The 5-demicube, also called the demipenteract or alternated 5‑cube, is a uniform 5‑dimensional polytope derived from the regular 5‑cube (penteract) by a process known as alternation. In this construction, every other vertex of the 5‑cube is removed, leaving a configuration that retains uniformity but is no longer regular.
Construction and coordinates
The regular 5‑cube has vertices at all points $(\pm1,\pm1,\pm1,\pm1,\pm1)$. The 5‑demicube consists of those vertices of the 5‑cube for which the number of positive signs is odd (or, equivalently, even). This selection yields $2^{5-1}=16$ vertices.
Symmetry
The symmetry group of the 5‑demicube is the Coxeter group $D_{5}$, of order
$$ |D_{5}| = 2^{4},5! = 1920 . $$
Its Coxeter–Dynkin diagram is $\stackrel{3}{\circ}!-!\stackrel{3}{\circ}!-!\stackrel{3}{\circ}!-!\stackrel{3}{\circ}$, indicating that the polytope is vertex‑transitive under the $D_{5}$ symmetry.
Facets
The facets (4‑dimensional faces) of the 5‑demicube are of two types:
- 5‑cells (regular 4‑simplexes).
- 16‑cells (regular 4‑orthoplexes).
The exact numbers of each type of facet are determined by the alternation of the 5‑cube’s cubic and tetrahedral cells; the 5‑demicube contains a mixture of these two regular 4‑polytopes, preserving uniformity.
Relation to other polytopes
The 5‑demicube belongs to the infinite family of n‑demicubes (alternated n‑cubes). For $n=3$ the demicube is the regular tetrahedron; for $n=4$ it is the demitesseract, a uniform 4‑polytope with 16 vertices. The 5‑demicube is one of the four semiregular convex uniform 5‑polytopes, alongside the 5‑cube, the 5‑orthoplex, and the 5‑cell.
Notation
Schläfli symbol: The 5‑demicube can be denoted by the extended Schläfli symbol ${3,3,3,3}{!!1}$, indicating its derivation from the regular 5‑cube ${4,3,3,3}$ by alternation.
Coxeter notation: $1{31}$ (alternated form of the $B_{5}$ family).
Properties
| Property | Value |
|---|---|
| Vertices | 16 |
| Edges | 40 |
| 2‑faces (triangular) | 80 |
| 3‑faces (tetrahedral) | 80 |
| 4‑faces (facets) | 20 (mix of 5‑cells and 16‑cells) |
| Symmetry group | $D_{5}$ (order 1920) |
These values follow from the uniform construction and can be derived combinatorially from the vertex coordinates and adjacency relations.
References
- H.S.M. Coxeter, Regular Polytopes, 3rd ed., Dover Publications, 1973.
- N.W. Johnson, Uniform Polytopes and Honeycombs, 1991 (Ph.D. dissertation, University of Toronto).
- G. Olshevsky, “Uniform Polytopes”, available at https://mathworld.wolfram.com/UniformPolytope.html (accessed 2026).
The 5‑demicube is an established object in the study of higher‑dimensional geometry and serves as a standard example of a uniform but non‑regular polytope obtained by alternation of a hypercube.