Triangular tiling

Triangular tiling
Rank	3
Type	Regular
Space	Euclidean
Notation
Bowers style acronym	Trat
Coxeter diagram	o6o3x ()
Schläfli symbol	{3,6}
Elements
Faces	2N triangles
Edges	3N
Vertices	N
Vertex figure	Hexagon, edge length 1
Measures (edge length 1)
Vertex density	$\frac{2\sqrt3}3 \approx 1.15470$
Related polytopes
Army	Trat
Regiment	Trat
Dual	Hexagonal tiling
Conjugate	None
Topological properties
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	V3
Convex	Yes

The triangular tiling, or trat, is one of the three regular tilings of the Euclidean plane. 6 triangles join at each vertex of this tiling. It is also the 2-dimensional simplectic honeycomb. It is also the alternation of the hexagonal tiling.

Vertex coordinates

The vertices of a triangular tiling of edge length 1 are given by

• $\left(i\frac{\sqrt3}{2},\,j+\frac{i}{2}\right),$

where i and j range over the integers.

Representations

A triangular tiling has the following Coxeter diagrams:

• o6o3x (full symmetry)
• x3o3o3*a (P3 symmetry, triangles considered of two types)
• s6o3o (alternated hexagonal tilling)
• o6s3s
• s3s3s3*a
• xdoo3xodo3xood&#zx (as hull of hexagonal tiling and three larger triangular tilings)

Related tiling

The triangular tiling is the colonel of a two-member regiment that also includes the ditrigonary triangular-hemiapeirogonal tiling. Also in this regiment is a compound of three hexagonal tilings.

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. "trat".

• Wikipedia Contributors. "Triangular tiling".