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.
| A2 lattice circle packing | A2* lattice circle packing |
|---|---|
| Hexagonal tilings | |
Read more about this topic: Triangular Tiling
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—St. Augustine (354–430)