List of A₈ polytopes

Orthographic projections
A₈ Coxeter plane
8-simplex

In 8-dimensional geometry, there are 135 uniform polytopes with A₈ symmetry. There is one self-dual regular form, the 8-simplex with 9 vertices.

Each can be visualized as symmetric orthographic projections in Coxeter planes of the A₈ Coxeter group, and other subgroups.

1 Graphs
2 References
3 External links
4 Notes

Graphs

Symmetric orthographic projections of these 135 polytopes can be made in the A₈, A₇, A₆, A₅, A₄, A₃, A₂ Coxeter planes. A_k has [k+1] symmetry.

These 135 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-Dynkin diagram Schläfli symbol Johnson name	A_k orthogonal projection graphs
#	Coxeter-Dynkin diagram Schläfli symbol Johnson name	A₈ [9]	A₇ [8]	A₆ [7]	A₅ [6]	A₄ [5]	A₃ [4]	A₂ [3]
1	t₀{3,3,3,3,3,3,3} 8-simplex
2	t₁{3,3,3,3,3,3,3} Rectified 8-simplex
3	t₂{3,3,3,3,3,3,3} Birectified 8-simplex
4	t₃{3,3,3,3,3,3,3} Trirectified 8-simplex
5	t_0,1{3,3,3,3,3,3,3} Truncated 8-simplex
6	t_0,2{3,3,3,3,3,3,3} Cantellated 8-simplex
7	t_1,2{3,3,3,3,3,3,3} Bitruncated 8-simplex
8	t_0,3{3,3,3,3,3,3,3} Runcinated 8-simplex
9	t_1,3{3,3,3,3,3,3,3} Bicantellated 8-simplex
10	t_2,3{3,3,3,3,3,3,3} Tritruncated 8-simplex
11	t_0,4{3,3,3,3,3,3,3} Stericated 8-simplex
12	t_1,4{3,3,3,3,3,3,3} Biruncinated 8-simplex
13	t_2,4{3,3,3,3,3,3,3} Tricantellated 8-simplex
14	t_3,4{3,3,3,3,3,3,3} Quadritruncated 8-simplex
15	t_0,5{3,3,3,3,3,3,3} Pentellated 8-simplex
16	t_1,5{3,3,3,3,3,3,3} Bistericated 8-simplex
17	t_2,5{3,3,3,3,3,3,3} Triruncinated 8-simplex
18	t_0,6{3,3,3,3,3,3,3} Hexicated 8-simplex
19	t_1,6{3,3,3,3,3,3,3} Bipentellated 8-simplex
20	t_0,7{3,3,3,3,3,3,3} Heptellated 8-simplex
21	t_0,1,2{3,3,3,3,3,3,3} Cantitruncated 8-simplex
22	t_0,1,3{3,3,3,3,3,3,3} Runcitruncated 8-simplex
23	t_0,2,3{3,3,3,3,3,3,3} Runcicantellated 8-simplex
24	t_1,2,3{3,3,3,3,3,3,3} Bicantitruncated 8-simplex
25	t_0,1,4{3,3,3,3,3,3,3} Steritruncated 8-simplex
26	t_0,2,4{3,3,3,3,3,3,3} Stericantellated 8-simplex
27	t_1,2,4{3,3,3,3,3,3,3} Biruncitruncated 8-simplex
28	t_0,3,4{3,3,3,3,3,3,3} Steriruncinated 8-simplex
29	t_1,3,4{3,3,3,3,3,3,3} Biruncicantellated 8-simplex
30	t_2,3,4{3,3,3,3,3,3,3} Tricantitruncated 8-simplex
31	t_0,1,5{3,3,3,3,3,3,3} Pentitruncated 8-simplex
32	t_0,2,5{3,3,3,3,3,3,3} Penticantellated 8-simplex
33	t_1,2,5{3,3,3,3,3,3,3} Bisteritruncated 8-simplex
