ABSTRACT. A tiling x of Rn is almost periodic if a copy of any patch in x occurs within a fixed distance from an arbitrary location in x. Periodic tilings are almost periodic, but aperiodic almost periodic tilings also exist; for example, the well known Penrose tilings have this property. This paper develops a generalized symmetry theory for almost periodic tilings which reduces in the periodic case to the classical theory of symmetry types. This approach to classification is based on a dynamical theory of tilings, which can be viewed as a continuous and multidimensional generalization of symbolic dynamics.

INTRODUCTION

The purpose of this this paper is to describe a natural generalization of the standard theory of symmetry types for periodic tilings to a larger class of tilings called almost periodic tilings. In particular, a tiling x of Rn is called almost periodic if a copy of any patch which occurs in x re-occurs within a bounded distance from an arbitrary location in x. Periodic tilings are clearly almost periodic since any patch occurs periodically, but there are also many aperiodic examples of almost periodic tilings—the most famous being the Penrose tilings, discovered in around 1974 by R. Penrose.

Ordinary symmetry theory is based on the notion of a symmetry group—the group of all rigid motions leaving an object invariant. The symmetry groups of periodic tilings are characterized by the fact that they contains a lattice of translations as a subgroup.