# Runcicantellated 5-simplex

(Redirected from Prismatorhombated hexateron)
Runcicantellated 5-simplex
Rank5
TypeUniform
Notation
Bowers style acronymPirx
Coxeter diagramx3o3x3x3o ()
Elements
Tera20 triangular duoprisms, 15 truncated tetrahedral prisms, 6 decachora, 6 prismatorhombated pentachora
Cells60+60 triangular prisms, 60 hexagonal prisms, 30+30 truncated tetrahedra, 15 cuboctahedra
Faces60+60+60 triangles, 90+180 squares, 120 hexagons
Edges180+180+180
Vertices180
Vertex figureSkewed rectangular scalene, base edge lengths 1 and 2, opposite edge length 1, lacing lengths 2 and 3
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {33}}{3}}\approx 1.91485}$
Hypervolume${\displaystyle {\frac {241{\sqrt {3}}}{10}}\approx 41.74242}$
Diteral anglesTuttip–trip–triddip: 135°
Deca–tut–tuttip: ${\displaystyle \arccos \left(-{\frac {\sqrt {10}}{5}}\right)\approx 129.23152^{\circ }}$
Prip–trip–triddip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}$
Prip–hip–tuttip: ${\displaystyle \arccos \left(-{\frac {\sqrt {10}}{10}}\right)\approx 108.43495^{\circ }}$
Prip–tut–deca: ${\displaystyle \arccos \left(-{\frac {1}{5}}\right)\approx 101.53696^{\circ }}$
Prip–co–prip: ${\displaystyle \arccos \left({\frac {1}{5}}\right)\approx 78.46304^{\circ }}$
Central density1
Number of external pieces47
Level of complexity40
Related polytopes
ArmyPirx
RegimentPirx
DualRhombiscalenic hecatonicoctacontateron
ConjugateNone
Abstract & topological properties
Flag count28800
Euler characteristic2
OrientableYes
Properties
SymmetryA5, order 720
ConvexYes
NatureTame

The runcicantellated 5-simplex, also called the prismatorhombated hexateron or pirx, is a convex uniform 5-polytope. It consists of 20 triangular duoprisms, 15 truncated tetrahedral prisms, 6 decachora, and 6 prismatorhombated pentachora. 2 prismatorhombated pentachora, 1 decachoron, 1 triangular duoprism, and 2 truncated tetrahedral prisms join at each vertex. As the name suggests, it is the runcicantellation of the 5-simplex.

## Vertex coordinates

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

• ${\displaystyle \left({\frac {3{\sqrt {2}}}{2}},\,{\sqrt {2}},\,{\sqrt {2}},\,{\frac {\sqrt {2}}{2}},\,0,\,0\right)}$.

## Representations

A runcicantellated 5-simplex has the following Coxeter diagrams:

• x3o3x3x3o () (full symmetry)
• oxxx3xxoo3xxux3ooox&#xt (A4 axial, decachoron-first)

## Related polytopes

The runcicantellated 5-simplex is the colonel of a three-member regiment that also includes the retroprismatorhombated hexateron and lepidoprismatorhombated hexateron.