# Diacosipentacontahexazetton

(Redirected from 8-orthoplex)
Diacosipentacontahexazetton Rank8
TypeRegular
SpaceSpherical
Notation
Bowers style acronymEk
Coxeter diagramo4o3o3o3o3o3o3x (               )
Schläfli symbol{3,3,3,3,3,3,4}
Bracket notation<IIIIIIII>
Elements
Zetta256 octaexa
Exa1024 heptapeta
Peta1792 hexatera
Tera1792 pentachora
Cells1120 tetrahedra
Faces448 triangles
Edges112
Vertices16
Vertex figureHecatonicosoctaexon, edge length 1
Measures (edge length 1)
Circumradius$\frac{\sqrt2}{2} \approx 0.70711$ Inradius$\frac14 = 0.25$ Hypervolume$\frac{1}{2520} \approx 0.00039683$ Dizettal angle$\arccos\left(-\frac34\right) \approx 138.59038º$ Height$\frac12 = 0.5$ Central density1
Number of pieces256
Level of complexity1
Related polytopes
ArmyEk
RegimentEk
DualOcteract
ConjugateNone
Abstract properties
Net count2642657228
Euler characteristic0
Topological properties
OrientableYes
Properties
SymmetryB8, order 10321920
ConvexYes
NatureTame

The Diacosipentacontahexazetton, or ek, also called the octacross or 8-orthoplex, is one of the 3 regular polyzetta. It has 256 regular octaexa as facets, joining 4 to a hexateron peak and 128 to a vertex in a hecatonicosoctaexal arrangement. It is the 8-dimensional orthoplex. It is also a hexadecachoric duotegum and square tetrategum.

## Vertex coordinates

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

• $\left(\pm\frac{\sqrt2}{2},\,0,\,0,\,0,\,0,\,0,\,0,\,0\right).$ ## Representations

A regular diacosipentacontahexazetton has the following Coxeter diagrams:

• o4o3o3o3o3o3o3x (full symmetry)
• o3o3o *b3o3o3o3o3x (D8 symmetry)
• xo3oo3oo3oo3oo3oo3ox&#x (A7 axial, octaexic antiprism)
• ooo4ooo3ooo3ooo3ooo3ooo3oxo&#xt (B7 axial, hecatonicosoctaexal tegum)