# Hexagonal tiling honeycomb

Hexagonal tiling honeycomb
Rank4
TypeRegular, paracompact
SpaceHyperbolic
Notation
Bowers style acronymHexa
Coxeter diagramx6o3o3o ()
Schläfli symbol{6,3,3}
Elements
Cells2N hexagonal tilings
FacesNM hexagons
Edges2NM
VerticesNM
Vertex figureTetrahedron, edge length 3
Measures (edge length 1)
Circumradius${\displaystyle i{\sqrt {2}}\approx 1.41421i}$
Related polytopes
ArmyHexah
RegimentHexah
DualTetrahedral honeycomb
Petrie dualPetrial hexagonal tiling honeycomb
Abstract & topological properties
OrientableYes
Properties
Symmetry[6,3,3]
ConvexYes

The hexagonal tiling honeycomb, also known as the order-3 hexagonal tiling honeycomb, is a paracompact regular tiling of 3D hyperbolic space. Each cell is a hexagonal tiling whose vertices lie on a horosphere, a surface in hyperbolic space that approaches a single ideal point at infinity. 3 hexagonal tilings meet at each edge, and 4 meet at each vertex.

It can be seen as a truncated triangular tiling honeycomb or bitruncated order-6 hexagonal tiling honeycomb.

This honeycomb can be alternated into an alternated hexagonal tiling honeycomb, which is uniform.

## Representations

The hexagonal tiling honeycomb has the following Coxeter diagrams:

• x6o3o3o () (full symmetry)
• x3x6o3o () (as truncated triangular tiling honeycomb)
• o6x3x6o () (as bitruncated order-6 hexagonal tiling honeycomb)
• o6x3x3x3*b () (half symmetry of bitruncated order-6 hexagonal tiling honeycomb)
• x3x3x3x3*a3*c *b3*d () (quarter symmetry of bitruncated order-6 hexagonal tiling honeycomb)

## Related polytopes

The hexagonal faces of the hexagonal tiling honeycomb form the regular skew polyhedron {6,4∣6}.