Rectified pentachoron
Rectified pentachoron  

Rank  4 
Type  Uniform 
Notation  
Bowers style acronym  Rap 
Coxeter diagram  o3x3o3o () 
Elements  
Cells  5 tetrahedra, 5 octahedra 
Faces  10+20 triangles 
Edges  30 
Vertices  10 
Vertex figure  Triangular prism, edge length 1 
Edge figure  tet 3 oct 3 oct 3 
Measures (edge length 1)  
Circumradius  
Hypervolume  
Dichoral angles  Oct–3–tet: 
Oct3oct:  
Height  
Central density  1 
Number of external pieces  10 
Level of complexity  3 
Related polytopes  
Army  Rap 
Regiment  Rap 
Dual  Joined pentachoron 
Conjugate  None 
Abstract & topological properties  
Flag count  360 
Euler characteristic  0 
Orientable  Yes 
Properties  
Symmetry  A_{4}, order 120 
Flag orbits  3 
Convex  Yes 
Nature  Tame 
The rectified pentachoron, or rap, also commonly called the rectified 5cell, is a convex uniform polychoron that consists of 5 regular tetrahedra and 5 regular octahedra. Two tetrahedra and three octahedra join at each triangular prismatic vertex. As the name suggests, it can be constructed by rectification of the pentachoron.
It is the vertex figure of the demipenteract.
It is also a convex segmentochoron (designated K4.5 in Richard Klitzing's list), formed as a tetrahedron atop an octahedron. If the edge lengths of this subsymmetrical variation are changed, the result is various nonuniform tetrahedron atop octahedron polychora with 4 triangle antipodiums and 4 triangle pyramids as sides.
It is also isogonal under the 52 step prism subsymmetry, and can be considered to be a 52 double step prism.
Gallery[edit  edit source]
Vertex coordinates[edit  edit source]
The vertices of a rectified pentachoron of edge length 1 are given by:
 ,
 ,
 ,
 ,
 ,
 ,
 .
Simpler coordinates can be given by all even sign changes of first 3 coordinates of:
 ,
and all permutations of the first three coordinates of:
 .
Much simpler coordinates can be given in five dimensions, as all permutations of:
 .
Representations[edit  edit source]
A rectified pentachoron has the following Coxeter diagrams:
 o3x3o3o () (full symmetry)
 xo3ox3oo&#x (A_{3} axial, as tetrahedron atop octahedron)
 oxo oxo3oox&#xt (A_{2}×A_{1} axial, vertexfirst)
 oxoo3xoxo&#xr (A_{2} axial)
 oxoox oxoxo&#xr (A_{1}×A_{1} axial)
Segmentochoron display[edit  edit source]

Tetrahedron atop octahedron
Related polychora[edit  edit source]
The rectified pentachoron is the colonel of a threemember regiment that also includes the facetorectified pentachoron and the prismatointercepted pentachoron.
The rectified pentachoron can be diminished by cutting off triangular prismatic pyramids, with each removing 2 tetrahedra and diminishing the octahedra down to square pyramids. If one pyramid is removed, the result is the triangular antifastegium. If two nonadjacent pyramids are removed, such that one of the octahedra gets reduced down to an equatorial square only, the result is the bidiminished rectified pentachoron.
An octahedral pyramid can be attached to one of the rectified pentachoron's octahedral cells to create the augmented rectified pentachoron, notable as a nonuniform Blind polytope.
Uniform polychoron compounds composed of rectified pentachora include:
 Rectified stellated decachoron (2)
 Compound of 12 rectified pentachora (12)
 Rectified medial hexacosichoron (120)
Isogonal derivatives[edit  edit source]
Substitution by vertices of these following elements will produce these convex isogonal polychora:
 Octahedron (5): Pentachoron
 Tetrahedron (5): Pentachoron
 Triangle (10): Rectified pentachoron
 Triangle (20): Semiuniform small prismatodecachoron without doubled symmetry
 Edge (30): Uniform small rhombated pentachoron
External links[edit  edit source]
 Bowers, Jonathan. "Category 3: Triangular Rectates" (#39).
 Bowers, Jonathan. "Pennic and Decaic Isogonals".
 Klitzing, Richard. "rap".
 Quickfur. "The Rectified 5cell".
 Wikipedia contributors. "Rectified 5cell".
 Hi.gher.Space Wiki Contributors. "Pyrorectichoron".