# 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${\displaystyle {\frac {4{\sqrt {3}}}{9}}\approx 0.76980}$
Related polytopes
ArmyHexat
RegimentHexat
DualTriangular tiling
Petrie dualPetrial hexagonal tiling
ConjugateNone
Abstract & topological properties
Flag count12N
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryV3
ConvexYes
NatureTame

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:

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

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.