Misplaced Pages

Orthographic projections in the D₈ Coxeter plane

8-demicube =

8-orthoplex =

In 8-dimensional geometry, there are 191 uniform polytopes with D₈ symmetry, 64 are unique, and 127 are shared with the B₈ symmetry. There are two regular forms, the 8-orthoplex, and 8-demicube with 16 and 128 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, A₅, A₃, Coxeter planes. A_k has symmetry, D_k has symmetry. B₈ is also included although only half of its symmetry exists in these polytopes.

These 64 polytopes are each shown in these 10 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
	B8	D8	D7	D6	D5	D4	D3	A7	A5	A3
1											= 8-demicube (hocto)
2											= cantic 8-cube (thocto)
3											= runcic 8-cube
4											= steric 8-cube
5											= pentic 8-cube
6											= hexic 8-cube
7											= heptic 8-cube
8											=
9											=
10											=
11											=
12											=
13											=
14											=
15											=
16											=
17											=
18											=
19											=
20											=
21											=
22											=
23											=
24											=
25											=
26											=
27											=
28											=
29											=
30											=
31											=
32											=
33											=
34											=
35											=
36											=
37											=
38											=
39											=
40											=
41											=
42											=
43											=
44											=
45											=
46											=
47											=
48											=
49											=
50											=
51											=
52											=
53											=
54											=
55											=
56											=
57											=
58											=
59											=
60											=
61											=
62											=
63											=
64											=

References

• 8-polytopes