Triangular Tiling - A2 Lattice and Circle Packings

A2 Lattice and Circle Packings

The vertex arrangement of the triangular tiling is called an A₂ lattice. It is the 2-dimensional case of a simplectic honeycomb.

The A₂* lattice (also called A₂3) 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%. Since the union of 3 A2 lattices is also an A2 lattice, the circle packing can be given with 3 colors of circles.

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.