Orthographic projections in the D7 Coxeter plane

7-demicube

7-orthoplex

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

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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

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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. "7D uniform polytopes (polyexa)".

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