A₇ polytope

Orthographic projections
A₇ Coxeter plane

7-simplex

In 7-dimensional geometry, there are 71 uniform polytopes with A₇ symmetry. There is one self-dual regular form, the 7-simplex with 8 vertices.

Each can be visualized as symmetric orthographic projections in Coxeter planes of the A₇ Coxeter group, and other subgroups.

Symmetric orthographic projections of these 71 polytopes can be made in the A₇, A₆, A₅, A₄, A₃, A₂ Coxeter planes. A_k has [k+1] symmetry. For even k and symmetrically ringed-diagrams, symmetry doubles to [2(k+1)].

These 71 polytopes are each shown in these 6 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.

#	Coxeter-Dynkin diagram Schläfli symbol Johnson name	Ak orthogonal projection graphs
		A7 [8]	A6 [7]	A5 [6]	A4 [5]	A3 [4]	A2 [3]
1	t0{3,3,3,3,3,3} 7-simplex
2	t1{3,3,3,3,3,3} Rectified 7-simplex
3	t2{3,3,3,3,3,3} Birectified 7-simplex
4	t3{3,3,3,3,3,3} Trirectified 7-simplex
5	t0,1{3,3,3,3,3,3} Truncated 7-simplex
6	t0,2{3,3,3,3,3,3} Cantellated 7-simplex
7	t1,2{3,3,3,3,3,3} Bitruncated 7-simplex
8	t0,3{3,3,3,3,3,3} Runcinated 7-simplex
9	t1,3{3,3,3,3,3,3} Bicantellated 7-simplex
10	t2,3{3,3,3,3,3,3} Tritruncated 7-simplex
11	t0,4{3,3,3,3,3,3} Stericated 7-simplex
12	t1,4{3,3,3,3,3,3} Biruncinated 7-simplex
13	t2,4{3,3,3,3,3,3} Tricantellated 7-simplex
14	t0,5{3,3,3,3,3,3} Pentellated 7-simplex
15	t1,5{3,3,3,3,3,3} Bistericated 7-simplex
16	t0,6{3,3,3,3,3,3} Hexicated 7-simplex
17	t0,1,2{3,3,3,3,3,3} Cantitruncated 7-simplex
18	t0,1,3{3,3,3,3,3,3} Runcitruncated 7-simplex
19	t0,2,3{3,3,3,3,3,3} Runcicantellated 7-simplex
20	t1,2,3{3,3,3,3,3,3} Bicantitruncated 7-simplex
21	t0,1,4{3,3,3,3,3,3} Steritruncated 7-simplex
22	t0,2,4{3,3,3,3,3,3} Stericantellated 7-simplex
23	t1,2,4{3,3,3,3,3,3} Biruncitruncated 7-simplex
24	t0,3,4{3,3,3,3,3,3} Steriruncinated 7-simplex
25	t1,3,4{3,3,3,3,3,3} Biruncicantellated 7-simplex
26	t2,3,4{3,3,3,3,3,3} Tricantitruncated 7-simplex
27	t0,1,5{3,3,3,3,3,3} Pentitruncated 7-simplex
28	t0,2,5{3,3,3,3,3,3} Penticantellated 7-simplex
29	t1,2,5{3,3,3,3,3,3} Bisteritruncated 7-simplex
30	t0,3,5{3,3,3,3,3,3} Pentiruncinated 7-simplex
31	t1,3,5{3,3,3,3,3,3} Bistericantellated 7-simplex
32	t0,4,5{3,3,3,3,3,3} Pentistericated 7-simplex
33	t0,1,6{3,3,3,3,3,3} Hexitruncated 7-simplex
34	t0,2,6{3,3,3,3,3,3} Hexicantellated 7-simplex
35	t0,3,6{3,3,3,3,3,3} Hexiruncinated 7-simplex
36	t0,1,2,3{3,3,3,3,3,3} Runcicantitruncated 7-simplex
37	t0,1,2,4{3,3,3,3,3,3} Stericantitruncated 7-simplex
38	t0,1,3,4{3,3,3,3,3,3} Steriruncitruncated 7-simplex
39	t0,2,3,4{3,3,3,3,3,3} Steriruncicantellated 7-simplex
40	t1,2,3,4{3,3,3,3,3,3} Biruncicantitruncated 7-simplex
41	t0,1,2,5{3,3,3,3,3,3} Penticantitruncated 7-simplex
42	t0,1,3,5{3,3,3,3,3,3} Pentiruncitruncated 7-simplex
43	t0,2,3,5{3,3,3,3,3,3} Pentiruncicantellated 7-simplex
44	t1,2,3,5{3,3,3,3,3,3} Bistericantitruncated 7-simplex
45	t0,1,4,5{3,3,3,3,3,3} Pentisteritruncated 7-simplex
46	t0,2,4,5{3,3,3,3,3,3} Pentistericantellated 7-simplex
47	t1,2,4,5{3,3,3,3,3,3} Bisteriruncitruncated 7-simplex
48	t0,3,4,5{3,3,3,3,3,3} Pentisteriruncinated 7-simplex
49	t0,1,2,6{3,3,3,3,3,3} Hexicantitruncated 7-simplex
50	t0,1,3,6{3,3,3,3,3,3} Hexiruncitruncated 7-simplex
51	t0,2,3,6{3,3,3,3,3,3} Hexiruncicantellated 7-simplex
52	t0,1,4,6{3,3,3,3,3,3} Hexisteritruncated 7-simplex
53	t0,2,4,6{3,3,3,3,3,3} Hexistericantellated 7-simplex
54	t0,1,5,6{3,3,3,3,3,3} Hexipentitruncated 7-simplex
55	t0,1,2,3,4{3,3,3,3,3,3} Steriruncicantitruncated 7-simplex
56	t0,1,2,3,5{3,3,3,3,3,3} Pentiruncicantitruncated 7-simplex
57	t0,1,2,4,5{3,3,3,3,3,3} Pentistericantitruncated 7-simplex
58	t0,1,3,4,5{3,3,3,3,3,3} Pentisteriruncitruncated 7-simplex
59	t0,2,3,4,5{3,3,3,3,3,3} Pentisteriruncicantellated 7-simplex
60	t1,2,3,4,5{3,3,3,3,3,3} Bisteriruncicantitruncated 7-simplex
61	t0,1,2,3,6{3,3,3,3,3,3} Hexiruncicantitruncated 7-simplex
62	t0,1,2,4,6{3,3,3,3,3,3} Hexistericantitruncated 7-simplex
63	t0,1,3,4,6{3,3,3,3,3,3} Hexisteriruncitruncated 7-simplex
64	t0,2,3,4,6{3,3,3,3,3,3} Hexisteriruncicantellated 7-simplex
65	t0,1,2,5,6{3,3,3,3,3,3} Hexipenticantitruncated 7-simplex
66	t0,1,3,5,6{3,3,3,3,3,3} Hexipentiruncitruncated 7-simplex
67	t0,1,2,3,4,5{3,3,3,3,3,3} Pentisteriruncicantitruncated 7-simplex
68	t0,1,2,3,4,6{3,3,3,3,3,3} Hexisteriruncicantitruncated 7-simplex
69	t0,1,2,3,5,6{3,3,3,3,3,3} Hexipentiruncicantitruncated 7-simplex
70	t0,1,2,4,5,6{3,3,3,3,3,3} Hexipentistericantitruncated 7-simplex
71	t0,1,2,3,4,5,6{3,3,3,3,3,3} Omnitruncated 7-simplex

• 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)".