The torus parametrization of quasiperiodic LI-classes

The torus parametrization of quasiperiodic local isomorphism classes is introduced and used to determine the number of elements in such a class with special symmetries or inflation properties. The method is explained in an illustrative fashion for some widely used tiling classes with golden mean rescaling, namely for the Fibonacci chain (1D), the triangle and Penrose patterns (2D) and for Kramer's and Danzer's icosahedral tilings (3D). We obtain a rather complete picture of the orbit structure within these classes, but discuss also various general results. 1. Introduction Given a pattern P in R n , one defines its local isomorphism class (LI-class for short) to be the set of all patterns P 0 locally indistinguishable from P in the sense that arbitrarily large parts or patches of P also appear in P 0 and vice versa 1 . For the comparison of patches we allow only translations, not general motions, see [5, 17] for details. (In [21, 17, 22] LI-classes are called species.) ...