# 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${\displaystyle \frac{3\sqrt2}4 \approx 1.06066017180}$
Dual cell volume${\displaystyle \frac{2\sqrt2}3 \approx 0.94280904158}$
Related polytopes
ArmyRich
RegimentRich
DualJoined cubic honeycomb
ConjugateNone
Abstract & topological properties
OrientableYes
Properties
SymmetryR4
ConvexYes

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:

• ${\displaystyle \left(\sqrt2i,\,±\frac{\sqrt2}{2}+\sqrt2j,\,±\frac{\sqrt2}{2}+\sqrt2k\right),}$

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

## Representations

A rectified cubic honeycomb has the following Coxeter diagrams:

• (full symmetry)
• (S4 symmetry, as rectified tetrahedral-octahedral honeycomb)
• (S4 symmetry)
• (P4 symmetry, as rectified cyclotetrahedral honeycomb)
• (as alternated faceting)
• (as alternated faceting)
• qo4ox3xo4oq&#zx

## Related polytopes

o4o3o4o truncations
Name OBSA Schläfli symbol CD diagram Picture
Cubic honeycomb chon {4,3,4}
Truncated cubic honeycomb tich t{4,3,4}
Rectified cubic honeycomb rich r{4,3,4}
Bitruncated cubic honeycomb batch 2t{4,3,4}
Rectified cubic honeycomb rich r{4,3,4}
Truncated cubic honeycomb tich t{4,3,4}
Cubic honeycomb chon {4,3,4}
Small rhombated cubic honeycomb srich rr{4,3,4}
Great rhombated cubic honeycomb grich tr{4,3,4}
Small rhombated cubic honeycomb srich rr{4,3,4}
Great rhombated cubic honeycomb grich tr{4,3,4}
Small prismated cubic honeycomb = Cubic honeycomb chon t0,3{4,3,4}
Prismatorhombated cubic honeycomb prich t0,1,3{4,3,4}
Prismatorhombated cubic honeycomb prich t0,1,3{4,3,4}
Great prismated cubic honeycomb gippich t0,1,2,3{4,3,4}