8-demicube

8-demicube
(OFF file)
Rank	8
Type	Uniform
Notation
Bowers style acronym	Hocto
Coxeter diagram	x3o3o3o3o3o3o *b3o ()
Elements
Zetta	128 7-simplices 16 7-demicubes
Exa	1024 6-simplices 112 6-demicubes
Peta	3584 5-simplices 448 5-demicubes
Tera	7168 pentachora 1120 hexadecachora
Cells	1792+8960 tetrahedra
Faces	7168 triangles
Edges	1792
Vertices	128
Vertex figure	Rectified 7-simplex, edge length 1
Measures (edge length 1)
Circumradius	1
Hypervolume	${\frac {157}{2520}}\approx 0.062302$
Dizettal angles	Hesa–hop–oca: $\arccos \left(-{\frac {\sqrt {2}}{4}}\right)\approx 110.70481^{\circ }$
	Hesa–hax–hesa: 90°
Height	${\frac {\sqrt {2}}{2}}\approx 0.70711$
Central density	1
Number of external pieces	144
Level of complexity	6
Related polytopes
Army	Hocto
Regiment	Hocto
Dual	Semistellated diacosipentacontahexazetton‎
Conjugate	None
Abstract & topological properties
Flag count	30965760
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	D8, order 5160960
Flag orbits	6
Convex	Yes
Nature	Tame

The 8-demicube, also called the demiocteract, hemiocteract, or hocto, is a convex uniform 8-polytope. It has 16 7-demicubes and 128 7-simplices as facets, with 8 of each at a vertex forming a rectified 7-simplex as the vertex figure. It is the 8-dimensional demihypercube and is formed by alternating the 8-cube. It is also a segmentozetton, as a 7-demicubic antiprism.

The 8-demicube can also be seen as the convex hull of two opposite hexadecachoric duoprisms, and also contains the vertices of a tetrahedral-5-demicubic duoprism. A unit 8-demicube can be vertex-inscribed into the 4₂₁ polytope.

Vertex coordinates

The vertices of an 8-demicube of edge length 1, centered at the origin, are given by all even sign changes of:

• $\left({\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}}\right)$.

Representations

An 8-demicube has the folowing Coxeter diagrams:

• x3o3o3o3o3o3o *b3o () (full symmetry)
• s4o3o3o3o3o3o3o () (as alternated octeract)
• xo3oo3ox *b3oo3oo3oo3oo&#x (D₇ axial, 7-demicubic antiprism)
• xooo3oooo3oxoo3oooo3ooxo3oooo3ooox&#xt (A₇ axial, 7-simplex-first)
• ooooo3oxooo3ooooo3ooxoo3ooooo3oooxo3ooooo&#xt (A₇ axial, vertex-first)
• xo3oo3ox *b3oo xo3oo3ox *f3oo&#zx (hull of two hexadecachoric duoprisms)

External links

• Klitzing, Richard. "hocto".
• Wikipedia contributors. "8-demicube".