# Demicube

(Redirected from Demihypercube)
n -demicube
Rankn
TypeUniform
Notation
Coxeter diagram...
Elements
Vertices${\displaystyle 2^{n-1}}$
Vertex figureRectified (n -1)-simplex, edge length 1
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {2n}}{4}}}$
Volume${\displaystyle {\frac {1-{\frac {2^{n-1}}{n!}}}{\sqrt {2^{n}}}}}$
Dihedral anglesDemicube-demicube-demicube: 90°
Demicube-simplex-simplex: ${\displaystyle \arccos \left(-{\frac {\sqrt {n}}{n}}\right)}$
Height${\displaystyle {\frac {\sqrt {2}}{2}}}$
Central density1
Number of external pieces${\displaystyle 2^{n-1}+2n}$
Level of complexity${\displaystyle n-2}$
Related polytopes
ConjugateNone
Abstract & topological properties
Flag count${\displaystyle n!*(n-2)*2^{n-1}}$
Euler characteristic0 if n  even
2 if n  odd
OrientableYes
Properties
SymmetryDn , order ${\displaystyle n!*2^{n-1}}$
ConvexYes
NatureTame

A demicube, demihypercube, or half measure polytope is one of an infinite family of convex uniform polytopes. The n -dimensional demicube, or simply the n -demicube, has ${\displaystyle 2^{n-1}}$ vertices. It can be formed by alternation of the n -hypercube, making its facets into (n -1)-demicube facets and half of its vertex figures into (n -1)-simplex facets.

## Examples

The demicubes up to 10D are the following:

Rank Name Picture Rank 3 Tetrahedron 7 Demihepteract 4 Hexadecachoron 8 Demiocteract 5 Demipenteract 9 Demienneract 6 Demihexeract 10 Demidekeract

## Vertex coordinates

Coordinates for the vertices of an n -demicube with edge length 1 are given by all even sign changes of:

• (2/4, 2/4, ..., 2/4).

## Measures

• The circumradius of an n -demicube of unit edge length is ${\displaystyle {\frac {\sqrt {2n}}{4}}}$.
• Its height from a demicube facet to the opposite demicube facet is ${\displaystyle {\frac {\sqrt {2}}{2}}}$, regardless of n .
• The angle between two demicube facet hyperplanes is 90°, and the angle between a demicube and a simplex facet hyperplane is ${\displaystyle \arccos \left(-{\frac {\sqrt {n}}{n}}\right)}$.