# Small rhombated cubic honeycomb

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.

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${\displaystyle 60\sqrt2-84 \approx 0.85281374239}$
Dual cell volume${\displaystyle \frac{7+5\sqrt2}{12} \approx 1.17258898432}$
Related polytopes
ArmySrich
RegimentSrich
DualNotch honeycomb
ConjugateQuasirhombated cubic honeycomb
Abstract & topological properties
OrientableYes
Properties
SymmetryR4
ConvexYes

## Vertex coordinates

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

• ${\displaystyle \left(±\frac12+(1+\sqrt2)i,\,±\frac12+(1+\sqrt2)j,\,±\frac{1+\sqrt2}{2}+(1+\sqrt2)k\right),}$

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

## Representations

A small rhombated cubic honeycomb has the following Coxeter diagrams:

• x4o3x4o (regular)
• x3o3x *b4x (S4 symmetry)
• s4x3o4x (as alternated faceting)

## 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}