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Concept
Triangular tiling
Summary
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.
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.
The vertex arrangement of the triangular tiling is called an A2 lattice. It is the 2-dimensional case of a simplectic honeycomb.
The A_ lattice (also called A_3) 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 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.
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