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, 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 [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.

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 [16/2]	D₈ [14]	D₇ [12]	D₆ [10]	D₅ [8]	D₄ [6]	D₃ [4]	A₇ [8]	A₅ [6]	A₃ [4]	Coxeter diagram Names
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
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References

v t e 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