Triangular tiling

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Triangular tiling

Type	Regular tiling
Faces	triangles
Edges	Infinite
Vertices	Infinite
Vertex configuration	3.3.3.3.3.3
Wythoff symbol	6 \| 2 3
Symmetry group	p6m
Dual	hexagonal tiling
Properties	planar, vertex-uniform
Vertex Figure

In geometry, the triangular tiling is a regular tiling of the Euclidean plane. It has Schläfli symbol of {3,6}.

The internal angle of the equilateral triangle is 60 degrees so six triangles at a point make a full 360 degrees. It is one of three regular tilings of the plane. The other two are the square tiling and the hexagonal tiling.

The planar tilings are related to polyhedra. Putting fewer triangles on a vertex leaves a gap and allows it to be folded into a pyramid. These can be expanded to Platonic solids: five, four and three triangles on a vertex define an icosahedron, octahedron, and tetrahedron respectively.

There are 9 distinct vertex-uniform colorings of a triangular tiling. (Naming the colors by indices on the 6 triangles around a vertex: 111111, 111112, 111212, 111213, 11222, 112122, 121212, 121213, 121314)

This tiling is topologically related as a part of sequence of regular polyhedra with vertex figure (3ⁿ).