# Stericated 5-simplex

(Redirected from Small cellidodecateron)
Stericated 5-simplex
Rank5
TypeUniform
Notation
Coxeter diagramx3o3o3o3x ()
Elements
Tera12 pentachora, 30 tetrahedral prisms, 20 triangular duoprisms
Cells60 tetrahedra, 120 triangular prisms
Faces120 triangles, 90 squares
Edges120
Vertices30
Vertex figureTetrahedral antiprism, edge lengths 1 (base) and 2 (sides)
Measures (edge length 1)
Hypervolume${\displaystyle {\frac {21{\sqrt {3}}}{40}}\approx 0.90933}$
Diteral anglesTepe–trip–triddip: 135°
Pen–tet–tepe: ${\displaystyle \arccos \left(-{\frac {\sqrt {10}}{5}}\right)\approx 129.23152^{\circ }}$
Central density1
Number of external pieces62
Level of complexity8
Related polytopes
DualTetrahedral-antitegmatic triacontateron
ConjugateNone
Abstract & topological properties
Flag count11520
Euler characteristic2
OrientableYes
Properties
SymmetryA5×2, order 1440
ConvexYes
NatureTame

The stericated 5-simplex, also called the small cellated hexateron, small cellidodecateron, or scad, is a convex uniform 5-polytope. It consists of 12 regular pentachora, 30 tetrahedral prisms, and 20 triangular duoprisms. 2 pentachora, 8 tetrahedral prisms, and 6 triangular duoprisms join at each vertex. It can be formed by expanding the facets of the 5-simplex outward.

## Vertex coordinates

The vertices of a stericated 5-simplex of edge length 1 can be given in 6 dimensions as all permutations of:

• ${\displaystyle \left({\sqrt {2}},\,{\frac {\sqrt {2}}{2}},\,{\frac {\sqrt {2}}{2}},\,{\frac {\sqrt {2}}{2}},\,{\frac {\sqrt {2}}{2}},\,0\right)}$.

## Representations

A stericated 5-simplex has the following Coxeter diagrams:

• x3o3o3o3x () (full symmetry)
• xxo3ooo3ooo3oxx&#xt (A4 axial, pentachoron-first)
• x(xo)o3o(xo)x x(ox)o3o(ox)x&#xt (A2×A2 axial, triangular duoprism-first)
• x(uo)x x(ox)o3o(oo)o3o(ox)x&#xt (A3×A1 axial, tetrahedral prism-first)

## Related polytopes

The stericated 5-simplex is the colonel of a regiment with 11 members. Of these 11, 7 have full symmetry, while the other 4 have single A5 symmetry only.