Diacosipentacontahexazetton

Diacosipentacontahexazetton
(OFF file)
Rank	8
Type	Regular
Space	Spherical
Notation
Bowers style acronym	Ek
Coxeter diagram	o4o3o3o3o3o3o3x ()
Schläfli symbol	{3,3,3,3,3,3,4}
Bracket notation	<IIIIIIII>
Elements
Zetta	256 octaexa
Exa	1024 heptapeta
Peta	1792 hexatera
Tera	1792 pentachora
Cells	1120 tetrahedra
Faces	448 triangles
Edges	112
Vertices	16
Vertex figure	Hecatonicosoctaexon, 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 density	1
Number of pieces	256
Level of complexity	1
Related polytopes
Army	Ek
Regiment	Ek
Dual	Octeract
Conjugate	None
Abstract properties
Net count	2642657228
Euler characteristic	0
Topological properties
Orientable	Yes
Properties
Symmetry	B8, order 10321920
Convex	Yes
Nature	Tame

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)

External links

• Klitzing, Richard. "ek".

• Wikipedia Contributors. "8-orthoplex".