Thresholds On Archimedean Lattices
This is a picture of the 11 Archimedean Lattices or uniform tilings, in which all polygons are regular and each vertex is surrounded by the same sequence of polygons. The notation (34, 6) for example means that every vertex is surrounded by four triangles and one hexagon. Drawings from . See also Uniform Tilings.
| Lattice | z | Site Percolation Threshold | Bond Percolation Threshold | |
|---|---|---|---|---|
| (3, 122 ) | 3 | 3 | 0.807900764... = (1 - 2 sin (π/18))1/2 | 0.740420800(2), 0.74042195(80),
0.74042081(10), 0.74042077(2) |
| cross (4, 6, 12) | 3 | 3 | 0.747806(4) | 0.69373383(72) |
| square octagon, bathroom tile, truncated square
(4, 82) |
3 | 3 | 0.729724(3) | 0.67680232(63) |
| honeycomb (63) | 3 | 3 | 0.6970413(10), 0.697043(3), | 0.652703645... = 1-2 sin (π/18), 1+ p3-3p2=0 |
| kagome (3, 6, 3, 6) | 4 | 4 | 0.652703645... = 1 - 2 sin(π/18) | 0.524404978(5), 0.52440499(2),
0.52440572..., 0.52440500(1), 0.52440516(10), 0.5244053(3) |
| ruby (3, 4, 6, 4) | 4 | 4 | 0.621819(3) | 0.52483258(53) |
| square (44) | 4 | 4 | 0.59274621(13), 0.59274621(33), 0.59274598(4), 0.59274605(3) | 1/2 |
| snub hexagonal, maple leaf (34,6 ) | 5 | 5 | 0.579498(3) | 0.43430621(50) |
| snub square, puzzle (32, 4, 3, 4 ) | 5 | 5 | 0.550806(3) | 0.41413743(46) |
| (33, 42) | 5 | 5 | 0.550213(3) | 0.41964191(43) |
| triangular (36) | 6 | 6 | 1/2 | 0.347296355... = 2 sin (π/18), 1+ p3-3p=0 |
Note: sometimes "hexagonal" is used in place of honeycomb, although in some fields, a triangular lattice is also called a hexagonal lattice). z = bulk coordination number.
Read more about this topic: Percolation Threshold