# 8-orthoplex

8-orthoplex
Rank8
TypeRegular
Notation
Bowers style acronymEk
Coxeter diagramo4o3o3o3o3o3o3x ()
Schläfli symbol{3,3,3,3,3,3,4}
Bracket notation<IIIIIIII>
Elements
Zetta256 7-simplices
Exa1024 6-simplices
Peta1792 5-simplices
Tera1792 pentachora
Cells1120 tetrahedra
Faces448 triangles
Edges112
Vertices16
Vertex figure7-orthoplex, edge length 1
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$
Inradius${\displaystyle {\frac {1}{4}}=0.25}$
Hypervolume${\displaystyle {\frac {1}{2520}}\approx 0.00039683}$
Dizettal angle${\displaystyle \arccos \left(-{\frac {3}{4}}\right)\approx 138.59038^{\circ }}$
Height${\displaystyle {\frac {1}{2}}=0.5}$
Central density1
Number of external pieces256
Level of complexity1
Related polytopes
ArmyEk
RegimentEk
Dual8-cube
ConjugateNone
Abstract & topological properties
Flag count10321920
Euler characteristic0
OrientableYes
Properties
SymmetryB8, order 10321920
ConvexYes
Net count2642657228
NatureTame

The 8-orthoplex, also called the octacross, diacosipentacontahexazetton, or ek, is a regular 8-polytope. It has 256 regular 7-simplices as facets, joining 4 to a 5-simplex peak and 128 to a vertex in a 7-orthoplecial arrangement. It is the 8-dimensional orthoplex. It is also a hexadecachoric duotegum and square tetrategum.

## Vertex coordinates

The vertices of a regular 8-orthoplex of edge length 1, centered at the origin, are given by all permutations of:

• ${\displaystyle \left(\pm {\frac {\sqrt {2}}{2}},\,0,\,0,\,0,\,0,\,0,\,0,\,0\right)}$.

## Representations

A regular 8-orthoplex has the following Coxeter diagrams:

• o4o3o3o3o3o3o3x () (full symmetry)
• o3o3o3o3o3o3x *b3o () (D8 symmetry)
• xo3oo3oo3oo3oo3oo3ox&#x (A7 axial, octaexic antiprism)
• ooo4ooo3ooo3ooo3ooo3ooo3oxo&#xt (B7 axial, hecatonicosoctaexal tegum)