Triangular tiling

Triangular tiling
Type	Regular tiling
Vertex configuration	3.3.3.3.3.3 (or 36)
Schläfli symbol(s)	{3,6} {3[3]}
Wythoff symbol(s)	6 | 3 2 3 | 3 3 | 3 3 3
Coxeter-Dynkin(s)
Symmetry	p6m, [6,3], *632 p3m1, [3[3]], *333 p3, [3[3]]+, 333
Dual	Hexagonal tiling
Properties	Vertex-transitive, edge-transitive, face-transitive
3.3.3.3.3.3 (or 36)

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, named from the triangular shape of the greek letter delta (Δ). The triangular tiling is roughly the kishextile.

It is one of three regular tilings of the plane. The other two are the square tiling and the hexagonal tiling.

This vertex arrangement is called the A2 lattice.^[1]

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)

Four of the colorings are generated by Wythoff constructions. Seven of the nine distinct colorings can be made as reductions of the four coloring: 121314. The remaining two, 111222 and 112122, have no Wythoff constructions.

Coloring indices	111111	121212	121314	121213
Coloring
Symmetry	*632 (p6m) [6,3]	*333 (p3m1) [3[3]]	333 (p3) [3[3]]+	3*3 (p31m) [6,3+]
Wythoff symbol	6 | 3 2	3 | 3 3	| 3 3 3
Coxeter-Dynkin

Coloring indices	111222	112122	111112	111212	111213
Coloring
Symmetry	2*22 (cmm) [∞,2+,∞]	2222 (p2) [∞,2,∞]+	*333 (p3m1) [3[3]]	*333 (p3m1) [3[3]]	333 (p3) [3[3]]+

Related polyhedra and tilings

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.

This tiling is topologically related as a part of sequence of regular polyhedra with Schläfli symbols {3,n}, continuing into the hyperbolic plane.

{3,3}

{3,4}

{3,5}

{3,6}

{3,7}

{3,8}

{3,9}

It is also topologically related as a part of sequence of Catalan solids with face configuration Vn.6.6, and also continuing into the hyperbolic plane.

V3.6.6

V4.6.6

V5.6.6

V6.6.6

V7.6.6

See also

• Tilings of regular polygons
• List of uniform tilings

Notes

1. ^ http://www2.research.att.com/~njas/lattices/A2.html

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 (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X.  p35
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 [1]

External links

• Weisstein, Eric W., "Triangular Grid" from MathWorld.
  • Weisstein, Eric W., "Regular tessellation" from MathWorld.
  • Weisstein, Eric W., "Uniform tessellation" from MathWorld.
• Richard Klitzing, 2D Euclidean tilings, x3o6o - trat - O2