# Rectified cubic honeycomb

Rectified cubic honeycomb
Rank4
TypeUniform
SpaceEuclidean
Notation
Bowers style acronymRich
Coxeter diagramo4x3o4o ()
Elements
CellsN octahedra, N cuboctahedra
Faces8N triangles, 3N squares
Edges12N
Vertices3N
Vertex figureSquare prism, edge lengths 1 (base) and 2 (sides)
Measures (edge length 1)
Vertex density${\displaystyle {\frac {3{\sqrt {2}}}{4}}\approx 1.06066}$
Dual cell volume${\displaystyle {\frac {2{\sqrt {2}}}{3}}\approx 0.94281}$
Related polytopes
ArmyRich
RegimentRich
DualJoined cubic honeycomb
ConjugateNone
Abstract & topological properties
OrientableYes
Properties
SymmetryR4
ConvexYes
NatureTame

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 tetrahedral-octahedral honeycomb.

## Vertex coordinates

The vertices of a rectified cubic honeycomb of edge length 1 are given by all permutations of:

• ${\displaystyle \left({\sqrt {2}}i,\,\pm {\frac {\sqrt {2}}{2}}+{\sqrt {2}}j,\,\pm {\frac {\sqrt {2}}{2}}+{\sqrt {2}}k\right)}$,

where i, j, and k range over the integers.

## Representations

A rectified cubic honeycomb has the following Coxeter diagrams:

• o4x3o4o () (full symmetry)
• o4x3o2o3*b () (S4 symmetry, as rectified tetrahedral-octahedral honeycomb)
• o4x3x2x3*b () (S4 symmetry)
• x3o3x3o3*a () (P4 symmetry, as rectified cyclotetrahedral honeycomb)
• s4x3o4o () (as alternated faceting)
• s4x3o2o3*b () (as alternated faceting)
• qo4ox3xo4oq&#zx