34	t_0,3,5{3,3,3,3,3,3,3} Pentiruncinated 8-simplex
35	t_1,3,5{3,3,3,3,3,3,3} Bistericantellated 8-simplex
36	t_2,3,5{3,3,3,3,3,3,3} Triruncitruncated 8-simplex
37	t_0,4,5{3,3,3,3,3,3,3} Pentistericated 8-simplex
38	t_1,4,5{3,3,3,3,3,3,3} Bisteriruncinated 8-simplex
39	t_0,1,6{3,3,3,3,3,3,3} Hexitruncated 8-simplex
40	t_0,2,6{3,3,3,3,3,3,3} Hexicantellated 8-simplex
41	t_1,2,6{3,3,3,3,3,3,3} Bipentitruncated 8-simplex
42	t_0,3,6{3,3,3,3,3,3,3} Hexiruncinated 8-simplex
43	t_1,3,6{3,3,3,3,3,3,3} Bipenticantellated 8-simplex
44	t_0,4,6{3,3,3,3,3,3,3} Hexistericated 8-simplex
45	t_0,5,6{3,3,3,3,3,3,3} Hexipentellated 8-simplex
46	t_0,1,7{3,3,3,3,3,3,3} Heptitruncated 8-simplex
47	t_0,2,7{3,3,3,3,3,3,3} Hepticantellated 8-simplex
48	t_0,3,7{3,3,3,3,3,3,3} Heptiruncinated 8-simplex
49	t_0,1,2,3{3,3,3,3,3,3,3} Runcicantitruncated 8-simplex
50	t_0,1,2,4{3,3,3,3,3,3,3} Stericantitruncated 8-simplex
51	t_0,1,3,4{3,3,3,3,3,3,3} Steriruncitruncated 8-simplex
52	t_0,2,3,4{3,3,3,3,3,3,3} Steriruncicantellated 8-simplex
53	t_1,2,3,4{3,3,3,3,3,3,3} Biruncicantitruncated 8-simplex
54	t_0,1,2,5{3,3,3,3,3,3,3} Penticantitruncated 8-simplex
55	t_0,1,3,5{3,3,3,3,3,3,3} Pentiruncitruncated 8-simplex
56	t_0,2,3,5{3,3,3,3,3,3,3} Pentiruncicantellated 8-simplex
57	t_1,2,3,5{3,3,3,3,3,3,3} Bistericantitruncated 8-simplex
58	t_0,1,4,5{3,3,3,3,3,3,3} Pentisteritruncated 8-simplex
59	t_0,2,4,5{3,3,3,3,3,3,3} Pentistericantellated 8-simplex
60	t_1,2,4,5{3,3,3,3,3,3,3} Bisteriruncitruncated 8-simplex
61	t_0,3,4,5{3,3,3,3,3,3,3} Pentisteriruncinated 8-simplex
62	t_1,3,4,5{3,3,3,3,3,3,3} Bisteriruncicantellated 8-simplex
63	t_2,3,4,5{3,3,3,3,3,3,3} Triruncicantitruncated 8-simplex
64	t_0,1,2,6{3,3,3,3,3,3,3} Hexicantitruncated 8-simplex
65	t_0,1,3,6{3,3,3,3,3,3,3} Hexiruncitruncated 8-simplex
66	t_0,2,3,6{3,3,3,3,3,3,3} Hexiruncicantellated 8-simplex
67	t_1,2,3,6{3,3,3,3,3,3,3} Bipenticantitruncated 8-simplex
68	t_0,1,4,6{3,3,3,3,3,3,3} Hexisteritruncated 8-simplex
69	t_0,2,4,6{3,3,3,3,3,3,3} Hexistericantellated 8-simplex
70	t_1,2,4,6{3,3,3,3,3,3,3} Bipentiruncitruncated 8-simplex
71	t_0,3,4,6{3,3,3,3,3,3,3} Hexisteriruncinated 8-simplex
72	t_1,3,4,6{3,3,3,3,3,3,3} Bipentiruncicantellated 8-simplex
73	t_0,1,5,6{3,3,3,3,3,3,3} Hexipentitruncated 8-simplex
74	t_0,2,5,6{3,3,3,3,3,3,3} Hexipenticantellated 8-simplex
