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${\displaystyle {\frac {528{\sqrt {2}}-600}{343}}\approx 0.42771}$
Dual cell volume${\displaystyle {\frac {25+22{\sqrt {2}}}{24}}\approx 2.33803}$
Related polytopes
ArmyGrich
RegimentGrich
DualSphenoidal honeycomb
ConjugateGreat quasirhombated cubic honeycomb
Abstract & topological properties
OrientableYes
Properties
SymmetryR4
ConvexYes
NatureTame

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:

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

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

Representations

A great rhombated cubic honeycomb has the following Coxeter diagrams:

• x4x3x4o () (regular)
• x4x3x2x3*b () (S4 symmetry)
• s4x3x4x () (as alternated faceting)