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Triangular tiling
From Wikipedia, the free encyclopedia
| 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}.
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.
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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 |
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 |
- 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
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