# 5-demicube

semiregular 5-polytope
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]
+
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 {3,32,1}.

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.

Fundamental convex regular and uniform polytopes in dimensions 2–10
An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Triangle Square p-gon Hexagon Pentagon
Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
5-simplex 5-orthoplex5-cube 5-demicube
6-simplex 6-orthoplex6-cube 6-demicube 122221
7-simplex 7-orthoplex7-cube 7-demicube 132231321
8-simplex 8-orthoplex8-cube 8-demicube 142241421
9-simplex 9-orthoplex9-cube 9-demicube
10-simplex 10-orthoplex10-cube 10-demicube
n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds