The countability of a tiling family and the periodicity of a tiling
The following interesting theorem is established: If the family of (\(d\)- dimensional) tilings (the species generated by a finite set of prototiles subject to a finite set of matching rules is countable, then it contains a periodic tiling. Or, conversely, any aperiodic set of prototiles (which, by definition, does not admit a periodic tiling) generates an uncountable species. This extends a result of L. Danzer that for any species of cardinality 1 the (unique) tiling is periodic. In view of known examples the most interesting open problem seems to be to determine whether there exists countably infinite species without a non-periodic element. (Note the following misprints: Page 408, line 12, should read `if \(U^ k \prec U^ 1\) and \(U^ 1 \prec U^ m\), then \(U^ k \prec U^ m\;\)', and page 410, line 4, should read: `with \(q\geq m-1\)'. Moreover, the paper would have deserved another revision since, unfortunately, some of the formulations are not as clear as they should be).
scientific article; zbMATH DE number 764234