Triangular tiling

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Triangular tiling

Type	Regular tiling
Vertex figure	3.3.3.3.3.3
Schläfli symbol	{3,6}
Wythoff symbol	6 \| 3 2 3 \| 3 3 \| 3 3 3
Coxeter-Dynkin
Symmetry	p6m
Dual	Hexagonal tiling
Properties	Vertex-transitive, edge-transitive, face-transitive
3.3.3.3.3.3

In geometry, the triangular tiling is one of the three regular tilings of the Euclidean plane. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees. The triangular tiling has Schläfli symbol of {3,6}.

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 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), and continues into the hyperbolic plane.

(3³)

(3⁴)

(3⁵)

(3⁶)

(3⁷)

It is also topologically related as a part of sequence of Catalan solids with face configuration V(n.6.6).

(V3.6.6)

(V4.6.6)

(V5.6.6)

(V6.6.6) tiling

(V7.6.6) tiling

[edit] See also

[edit] References

Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p.296, Table II: Regular honeycombs
Grünbaum, Branko ; and Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-716-71193-1. (Chapter 2.1: Regular and uniform tilings, p.58-65)
Williams, Robert The Geometrical Foundation of Natural Structure: A Source Book of Design New York: Dover, 1979. p35