# Hexagonal tiling

Hexagonal tiling Rank3
TypeRegular
SpaceEuclidean
Notation
Bowers style acronymHexat
Coxeter diagramx6o3o (     )
Schläfli symbol{6,3}
Elements
FacesN hexagons
Edges3N
Vertices2N
Vertex figureEquilateral triangle, edge length 3
Measures (edge length 1)
Vertex density$\frac{4\sqrt3}9 \approx 0.76980$ Related polytopes
ArmyHexat
RegimentHexat
DualTriangular tiling
Petrie dualPetrial hexagonal tiling
ConjugateNone
Topological properties
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryV3
ConvexYes

The hexagonal tiling, or hexat, is one of the three regular tilings of the Euclidean plane. 3 hexagons join at each vertex of this tiling. It can also be formed as a truncation of the triangular tiling.

## Vertex coordinates

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

• $\left(3i\pm\frac12,\,\sqrt3j+\frac{\sqrt3}{2}\right),$ • $\left(3i\pm1,\,\sqrt3j\right),$ where i and j range over the integers.

## Representations

A hexagonal tiling has the following Coxeter diagrams:

• x6o3o (full symmetry)
• o6x3x (as truncated triangular tiling)
• x3x3x3*a ((P3 symmetry, as omnitruncated cyclotriangular tiling)
• s6x3x (additional alternated faceting form)
• x∞s2s∞o (as alternated faceting from the square tiling)
• uBxx3uxBx3uxxB&#zx (B = 4)
• ho3oo3oh&#zx (hull of two opposite triangular tilings)

## In vertex figures

The hexagonal tiling appears as the vertex figure of the hyperbolic triangular tiling honeycomb.