# 8-demicube

8-demicube
Rank8
TypeUniform
Notation
Bowers style acronymHocto
Coxeter diagramx3o3o3o3o3o3o *b3o ()
Elements
Zetta
Exa
Peta
Tera
Cells1792+8960 tetrahedra
Faces7168 triangles
Edges1792
Vertices128
Vertex figureRectified octaexon, edge length 1
Measures (edge length 1)
Hypervolume${\displaystyle {\frac {157}{2520}}\approx 0.062302}$
Dizettal anglesHesa–hop–oca: ${\displaystyle \arccos \left(-{\frac {\sqrt {2}}{4}}\right)\approx 110.70481^{\circ }}$
Hesa–hax–hesa: 90°
Height${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$
Central density1
Number of external pieces144
Level of complexity6
Related polytopes
ArmyHocto
RegimentHocto
DualSemistellated diacosipentacontahexazetton‎
ConjugateNone
Abstract & topological properties
Flag count30965760
Euler characteristic0
OrientableYes
Properties
SymmetryD8, order 5160960
Flag orbits6
ConvexYes
NatureTame

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:

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

## 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)