Triangular tiling

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Triangular tiling

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

Conway calls it a deltille.

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.

1 Uniform colorings
2 Related polyhedra
3 See also
4 References
5 External links

[edit] Uniform colorings

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, 111222, 112122, 121212, 121213, 121314)

Here's three of the colorings, generated by Wythoff constructions. Six of nine of them can be made as reductions to the four coloring: 121314. The remaining two, 111222 and 122122, have no Wythoff constructions.

111111	121212	121314
6 \| 3 2	3 \| 3 3	\| 3 3 3

[edit] Related polyhedra

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