Runcicantellated 5simplex
Runcicantellated 5simplex  

Rank  5 
Type  Uniform 
Notation  
Bowers style acronym  Pirx 
Coxeter diagram  x3o3x3x3o () 
Elements  
Tera  20 triangular duoprisms, 15 truncated tetrahedral prisms, 6 decachora, 6 prismatorhombated pentachora 
Cells  60+60 triangular prisms, 60 hexagonal prisms, 30+30 truncated tetrahedra, 15 cuboctahedra 
Faces  60+60+60 triangles, 90+180 squares, 120 hexagons 
Edges  180+180+180 
Vertices  180 
Vertex figure  Skewed rectangular scalene, base edge lengths 1 and √2, opposite edge length 1, lacing lengths √2 and √3 
Measures (edge length 1)  
Circumradius  
Hypervolume  
Diteral angles  Tuttip–trip–triddip: 135° 
Deca–tut–tuttip:  
Prip–trip–triddip:  
Prip–hip–tuttip:  
Prip–tut–deca:  
Prip–co–prip:  
Central density  1 
Number of external pieces  47 
Level of complexity  40 
Related polytopes  
Army  Pirx 
Regiment  Pirx 
Dual  Rhombiscalenic hecatonicoctacontateron 
Conjugate  None 
Abstract & topological properties  
Flag count  28800 
Euler characteristic  2 
Orientable  Yes 
Properties  
Symmetry  A_{5}, order 720 
Convex  Yes 
Nature  Tame 
The runcicantellated 5simplex, also called the prismatorhombated hexateron or pirx, is a convex uniform 5polytope. 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 5simplex.
Vertex coordinates[edit  edit source]
The vertices of a runcicantellated 5simplex of edge length 1 can be given in 6 dimensions as all permutations of:
 .
Representations[edit  edit source]
A runcicantellated 5simplex has the following Coxeter diagrams:
 x3o3x3x3o () (full symmetry)
 oxxx3xxoo3xxux3ooox&#xt (A_{4} axial, decachoronfirst)
Gallery[edit  edit source]

A_{4} orthographic projection

A_{3}

A_{2}
Related polytopes[edit  edit source]
The runcicantellated 5simplex is the colonel of a threemember regiment that also includes the retroprismatorhombated hexateron and lepidoprismatorhombated hexateron.
External links[edit  edit source]
 Bowers, Jonathan. "Category 13: Prismatorhombates and Kin" (#829).
 Klitzing, Richard. "pirx".
 Wikipedia contributors. "Runcicantellated 5simplex".