# Order-4 hexagonal tiling honeycomb

Order-4 hexagonal tiling honeycomb
Rank4
TypeRegular, paracompact
SpaceHyperbolic
Notation
Bowers style acronymShexah
Coxeter diagramx6o3o4o ()
Schläfli symbol{6,3,4}
Elements
Cells4N hexagonal tilings
Faces2NM hexagons
Edges3NM
VerticesNM
Vertex figureOctahedron, edge length 3
Measures (edge length 1)
Circumradius${\displaystyle {\frac {i{\sqrt {2}}}{2}}\approx 0.70711i}$
Related polytopes
ArmyShexah
RegimentShexah
DualOrder-6 cubic honeycomb
Petrie dualPetrial order-4 hexagonal tiling honeycomb
Abstract & topological properties
OrientableYes
Properties
Symmetry[6,3,4]
ConvexYes

The order-4 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. 4 hexagonal tilings meet at each edge, and 8 meet at each vertex.

## Representations

An order-4 hexagonal tiling has the following Coxeter diagrams:

• x6o3o4o () (full symmetry)
• o3o6x *b3o () (half symmetry, 2 types of cells)
• x3x6o3o6*a () (has a triangular antiprism verf)

## Related polytopes

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