Triangular tiling
Regular tiling of the plane
| Triangular tiling | |
|---|---|
| Type | Regular tiling |
| Vertex configuration | 3.3.3.3.3.3 (or 36) |
| Face configuration | V6.6.6 (or V63) |
| Schläfli symbol(s) | {3,6} {3[3]} |
| Wythoff symbol(s) | 6 | 3 2 3 | 3 3 | 3 3 3 |
| Coxeter diagram(s) | = |
| Symmetry | p6m, [6,3], (*632) |
| Rotation symmetry | p6, [6,3]+, (632) p3, [3[3]]+, (333) |
| Dual | Hexagonal tiling |
| Properties | Vertex-transitive, edge-transitive, face-transitive |
In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. 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}.
English mathematician John Conway called it a deltille, named from the triangular shape of the Greek letter delta (Δ). The triangular tiling can also be called a kishextille by a kis operation that adds a center point and triangles to replace the faces of a hextille.
It is one of three regular tilings of the plane. The other two are the square tiling and the hexagonal tiling.
01Uniform 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) Three of them can be derived from others by repeating colors: 111212 and 111112 from 121213 by combining 1 and 3, while 111213 is reduced from 121314.
There is one class of Archimedean colorings, 111112, (marked with a *) which is not 1-uniform, containing alternate rows of triangles where every third is colored. The example shown is 2-uniform, but there are infinitely many such Archimedean colorings that can be created by arbitrary horizontal shifts of the rows.
| 111111 | 121212 | 111222 | 112122 | 111112(*) |
| p6m (*632) | p3m1 (*333) | cmm (2*22) | p2 (2222) | p2 (2222) |
| 121213 | 111212 | 111112 | 121314 | 111213 |
| p31m (3*3) | p3 (333) | |||
02A2 lattice and circle packings
The vertex arrangement of the triangular tiling is called an A2 lattice. It is the 2-dimensional case of a simplicial honeycomb.
The A*
2 lattice (also called A3
2) can be constructed by the union of all three A2 lattices, and equivalent to the A2 lattice.
- + + = dual of =
The vertices of the triangular tiling are the centers of the densest possible circle packing. Every circle is in contact with 6 other circles in the packing (kissing number). The packing density is π⁄√12 or 90.69%. The voronoi cell of a triangular tiling is a hexagon, and so the voronoi tessellation, the hexagonal tiling, has a direct correspondence to the circle packings.
03Geometric variations
Triangular tilings can be made with the equivalent {3,6} topology as the regular tiling (6 triangles around every vertex). With identical faces (face-transitivity) and vertex-transitivity, there are 5 variations. Symmetry given assumes all faces are the same color.
Sources and credits
This article is adapted from the Wikipedia article “Triangular tiling”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.
Images, from Wikimedia Commons:
- Tiling 3 simple.svg by Watchduck You can name the author as "T. Piesk", "Tilman Piesk" or "Watchduck"., CC BY 4.0
- Triangular tiling 4-color.svg by Tomruen, CC BY-SA 4.0
- Compound 3 triangular tilings.svg by Д.Ильин: vectorization, CC0
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