# Small rhombated cubic honeycomb

Small rhombated cubic honeycomb
Rank4
Typeuniform
SpaceEuclidean
Notation
Bowers style acronymSrich
Coxeter diagramx4o3x4o ()
Elements
Cells3N cubes, N cuboctahedra, N small rhombicuboctahedra
Faces8N triangles, 3N+6N+12N squares
Edges12N+24N
Vertices12N
Vertex figureRectangular wedge, edge lengths 1 (two edges of base) and 2 (remaining edges)
Measures (edge length 1)
Vertex density${\displaystyle 60{\sqrt {2}}-84\approx 0.85281}$
Dual cell volume${\displaystyle {\frac {7+5{\sqrt {2}}}{12}}\approx 1.17259}$
Related polytopes
ArmySrich
RegimentSrich
DualNotch honeycomb
ConjugateQuasirhombated cubic honeycomb
Abstract & topological properties
OrientableYes
Properties
SymmetryR4
ConvexYes
NatureTame

The small rhombated cubic honeycomb, or srich, also known as the cantellated cubic honeycomb, is a convex uniform honeycomb. 1 cuboctahedron, 2 small rhombicuboctahedra, and 2 cubes join at each vertex of this honeycomb. As the name suggests, it is the cantellation of the cubic honeycomb.

## Vertex coordinates

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

• ${\displaystyle \left(\pm {\frac {1}{2}}+(1+{\sqrt {2}})i,\,\pm {\frac {1}{2}}+(1+{\sqrt {2}})j,\,\pm {\frac {1+{\sqrt {2}}}{2}}+(1+{\sqrt {2}})k\right)}$,

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

## Representations

A small rhombated cubic honeycomb has the following Coxeter diagrams:

• x4o3x4o () (regular)
• x4o3x2x3*b () (S4 symmetry)
• s4x3o4x () (as alternated faceting)