8-cube

8-cube
(OFF file)
Rank	8
Type	Regular
Notation
Bowers style acronym	Octo
Coxeter diagram	x4o3o3o3o3o3o3o ()
Schläfli symbol	{4,3,3,3,3,3,3}
Tapertopic notation	11111111
Toratopic notation	IIIIIIII
Bracket notation	[IIIIIIII]
Elements
Zetta	16 7-cubes
Exa	112 6-cubes
Peta	448 5-cubes
Tera	1120 tesseracts
Cells	1792 cubes
Faces	1792 squares
Edges	1024
Vertices	256
Vertex figure	7-simplex, edge length √2
Measures (edge length 1)
Circumradius	${\sqrt {2}}\approx 1.41421$
Inradius	${\frac {1}{2}}=0.5$
Hypervolume	1
Dizettal angle	90°
Height	1
Central density	1
Number of external pieces	16
Level of complexity	1
Related polytopes
Army	Octo
Regiment	Octo
Dual	8-orthoplex
Conjugate	None
Abstract & topological properties
Flag count	10321920
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	B8, order 10321920
Convex	Yes
Net count	2642657228
Nature	Tame

The 8-cube, also called the octeract, or octo, or hexadecazetton, is one of the 3 regular 8-polytopes. It has 16 7-cubes as facets, joining 8 to a vertex.

It is the 8-dimensional hypercube. It is a tesseractic duoprism and square tetraprism.

It can be alternated into a 8-demicube, which is uniform.

Vertex coordinates

The vertices of an 8-cube of edge length 1, centered at the origin, are given by:

• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right)$.

Representations

An 8-cube has the following Coxeter diagrams:

• x4o3o3o3o3o3o3o () (full symmetry)
• x x4o3o3o3o3o3o () (hepteractic prism)
• xx4oo3oo3oo3oo3oo3oo&#x (B₇ axial, hepteract atop hepteract)
• x4o3o3o x4o3o3o () (tesseractic duoprism)
• x4o x4o x4o x4o () (square tetraprism)

External links

• Klitzing, Richard. "octo".
• Wikipedia contributors. "8-cube".