorhombated demihepteract (Coprahesa)
21 =
Hexiruncicantic 7-cube
Terigreatorhombated demihepteract (Tugrohesa)
22 =
Hexistericantic 7-cube
Teriprismatotruncated demihepteract (Tupthesa)
23 =
Hexisteriruncic 7-cube
Teriprismatorhombated demihepteract (Tuprohesa)
24 =
Hexipenticantic 7-cube
Tericellitruncated demihepteract (Tucothesa)
25 =
Hexipentiruncic 7-cube
Tericellirhombated demihepteract (Tucrohesa)
26 =
Hexipentisteric 7-cube
Tericelliprismated demihepteract (Tucophesa)
27 =
Pentisteriruncicantic 7-cube
Great cellated demihepteract (Gochesa)
28 =
Hexisteriruncicantic 7-cube
Terigreatoprimated demihepteract (Tugphesa)
29 =
Hexipentiruncicantic 7-cube
Tericelligreatorhombated demihepteract (Tucagrohesa)
30 =
Hexipentistericantic 7-cube
Tericelliprismatotruncated demihepteract (Tucpathesa)
31 =
Hexipentisteriruncic 7-cube
Tericellprismatorhombated demihepteract (Tucprohesa)
32 =
Hexipentisteriruncicantic 7-cube
Great terated demihepteract (Guthesa)

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. "7D uniform polytopes (polyexa)".

Notes

1. ^ Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter
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 OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 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