# Great rhombi-tetrahedral-octahedral honeycomb

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Great rhombi-tetrahedral-octahedral honeycomb
Rank4
Typeuniform
SpaceEuclidean
Notation
Bowers style acronymGratoh
Coxeter diagramx3x3o *b4x ()
Elements
Cells2N truncated tetrahedra, N truncated cubes, N great rhombicuboctahedra
Faces8N triangles, 6N squares, 8N hexagons, 6N octagons
Edges12N+12N+24N
Vertices24N
Vertex figureSphenoid, edge lengths 1 (1), 2 (1), 3 (2), and 2+2 (2)
Measures (edge length 1)
Vertex density${\displaystyle \frac{1080\sqrt2-1392}{343} \approx 0.39460830135}$
Dual cell volume${\displaystyle \frac{58+45\sqrt2}{48} \approx 2.53415854806}$
Related polytopes
ArmyGratoh
RegimentGratoh
DualHalf sphenoidal honeycomb
ConjugateGreat quasirhombi-tetrahedral-octahedral honeycomb
Abstract & topological properties
OrientableYes
Properties
SymmetryS4
ConvexYes

The great rhombi-tetrahedral-octahedral 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 tetrahedral-octahedral honeycomb.

## Representations

A great rhombated tetrahedral-octahedral honeycomb has the following Coxeter diagrams:

• x3x3o *b4x (full symmetry)
• s4o3x4x (as alternated faceting)