D₈ polytope

Orthographic projections in the D₈ Coxeter plane
8-demicube =	8-orthoplex =

In 8-dimensional geometry, there are 191 uniform polytopes with D₈ symmetry, of which 64 are unique and 127 are shared with the B₈ symmetry. There is one regular form, the 8-orthoplex with 16 vertices respectively.

They can be visualized as symmetric orthographic projections in Coxeter planes of the D₈ Coxeter group, and other subgroups.

Graphs

Symmetric orthographic projections of these 64 polytopes can be made in the D₈, D₇, D₆, D₅, D₄, D₃, A₇, A₅, A₃, Coxeter planes. A_k has [k+1] symmetry, D_k has [2(k-1)] symmetry. B₈ is also included although only half of its [16] symmetry exists in these polytopes.

Each of these 64 polytopes is shown in these 10 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position in progressive order: red, orange, yellow, green, cyan, blue, purple, magenta, red-violet.

#	Coxeter plane graphs										Coxeter diagram Name
#	B₈ [16/2]	D₈ [14]	D₇ [12]	D₆ [10]	D₅ [8]	D₄ [6]	D₃ [4]	A₇ [8]	A₅ [6]	A₃ [4]	Coxeter diagram Name
1											= 8-demicube
2											= Cantic 8-cube
3											= Runcic 8-cube
4											= Steric 8-cube
5											= Pentic 8-cube
6											= Hexic 8-cube
7											= Heptic 8-cube
8											= Runcicantic 8-cube
9											= Stericantic 8-cube
10											= Steriruncic 8-cube
11											= Penticantic 8-cube
12											= Pentiruncic 8-cube
13											= Pentisteric 8-cube
14											= Hexicantic 8-cube
15											= Hexiruncic 8-cube
16											= Hexisteric
17											= Hexipentic 8-cube
18											= Hepticantic 8-cube
19											= Heptiruncic 8-cube
20											= Heptisteric 8-cube
21											= Heptipentic 8-cube
22											= Heptihexic 8-cube
23											= Steriruncicantic 8-cube
24											= Pentiruncicantic 8-cube
25											= Pentistericantic 8-cube
26											= Pentisteriruncic 8-cube
27											= Hexiruncicantic 8-cube
28											= Hexistericantic 8-cube
29											= Hexisteriruncic 8-cube
30											= Hexipenticantic 8-cube
31											= Hexipentiruncic 8-cube
32											= Hexipentisteric 8-cube
33											= Heptiruncicantic 8-cube
34											= Heptistericantic 8-cube
35											= Heptisteriruncic 8-cube
36											= Heptipenticantic 8-cube
37											= Heptipentiruncic 8-cube
38											= Heptipentisteric 8-cube
39											= Heptihexicantic 8-cube
40											= Heptihexiruncic 8-cube
41											= Heptihexisteric 8-cube
42											= Heptihexipentic 8-cube
43											= Pentisteriruncicantic 8-cube
44											= Hexisteriruncicantic 8-cube
45											= Hexipentiruncicantic 8-cube
46											= Hexipentistericantic 8-cube
47											= Hexipentisteriruncic 8-cube
48											= Heptisteriruncicantic 8-cube
49											= Heptipentiruncicantic 8-cube
50											= Heptipentistericantic 8-cube
51											= Heptipentisteriruncic 8-cube
52											= Heptihexiruncicantic 8-cube
53											= Heptihexistericantic 8-cube
54											= Heptihexisteriruncic 8-cube
55											= Heptihexipenticantic 8-cube
56											= Heptihexipentiruncic 8-cube
57											= Heptihexipentisteric 8-cube
58											= Hexipentisteriruncicantic 8-cube
59											= Heptipentisteriruncicantic 8-cube
60											= Heptihexisteriruncicantic 8-cube
61											= Heptihexipentiruncicantic 8-cube
62											= Heptihexipentistericantic 8-cube
63											= Heptihexipentisteriruncic 8-cube
64											= Heptihexipentisteriruncicantic 8-cube

References

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	$A n$	$B n$	$I 2 (p)$ / $D n$	$E 6$ / $E 7$ / $E 8$ / $F 4$ / $G 2$	$H n$
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	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
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds • Polytope operations