Hexagonal tiling

Hexagonal tiling
Rank	3
Type	Regular
Space	Euclidean
Notation
Bowers style acronym	Hexat
Coxeter diagram	x6o3o ()
Schläfli symbol	{6,3}
Elements
Faces	N hexagons
Edges	3N
Vertices	2N
Vertex figure	Equilateral triangle, edge length √3
Measures (edge length 1)
Vertex density	$\frac{4\sqrt3}9 \approx 0.76980$
Related polytopes
Army	Hexat
Regiment	Hexat
Dual	Triangular tiling
Petrie dual	Petrial hexagonal tiling
Conjugate	None
Topological properties
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	V3
Convex	Yes

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.

Related polytopes

o6o3o truncations

Name	OBSA	Schläfli symbol	CD diagram	Picture
Hexagonal tiling	hexat	{6,3}	x6o3o
Truncated hexagonal tiling	toxat	t{6,3}	x6x3o
Trihexagonal tiling	that	r{6,3}	o6x3o
Truncated triangular tiling = Hexagonal tiling	hexat	t{3,6}	o6x3x
Triangular tiling	trat	{3,6}	o6o3x
Small rhombitrihexagonal tiling	srothat	rr{6,3}	x6o3x
Great rhombitrihexagonal tiling	grothat	tr{6,3}	x6x3x
Snub trihexagonal tiling	snathat	sr{6,3}	s6s3s

External links

• Klitzing, Richard. "hexat".

• Wikipedia Contributors. "Hexagonal tiling".