5-demicube

Demipenteract (5-demicube)
Petrie polygon projection
Type	Uniform 5-polytope
Family (Dn)	5-demicube
Families (En)	k21 polytope 1k2 polytope
Coxeter symbol	121
Schläfli symbols	{3,32,1} = h{4,33} s{2,4,3,3} or h{2}h{4,3,3} sr{2,2,4,3} or h{2}h{2}h{4,3} h{2}h{2}h{2}h{4} s{21,1,1,1} or h{2}h{2}h{2}s{2}
Coxeter diagrams	=
4-faces	26	10 {31,1,1} 16 {3,3,3}
Cells	120	40 {31,0,1} 80 {3,3}
Faces	160	{3}
Edges	80
Vertices	16
Vertex figure	rectified 5-cell
Petrie polygon	Octagon
Symmetry	D5, [32,1,1] = [1+,4,33] [24]+
Properties	convex

In five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube (penteract) with alternated vertices removed.

It was discovered by Thorold Gosset. Since it was the only semiregular 5-polytope (made of more than one type of regular facets), he called it a 5-ic semi-regular. E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM₅ for a 5-dimensional half measure polytope.

Coxeter named this polytope as 1₂₁ from its Coxeter diagram, which has branches of length 2, 1 and 1 with a ringed node on one of the short branches, and Schläfli symbol $\left\{3{\begin{array}{l}3,3\\3\end{array}}\right\}$ or {3,3^2,1}.

It exists in the k₂₁ polytope family as 1₂₁ with the Gosset polytopes: 2₂₁, 3₂₁, and 4₂₁.

The graph formed by the vertices and edges of the demipenteract is sometimes called the Clebsch graph, though that name sometimes refers to the folded cube graph of order five instead.