Rectified 5simplex
The rectified 5simplex, also called the rectified hexateron or rix, is a convex uniform 5polytope. It consists of 6 regular pentachora and 6 rectified pentachora. Two pentachora and 4 rectified pentachora join at each tetrahedral prismatic vertex. As the name suggests, it is the rectification of the 5simplex.
Rectified 5simplex  

Rank  5 
Type  Uniform 
Notation  
Bowers style acronym  Rix 
Coxeter diagram  o3x3o3o3o () 
Elements  
Tera  6 pentachora, 6 rectified pentachora 
Cells  30 tetrahedra, 15 octahedra 
Faces  20+60 triangles 
Edges  60 
Vertices  15 
Vertex figure  Tetrahedral prism, edge length 1 
Measures (edge length 1)  
Circumradius  
Hypervolume  
Diteral angles  Rap–tet–pen: 
Rap–oct–rap:  
Height  
Central density  1 
Number of external pieces  12 
Level of complexity  4 
Related polytopes  
Army  Rix 
Regiment  Rix 
Dual  Joined hexateron 
Conjugate  None 
Abstract & topological properties  
Flag count  2880 
Euler characteristic  2 
Orientable  Yes 
Properties  
Symmetry  A_{5}, order 720 
Flag orbits  4 
Convex  Yes 
Nature  Tame 
It is also a convex segmentoteron, as a pentachoron atop rectified pentachoron.
Vertex coordinates edit
The vertices of a rectified 5simplex of edge length 1 are given by:
 ,
 ,
 ,
 ,
 ,
 ,
 ,
 ,
 ,
 ,
 .
Much simpler coordinates can be given in six dimensions, as all permutations of:
 .
Representations edit
A rectified 5simplex has the following Coxeter diagrams:
 o3x3o3o3o ( ) (full symmetry)
 xo3ox3oo3oo&#x (A_{4} axial, pentachoron atop rectified pentachoron)
 oxo oxo3oox3ooo&#xt (A_{3}×A_{1} symmetry, vertexfirst)
 x(oo)x3o(ox)o3o(oo)o&#xt (A_{3} symmetry, tetrahedronfirst)
 oooo3ooxo3oxox&#xr (A_{3} symmetry)
 xoo3oxo oxo3oox&#xt (A_{2}×A_{2} axial, trianglefirst)
 oxoox ooxoo3oxoxo&#xr (A_{2}×A_{1} symmetry)
Gallery edit

A_{4} orthographic projection

A_{3}

A_{2}
Related polytopes edit
A rectified 5simplex can be diminished by removing tetrahedral prismatic pyramids. 3 such pyramids from nonadjacent vertices can be removed to form a scaliform tridiminished rectified hexateron.
The rectified 5simplex is the colonel of a 7member regiment. Its other members include the cellibiprismatointercepted hexateron, facetorectified hexateron, cellihexateron, biprismatodishexateron, cellintercepted dishexateron, and spinobiprismatohexateron.
External links edit
 Bowers, Jonathan. "Category 3: Rectates" (#25).
 Klitzing, Richard. "rix".
 Wikipedia contributors. "Rectified 5simplex".