Small rhombated pentachoron
Small rhombated pentachoron  

Rank  4 
Type  Uniform 
Notation  
Bowers style acronym  Srip 
Coxeter diagram  x3o3x3o () 
Elements  
Cells  5 octahedra, 10 triangular prisms, 5 cuboctahedra 
Faces  10+20+20 triangles, 30 squares 
Edges  30+60 
Vertices  30 
Vertex figure  Square wedge, edge lengths 1 (base square and top edge) and √2 (side edges) 
Measures (edge length 1)  
Circumradius  
Hypervolume  
Dichoral angles  Oct–3–trip: 
Co–4–trip:  
Co–3–oct:  
Co–3–co:  
Central density  1 
Number of external pieces  20 
Level of complexity  9 
Related polytopes  
Army  Srip 
Regiment  Srip 
Dual  Notched triacontachoron 
Conjugate  None 
Abstract & topological properties  
Flag count  1080 
Euler characteristic  0 
Orientable  Yes 
Properties  
Symmetry  A_{4}, order 120 
Flag orbits  9 
Convex  Yes 
Nature  Tame 
The small rhombated pentachoron, or srip, also commonly called the cantellated 5cell or cantellated pentachoron, is a convex uniform polychoron that consists of 5 regular octahedra, 10 triangular prisms, and 5 cuboctahedra. 1 octahedron, 2 triangular prisms, and 2 cuboctahedra join at each vertex. As one of its names suggests, it can be formed by cantellating the pentachoron. It can also be formed by rectification of the rectified pentachoron.
Gallery[edit  edit source]

Net
Vertex coordinates[edit  edit source]
The vertices of a small rhombated pentachoron of edge length 1 are given by:
 ,
 ,
 ,
 ,
 ,
 ,
 ,
 ,
 ,
 ,
 ,
 ,
 ,
 ,
 ,
 .
Much simpler coordinates can be given in five dimensions, as all permutations of:
 .
Representations[edit  edit source]
A small rhombated pentachoron has the following Coxeter diagrams:
 x3o3x3o () (full symmetry)
 oxx3xxo3oox&#xt (A_{3} axial, octahedronfirst)
 x(uo)xo x(ou)xx3o(xo)xo&#xt (A_{2}×A_{1} axial, triangular prismfirst)
Semiuniform variant[edit  edit source]
The small rhombated pentachoron has a semiuniform variant of the form x3o3y3o that maintains its full symmetry. This variant uses 5 octahedra of size y, 5 rhombitetratetrahedra of form x3o3y, and 10 triangular prisms of form x y3o as cells, with 2 edge lengths.
With edges of length a (surrounds 2 rhombitetratetrahedra) and b (of octahedra), its circumradius is given by and its hypervolume is given by .
Related polychora[edit  edit source]
The small rhombated pentachoron is the colonel of the largest regiment of uniform polychora with A_{4} symmetry, which has a total of 7 members. Its facetings include the retrosphenoverted trispentachoron, small rhombic dispentachoron, pseudorhombic prismatopentachoron, grand rhombic prismatopentachoron, prismatopentintercepted dispentachoron, and prismatointercepted prismatodispentachoron.
When viewed in A_{3} axial symmetry, the small rhombated pentachoron can be cut into 2 segmentochora, namely cuboctahedron atop truncated tetrahedron and octahedron atop truncated tetrahedron, join at the truncated tetrahedral bases.
The triangular pucofastegium occurs as the trianglefirst cap of the small rhombated pentachoron.
Uniform polychoron compounds composed of small rhombated pentachora include:
External links[edit  edit source]
 Bowers, Jonathan. "Category 6: Sphenoverts" (#133).
 Bowers, Jonathan. "Pennic and Decaic Isogonals".
 Klitzing, Richard. "srip".
 Quickfur. "The Cantellated 5cell".
 Wikipedia contributors. "Cantellated 5cell".