75	t_1,2,5,6{3,3,3,3,3,3,3} Bipentisteritruncated 8-simplex
76	t_0,3,5,6{3,3,3,3,3,3,3} Hexipentiruncinated 8-simplex
77	t_0,4,5,6{3,3,3,3,3,3,3} Hexipentistericated 8-simplex
78	t_0,1,2,7{3,3,3,3,3,3,3} Hepticantitruncated 8-simplex
79	t_0,1,3,7{3,3,3,3,3,3,3} Heptiruncitruncated 8-simplex
80	t_0,2,3,7{3,3,3,3,3,3,3} Heptiruncicantellated 8-simplex
81	t_0,1,4,7{3,3,3,3,3,3,3} Heptisteritruncated 8-simplex
82	t_0,2,4,7{3,3,3,3,3,3,3} Heptistericantellated 8-simplex
83	t_0,3,4,7{3,3,3,3,3,3,3} Heptisteriruncinated 8-simplex
84	t_0,1,5,7{3,3,3,3,3,3,3} Heptipentitruncated 8-simplex
85	t_0,2,5,7{3,3,3,3,3,3,3} Heptipenticantellated 8-simplex
86	t_0,1,6,7{3,3,3,3,3,3,3} Heptihexitruncated 8-simplex
87	t_0,1,2,3,4{3,3,3,3,3,3,3} Steriruncicantitruncated 8-simplex
88	t_0,1,2,3,5{3,3,3,3,3,3,3} Pentiruncicantitruncated 8-simplex
89	t_0,1,2,4,5{3,3,3,3,3,3,3} Pentistericantitruncated 8-simplex
90	t_0,1,3,4,5{3,3,3,3,3,3,3} Pentisteriruncitruncated 8-simplex
91	t_0,2,3,4,5{3,3,3,3,3,3,3} Pentisteriruncicantellated 8-simplex
92	t_1,2,3,4,5{3,3,3,3,3,3,3} Bisteriruncicantitruncated 8-simplex
93	t_0,1,2,3,6{3,3,3,3,3,3,3} Hexiruncicantitruncated 8-simplex
94	t_0,1,2,4,6{3,3,3,3,3,3,3} Hexistericantitruncated 8-simplex
95	t_0,1,3,4,6{3,3,3,3,3,3,3} Hexisteriruncitruncated 8-simplex
96	t_0,2,3,4,6{3,3,3,3,3,3,3} Hexisteriruncicantellated 8-simplex
97	t_1,2,3,4,6{3,3,3,3,3,3,3} Bipentiruncicantitruncated 8-simplex
98	t_0,1,2,5,6{3,3,3,3,3,3,3} Hexipenticantitruncated 8-simplex
99	t_0,1,3,5,6{3,3,3,3,3,3,3} Hexipentiruncitruncated 8-simplex
100	t_0,2,3,5,6{3,3,3,3,3,3,3} Hexipentiruncicantellated 8-simplex
101	t_1,2,3,5,6{3,3,3,3,3,3,3} Bipentistericantitruncated 8-simplex
102	t_0,1,4,5,6{3,3,3,3,3,3,3} Hexipentisteritruncated 8-simplex
103	t_0,2,4,5,6{3,3,3,3,3,3,3} Hexipentistericantellated 8-simplex
104	t_0,3,4,5,6{3,3,3,3,3,3,3} Hexipentisteriruncinated 8-simplex
105	t_0,1,2,3,7{3,3,3,3,3,3,3} Heptiruncicantitruncated 8-simplex
106	t_0,1,2,4,7{3,3,3,3,3,3,3} Heptistericantitruncated 8-simplex
107	t_0,1,3,4,7{3,3,3,3,3,3,3} Heptisteriruncitruncated 8-simplex
108	t_0,2,3,4,7{3,3,3,3,3,3,3} Heptisteriruncicantellated 8-simplex
109	t_0,1,2,5,7{3,3,3,3,3,3,3} Heptipenticantitruncated 8-simplex
110	t_0,1,3,5,7{3,3,3,3,3,3,3} Heptipentiruncitruncated 8-simplex
111	t_0,2,3,5,7{3,3,3,3,3,3,3} Heptipentiruncicantellated 8-simplex
