Atomistic octagonal random-tiling model

Abstract

A random-tiling model with octagonal symmetry is presented. The tiles are the square and various hexagons. When each vertex of the tiling is decorated by a disc, a valid disc packing is formed, which is an octagonal disc packing of maximum density under certain constraints. A simple inflation rule is given that produces one member of the random-tiling ensemble. The space group of this tiling is non-symmorphic. The projection description of this tiling involves a fractal acceptance domain with fourfold symmetry for the even nodes of a four-dimensional cubic lattice and the same acceptance domain rotated by pi /4 in perpendicular space for the odd nodes. Other tilings can be generated by an inflation rule with constrained randomness. Additional members of the random-tiling ensemble can be created from inflation-generated tilings by a set of update moves that rearrange the positions of the discs along closed loops. An update move does not always conserve the number of each kind of tile.