Demicube

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n -demicube
Rankn 
TypeUniform
Notation
Coxeter diagram...
Elements
Vertices
Vertex figureRectified (n -1)-simplex, edge length 1
Measures (edge length 1)
Circumradius
Volume
Dihedral anglesDemicube-demicube-demicube: 90°
 Demicube-simplex-simplex:
Height
Central density1
Number of external pieces
Level of complexity
Related polytopes
ConjugateNone
Abstract & topological properties
Flag count
Euler characteristic0 if n  even
2 if n  odd
OrientableYes
Properties
SymmetryDn , order
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 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[edit | edit source]

The demicubes up to 10D are the following:

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

Vertex coordinates[edit | edit source]

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[edit | edit source]

  • The circumradius of an n -demicube of unit edge length is .
  • Its height from a demicube facet to the opposite demicube facet is , regardless of n .
  • The angle between two demicube facet hyperplanes is 90°, and the angle between a demicube and a simplex facet hyperplane is .

External links[edit | edit source]