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A6 polytope

[[File:6-simplex_t0.svg160px]][6-simplex](6-simplex)

In 6-dimensional geometry, there are 35 uniform polytopes with A6 symmetry. There is one self-dual regular form, the 6-simplex with 7 vertices.

Each can be visualized as symmetric orthographic projections in Coxeter planes of the A6 Coxeter group, and other subgroups.

Graphs

Symmetric orthographic projections of these 35 polytopes can be made in the A6, A5, A4, A3, A2 Coxeter planes. Ak graphs have [k+1] symmetry. For even k and symmetric ringed diagrams, symmetry doubles to [2(k+1)].

These 35 polytopes are each shown in these 5 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.

#A6[7]A5[6]A4[5]A3[4]A2[3]Coxeter-Dynkin diagramSchläfli symbolName
1[[File:6-simplex t0.svg80px]][[File:6-simplex t0 A5.svg80px]]
2[[File:6-simplex t1.svg80px]][[File:6-simplex t1 A5.svg80px]]
3[[File:6-simplex t01.svg80px]][[File:6-simplex t01 A5.svg80px]]
4[[File:6-simplex t2.svg80px]][[File:6-simplex t2 A5.svg80px]]
5[[File:6-simplex t02.svg80px]][[File:6-simplex t02 A5.svg80px]]
6[[File:6-simplex t12.svg80px]][[File:6-simplex t12 A5.svg80px]]
7[[File:6-simplex t012.svg80px]][[File:6-simplex t012 A5.svg80px]]
8[[File:6-simplex t03.svg80px]][[File:6-simplex t03 A5.svg80px]]
9[[File:6-simplex t13.svg80px]][[File:6-simplex t13 A5.svg80px]]
10[[File:6-simplex t013.svg80px]][[File:6-simplex t013 A5.svg80px]]
11[[File:6-simplex t23.svg80px]][[File:6-simplex t23 A5.svg80px]]
12[[File:6-simplex t023.svg80px]][[File:6-simplex t023 A5.svg80px]]
13[[File:6-simplex t123.svg80px]][[File:6-simplex t123 A5.svg80px]]
14[[File:6-simplex t0123.svg80px]][[File:6-simplex t0123 A5.svg80px]]
15[[File:6-simplex t04.svg80px]][[File:6-simplex t04 A5.svg80px]]
16[[File:6-simplex t14.svg80px]][[File:6-simplex t14 A5.svg80px]]
17[[File:6-simplex t014.svg80px]][[File:6-simplex t014 A5.svg80px]]
18[[File:6-simplex t024.svg80px]][[File:6-simplex t024 A5.svg80px]]
19[[File:6-simplex t124.svg80px]][[File:6-simplex t124 A5.svg80px]]
20[[File:6-simplex t0124.svg80px]][[File:6-simplex t0124 A5.svg80px]]
21[[File:6-simplex t034.svg80px]][[File:6-simplex t034 A5.svg80px]]
22[[File:6-simplex t0134.svg80px]][[File:6-simplex t0134 A5.svg80px]]
23[[File:6-simplex t0234.svg80px]][[File:6-simplex t0234 A5.svg80px]]
24[[File:6-simplex t1234.svg80px]][[File:6-simplex t1234 A5.svg80px]]
25[[File:6-simplex t01234.svg80px]][[File:6-simplex t01234 A5.svg80px]]
26[[File:6-simplex t05.svg80px]][[File:6-simplex t05 A5.svg80px]]
27[[File:6-simplex t015.svg80px]][[File:6-simplex t015 A5.svg80px]]
28[[File:6-simplex t025.svg80px]][[File:6-simplex t025 A5.svg80px]]
29[[File:6-simplex t0125.svg80px]][[File:6-simplex t0125 A5.svg80px]]
30[[File:6-simplex t0135.svg80px]][[File:6-simplex t0135 A5.svg80px]]
31[[File:6-simplex t0235.svg80px]][[File:6-simplex t0235 A5.svg80px]]
32[[File:6-simplex t01235.svg80px]][[File:6-simplex t01235 A5.svg80px]]
33[[File:6-simplex t0145.svg80px]][[File:6-simplex t0145 A5.svg80px]]
34[[File:6-simplex t01245.svg80px]][[File:6-simplex t01245 A5.svg80px]]
35[[File:6-simplex t012345.svg80px]][[File:6-simplex t012345 A5.svg80px]]

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,
    • (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

Notes

References

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