Demicube

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 $2^{n-1}$ 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.

n -demicube
Rank	n
Type	Uniform
Notation
Coxeter diagram	...
Elements
Vertices	$2^{n-1}$
Vertex figure	Rectified (n -1)-simplex, edge length 1
Measures (edge length 1)
Circumradius	${\frac {\sqrt {2n}}{4}}$
Volume	${\frac {1-{\frac {2^{n-1}}{n!}}}{\sqrt {2^{n}}}}$
Dihedral angles	Demicube-demicube-demicube: 90°
	Demicube-simplex-simplex: $\arccos \left(-{\frac {\sqrt {n}}{n}}\right)$
Height	${\frac {\sqrt {2}}{2}}$
Central density	1
Number of external pieces	$2^{n-1}+2n$
Level of complexity	$n-2$
Related polytopes
Conjugate	None
Abstract & topological properties
Flag count	$n!*(n-2)*2^{n-1}$
Euler characteristic	0 if n  even 2 if n  odd
Orientable	Yes
Properties
Symmetry	Dn , order $n!*2^{n-1}$
Convex	Yes
Nature	Tame

Examples

The demicubes up to 10D are the following:

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

External links

• Wikipedia contributors. "Demihypercube".
• Klitzing, Richard. Demihypercube D_n