112	t_0,1,4,5,7{3,3,3,3,3,3,3} Heptipentisteritruncated 8-simplex
113	t_0,1,2,6,7{3,3,3,3,3,3,3} Heptihexicantitruncated 8-simplex
114	t_0,1,3,6,7{3,3,3,3,3,3,3} Heptihexiruncitruncated 8-simplex
115	t_0,1,2,3,4,5{3,3,3,3,3,3,3} Pentisteriruncicantitruncated 8-simplex
116	t_0,1,2,3,4,6{3,3,3,3,3,3,3} Hexisteriruncicantitruncated 8-simplex
117	t_0,1,2,3,5,6{3,3,3,3,3,3,3} Hexipentiruncicantitruncated 8-simplex
118	t_0,1,2,4,5,6{3,3,3,3,3,3,3} Hexipentistericantitruncated 8-simplex
119	t_0,1,3,4,5,6{3,3,3,3,3,3,3} Hexipentisteriruncitruncated 8-simplex
120	t_0,2,3,4,5,6{3,3,3,3,3,3,3} Hexipentisteriruncicantellated 8-simplex
121	t_1,2,3,4,5,6{3,3,3,3,3,3,3} Bipentisteriruncicantitruncated 8-simplex
122	t_0,1,2,3,4,7{3,3,3,3,3,3,3} Heptisteriruncicantitruncated 8-simplex
123	t_0,1,2,3,5,7{3,3,3,3,3,3,3} Heptipentiruncicantitruncated 8-simplex
124	t_0,1,2,4,5,7{3,3,3,3,3,3,3} Heptipentistericantitruncated 8-simplex
125	t_0,1,3,4,5,7{3,3,3,3,3,3,3} Heptipentisteriruncitruncated 8-simplex
126	t_0,2,3,4,5,7{3,3,3,3,3,3,3} Heptipentisteriruncicantellated 8-simplex
127	t_0,1,2,3,6,7{3,3,3,3,3,3,3} Heptihexiruncicantitruncated 8-simplex
128	t_0,1,2,4,6,7{3,3,3,3,3,3,3} Heptihexistericantitruncated 8-simplex
129	t_0,1,3,4,6,7{3,3,3,3,3,3,3} Heptihexisteriruncitruncated 8-simplex
130	t_0,1,2,5,6,7{3,3,3,3,3,3,3} Heptihexipenticantitruncated 8-simplex
131	t_{0,1,2,3,4,5,6}{3,3,3,3,3,3,3} Hexipentisteriruncicantitruncated 8-simplex
132	t_{0,1,2,3,4,5,7}{3,3,3,3,3,3,3} Heptipentisteriruncicantitruncated 8-simplex
133	t_{0,1,2,3,4,6,7}{3,3,3,3,3,3,3} Heptihexisteriruncicantitruncated 8-simplex
134	t_{0,1,2,3,5,6,7}{3,3,3,3,3,3,3} Heptihexipentiruncicantitruncated 8-simplex
135	t_{0,1,2,3,4,5,6,7}{3,3,3,3,3,3,3} Omnitruncated 8-simplex

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, editied 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

External links

Richard Klitzing, 8D, uniform polytopes (polyzetta)

Notes

^ Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter


Family	A_n	BC_n	D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
Regular polygon	Triangle	Square		Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	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
n-polytopes	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes

List of A8 polytopes

Contents

Graphs

References

External links

Notes

List of A₈ polytopes