# Trihexagonal prismatic honeycomb

Trihexagonal prismatic honeycomb Rank4
Typeuniform
SpaceEuclidean
Notation
Bowers style acronymThiph
Coxeter diagramx∞o o6x3o
Elements
Cells2N triangular prisms, N hexagonal prisms
Faces2N triangles, 6N squares, N hexagons
Edges3N+6N
Vertices3N
Vertex figureRectangular tegum, edge lengths 1 and 3 (equatorial) and 2 (sides)
Related polytopes
ArmyThiph
RegimentThiph
DualRhombic prismatic honeycomb
ConjugateNone
Abstract & topological properties
OrientableYes
Properties
SymmetryV3❘W2
ConvexYes

The trihexagonal prismatic honeycomb, or thiph, is a convex uniform honeycomb. 4 triangular prisms and 4 hexagonal prisms join at each vertex of this honeycomb. As the name suggests, it is the honeycomb product of the trihexagonal tiling and the apeirogon.

## Vertex coordinates

The vertices of a trihexagonal prismatic honeycomb of edge length 1 are given by:

• $\left(\sqrt3i,\,i+2j+1,\,k\right),$ • $\left(\sqrt3i+\frac{\sqrt3}{2},\,j+\frac12,\,k\right).$ Where i, j, and k range over the integers.

## Representations

A trihexagonal prismatic honeycomb has the following Coxeter diagrams:

• x∞o o6x3o
• x∞x o6x3o
• x∞o x3x3o3*a
• x∞x x3x3o3*a