Cantic 8-cube

Cantic 8-cube
D8 Coxeter plane projection
Type	uniform 8-polytope
Schläfli symbol	t_0,1{3,3^5,1} h₂{4,3,3,3,3,3,3}
Coxeter-Dynkin diagram
7-faces	16 truncated 7-demicubes 128 truncated 7-simplexes 128 rectified 7-simplexes
6-faces	112 truncated 6-demicubes 1024 truncated 6-simplexes 1024 rectified 6-simplexes 1024 6-simplexes
5-faces	448 truncated 5-demicubes 3584 truncated 5-simplexes 3584 rectified 5-simplexes 7168 5-simplexes
4-faces	1120 truncated 16-cells 7168 truncated 5-cells 7168 rectified 5-cells 21504 5-cells
Cells	1792 truncated tetrahedra 8960 truncated tetrahedra 8960 octahedra 35840 tetrahedra
Faces	7168 hexagons 7168 triangles 35840 triangles
Edges	1792 segments 21504 segments
Vertices	3584
Vertex figure	( )v{ }x{3,3,3,3}
Coxeter groups	D₈, [3^5,1,1]
Properties	convex

In eight-dimensional geometry, a cantic 8-cube or truncated 8-demicube is a uniform 8-polytope, being a truncation of the 8-demicube.

Alternate names

The Cartesian coordinates for the vertices of a truncated 8-demicube centered at the origin and edge length 6√2 are coordinate permutations:

(±1,±1,±3,±3,±3,±3,±3,±3)

with an odd number of plus signs.

Orthographic projections
Coxeter plane	B₈	D₈	D₇	D₆	D₅
Graph
Dihedral symmetry	[16/2]	[14]	[12]	[10]	[8]
Coxeter plane	D₄	D₃	A₇	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]	[8]	[6]	[4]

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 Ivić Weiss, Wiley-Interscience Publication, 1995, wiley.com, ISBN 978-0-471-01003-6
  - (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]
Norman Johnson Uniform Polytopes, Manuscript (1991)
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
Klitzing, Richard. "8D uniform polytopes (polyzetta) with acronyms". x3x3o *b3o3o3o3o3o - thocto

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 • Polytope operations