# Bicantellated 5-simplex

(Redirected from Small birhombidodecateron)
Bicantellated 5-simplex
Rank5
TypeUniform
Notation
Bowers style acronymSibrid
Coxeter diagramo3x3o3x3o ()
Elements
Tera20 triangular duoprisms, 12 small rhombated pentachora
Cells120 triangular prisms, 30 octahedra, 30 cuboctahedra
Faces120+120 triangles, 180 squares
Edges360
Vertices90
Vertex figureTetragonal disphenoidal prism, edge lengths 1 (base squares) and 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {2}}\approx 1.41421}$
Hypervolume${\displaystyle {\frac {83{\sqrt {3}}}{20}}\approx 7.18801}$
Diteral anglesSrip–trip–triddip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}$
Srip–co–srip: ${\displaystyle \arccos \left(-{\frac {1}{5}}\right)\approx 101.53696^{\circ }}$
Srip–oct–srip: ${\displaystyle \arccos \left({\frac {1}{5}}\right)\approx 78.46304^{\circ }}$
Central density1
Number of external pieces32
Level of complexity12
Related polytopes
ArmySibrid
RegimentSibrid
DualDisphenoidal tegmatic enneacontateron
ConjugateNone
Abstract & topological properties
Flag count17280
Euler characteristic2
OrientableYes
Properties
SymmetryA5×2, order 1440
ConvexYes
NatureTame

The bicantellated 5-simplex, also called the small birhombated hexateron, small birhombidodecateron, or sibrid, is a convex uniform 5-polytope. It consists of 20 triangular duoprisms and 12 small rhombated pentachora. 2 triangular duoprisms and 4 small rhombated pentachora join at each vertex. As the name suggests, it is the bicantellation of the 5-simplex. It is also the rectification of the birectified 5-simplex.

## Vertex coordinates

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

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

## Representations

A bicantellated 5-simplex has the following Coxeter diagrams:

• o3x3o3x3o () (full symmetry)
• xoo3oxx3xxo3oox&#xt (A4 axial, small rhombated pentachoron-first)

## Related polytopes

The regiment of the bicantellated 5-simplex includes a total of 23 members. Of these, 15 have full symmetry, while the remaining 8 have single A5 symmetry only.