# 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${\displaystyle 60\sqrt2-84 \approx 0.85281374239}$
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

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:

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

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}