Great rhombitrihexagonal prismatic honeycomb

Great rhombitrihexagonal prismatic honeycomb
Rank	4
Type	uniform
Space	Euclidean
Notation
Bowers style acronym	Grothaph
Coxeter diagram	x∞o x6x3x ()
Elements
Cells	3N cubes, 2N hexagonal prisms, N dodecagonal prisms
Faces	3N+6N+6N+6N squares, 2N hexagons, N dodecagons
Edges	6N+6N+6N+12N
Vertices	12N
Vertex figure	Scalene notch, edge lengths √3 and (√3+√6)/2 (two edges of equatorial triangle) and √2 (remaining edges)
Related polytopes
Army	Grothaph
Regiment	Grothaph
Dual	Disdyakis rhombic prismatic honeycomb
Conjugate	Quasitruncated trihexagonal prismatic honeycomb
Abstract & topological properties
Orientable	Yes
Properties
Symmetry	V3❘W2
Convex	Yes

The great rhombitrihexagonal prismatic honeycomb, or grothaph, also known as the omnitruncated trihexagonal prismatic honeycomb, or otathaph, is a convex uniform honeycomb. 2 cubes, 2 hexagonal prisms, and 2 dodecagonal prisms join at each vertex of this honeycomb. As the name suggests, it is the honeycomb product of the great rhombitrihexagonal tiling and the apeirogon.

This honeycomb can be alternated into a snub trihexagonal antiprismatic honeycomb, although it cannot be made uniform. The dodecagons can also be alternated into long ditrigons to create an edge-snub trihexagonal prismatic honeycomb, which is also nonuniform.

Representations

A great rhombitrihexagonal prismatic honeycomb has the following Coxeter diagrams:

• x∞o x6x3x (full symmetry)
• x∞x x6x3x ()

External links

• Klitzing, Richard. "grothaph".

• Wikipedia Contributors. "Truncated trihexagonal prismatic honeycomb".