Rectified cubic honeycomb
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Rectified cubic honeycomb  

Rank  4 
Type  Uniform 
Space  Euclidean 
Notation  
Bowers style acronym  Rich 
Coxeter diagram  o4x3o4o () 
Elements  
Cells  N octahedra, N cuboctahedra 
Faces  8N triangles, 3N squares 
Edges  12N 
Vertices  3N 
Vertex figure  Square prism, edge lengths 1 (base) and √2 (sides) 
Measures (edge length 1)  
Vertex density  
Dual cell volume  
Related polytopes  
Army  Rich 
Regiment  Rich 
Dual  Joined cubic honeycomb 
Conjugate  None 
Abstract & topological properties  
Orientable  Yes 
Properties  
Symmetry  R_{4} 
Convex  Yes 
Nature  Tame 
The rectified cubic honeycomb, or rich, is a convex uniform honeycomb. 2 octahedra and 4 cuboctahedra join at each vertex of this honeycomb. As the name suggests, it is the rectification of the cubic honeycomb. It is also the rectification of the tetrahedraloctahedral honeycomb.
Vertex coordinates[edit  edit source]
The vertices of a rectified cubic honeycomb of edge length 1 are given by all permutations of:
 ,
where i, j, and k range over the integers.
Representations[edit  edit source]
A rectified cubic honeycomb has the following Coxeter diagrams:
 o4x3o4o () (full symmetry)
 o4x3o2o3*b () (S_{4} symmetry, as rectified tetrahedraloctahedral honeycomb)
 o4x3x2x3*b () (S_{4} symmetry)
 x3o3x3o3*a () (P_{4} symmetry, as rectified cyclotetrahedral honeycomb)
 s4x3o4o () (as alternated faceting)
 s4x3o2o3*b () (as alternated faceting)
 qo4ox3xo4oq&#zx
Gallery[edit  edit source]

Wireframe

External links[edit  edit source]
 Klitzing, Richard. "rich".
 Wikipedia contributors. "Rectified cubic honeycomb".
 Binnendyk, Eric. "Category 3: Rich regiment" (#22).