Great rhombated cubic honeycomb
Great rhombated cubic honeycomb  

Rank  4 
Type  Uniform 
Space  Euclidean 
Notation  
Bowers style acronym  Grich 
Coxeter diagram  x4x3x4o () 
Elements  
Cells  3N cubes, N truncated octahedra, N great rhombicuboctahedra 
Faces  6N+12N squares, 8N hexagons, 3N octagons 
Edges  12N+12N+24N 
Vertices  24N 
Vertex figure  Sphenoid, edge lengths √2 (3), √3 (2), and √2+√2 (1) 
Measures (edge length 1)  
Vertex density  
Dual cell volume  
Related polytopes  
Army  Grich 
Regiment  Grich 
Dual  Sphenoidal honeycomb 
Conjugate  Great quasirhombated cubic honeycomb 
Abstract & topological properties  
Orientable  Yes 
Properties  
Symmetry  R_{4} 
Convex  Yes 
Nature  Tame 
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[edit  edit source]
The vertices of a great rhombated cubic honeycomb of edge length 1 are given by all permutations of:
 ,
Where i, j, and k range over the integers.
Representations[edit  edit source]
A great rhombated cubic honeycomb has the following Coxeter diagrams:
 x4x3x4o () (regular)
 x4x3x2x3*b () (S_{4} symmetry)
 s4x3x4x () (as alternated faceting)
Gallery[edit  edit source]

Wireframe


The four cells meeting at each vertex

Orthogonal projection into Euclidean plane with V_{3} symmetry
External links[edit  edit source]
 Klitzing, Richard. "grich".
 Wikipedia contributors. "Cantitruncated cubic honeycomb".
 Binnendyk, Eric. "Category 5: Greater Truncates" (#92).