Hexagonal prismatic honeycomb

Hexagonal prismatic honeycomb
Rank4
TypeUniform
SpaceEuclidean
Notation
Bowers style acronymHiph
Coxeter diagramxØo2x6o3o ()
Elements
CellsN hexagonal prisms
Faces3N squares, N hexagons
Edges2N+3N
Vertices2N
Vertex figureTriangular tegum, edge lengths 3 (equatorial) and 2 (sides)
Related polytopes
ArmyHiph
RegimentHiph
DualTriangular prismatic honeycomb
ConjugateNone
Abstract & topological properties
OrientableYes
Properties
SymmetryV3❘W2
ConvexYes
NatureTame

The hexagonal prismatic honeycomb, or hiph, is a convex noble uniform honeycomb. 6 hexagonal prisms join at each vertex of this honeycomb. It is the honeycomb product of the hexagonal tiling and the apeirogon.

This honeycomb can be alternated into a gyrated tetrahedral-octahedral honeycomb, which can be made uniform.

Vertex coordinates

Coordinates for the vertices of a hexagonal prismatic honeycomb of edge length 1 are given by:

• ${\displaystyle \left(3i\pm {\frac {1}{2}},\,{\sqrt {3}}j+{\frac {\sqrt {3}}{2}},\,k\right)}$,
• ${\displaystyle \left(3i\pm 1,\,{\sqrt {3}}j,\,k\right)}$,

where i , j , and k  range over the integers.

Representations

A hexagonal prismatic honeycomb has the following Coxeter diagrams:

• xØo2x6o3o ()
• xØx2x6o3o ()
• xØo2o6x3x ()
• xØx2o6x3x ()
• xØo2x3x3x3*c ()
• xØx2x3x3x3*c ()