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	*632 and *333
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}.

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.

Contents 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 are three of the colorings generated by Wythoff constructions. Six of the nine distinct colorings can be made as reductions of 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

• Tilings of regular polygons
• List of uniform tilings

[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

[edit] External links

• Weistein, Eric W., "Regular tessellation" from MathWorld.
• Weistein, Eric W., "Triangular Grid" from MathWorld.
• Weistein, Eric W., "Uniform tessellation" from MathWorld.