# 8-cube

8-cube
Rank8
TypeRegular
Notation
Bowers style acronymOcto
Coxeter diagramx4o3o3o3o3o3o3o ()
Schläfli symbol{4,3,3,3,3,3,3}
Tapertopic notation11111111
Toratopic notationIIIIIIII
Bracket notation[IIIIIIII]
Elements
Zetta16 7-cubes
Exa112 6-cubes
Peta448 5-cubes
Tera1120 tesseracts
Cells1792 cubes
Faces1792 squares
Edges1024
Vertices256
Vertex figure7-simplex, edge length 2
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {2}}\approx 1.41421}$
Inradius${\displaystyle {\frac {1}{2}}=0.5}$
Hypervolume1
Dizettal angle90°
Height1
Central density1
Number of external pieces16
Level of complexity1
Related polytopes
ArmyOcto
RegimentOcto
Dual8-orthoplex
ConjugateNone
Abstract & topological properties
Flag count10321920
Euler characteristic0
OrientableYes
Properties
SymmetryB8, order 10321920
ConvexYes
Net count2642657228[1]
NatureTame

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:

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

## 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)