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
Famous quotes containing the word circle:
“... [a] girl one day flared out and told the principal “the only mission opening before a girl in his school was to marry one of those candidates [for the ministry].” He said he didn’t know but it was. And when at last that same girl announced her desire and intention to go to college it was received with about the same incredulity and dismay as if a brass button on one of those candidate’s coats had propounded a new method for squaring the circle or trisecting the arc.”
—Anna Julia Cooper (1859–1964)