# Great rhombated cubic honeycomb

Great rhombated cubic honeycomb
Rank4
TypeUniform
SpaceEuclidean
Notation
Bowers style acronymGrich
Coxeter diagramx4x3x4o ()
Elements
Cells3N cubes, N truncated octahedra, N great rhombicuboctahedra
Faces6N+12N squares, 8N hexagons, 3N octagons
Edges12N+12N+24N
Vertices24N
Vertex figureSphenoid, edge lengths 2 (3), 3 (2), and 2+2 (1)
Measures (edge length 1)
Vertex density${\displaystyle \frac{528\sqrt2-600}{343} \approx 0.42771067327}$
Dual cell volume${\displaystyle \frac{25+22\sqrt2}{24} \approx 2.33802909884}$
Related polytopes
ArmyGrich
RegimentGrich
DualSphenoidal honeycomb
ConjugateGreat quasirhombated cubic honeycomb
Topological properties
OrientableYes
Properties
SymmetryR4
ConvexYes

The great rhombated cubic honeycomb, or grich, also known as the cantitruncated cubic honeycomb, is a convex uniform honeycomb. 1 truncated octahedron, 1 cube, and 2 great rhombicuboctahedra join at each vertex of this honeycomb. As the name suggests, it is the cantitruncation of the cubic honeycomb.

This honeycomb can be alternated into a snub rectified cubic honeycomb, although it cannot be made uniform. The octagons can also be alternated into long rectangles to create a cantic snub cubic honeycomb, which is also nonuniform.

## Vertex coordinates

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

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

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

## Representations

A great rhombated cubic honeycomb has the following Coxeter diagrams:

• x4x3x4o (regular)
• x3x3x *b4x (S4 symmetry)
• s4x3x4x (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}