Great rhombitetrahedraloctahedral honeycomb
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Great rhombitetrahedraloctahedral honeycomb  

Rank  4 
Type  uniform 
Space  Euclidean 
Notation  
Bowers style acronym  Gratoh 
Coxeter diagram  x3x3o *b4x () 
Elements  
Cells  2N truncated tetrahedra, N truncated cubes, N great rhombicuboctahedra 
Faces  8N triangles, 6N squares, 8N hexagons, 6N octagons 
Edges  12N+12N+24N 
Vertices  24N 
Vertex figure  Sphenoid, edge lengths 1 (1), √2 (1), √3 (2), and √2+√2 (2) 
Measures (edge length 1)  
Vertex density  
Dual cell volume  
Related polytopes  
Army  Gratoh 
Regiment  Gratoh 
Dual  Half sphenoidal honeycomb 
Conjugate  Great quasirhombitetrahedraloctahedral honeycomb 
Abstract & topological properties  
Orientable  Yes 
Properties  
Symmetry  S_{4} 
Convex  Yes 
Nature  Tame 
The great rhombitetrahedraloctahedral honeycomb, or gratoh, also known as the runcicantic cubic honeycomb, is a convex uniform honeycomb. 1 truncated tetrahedron, 1 truncated cube, and 2 great rhombicuboctahedra join at each vertex of this honeycomb. It can be formed as an alternated faceting from the prismatorhombated cubic honeycomb, or alternately as the bitruncation of the tetrahedraloctahedral honeycomb.
Representations[edit  edit source]
A great rhombated tetrahedraloctahedral honeycomb has the following Coxeter diagrams:
 x3x3o *b4x (full symmetry)
 s4o3x4x (as alternated faceting)
Gallery[edit  edit source]

Wireframe

External links[edit  edit source]
 Klitzing, Richard. "gratoh".
 Wikipedia contributors. "Runcicantic cubic honeycomb".
 Binnendyk, Eric. "Category 5: Greater Truncates" (#93).