Tilings and patterns
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A d-fold uniform covering of Rn by elementary shapes is a level d multiple tiling of Rn. The set of level values for which a prototile (a model shape) admits a multiple tiling is called the level semigroup of the prototile. In this paper we discuss the existence of prototiles with nontrivial level semigroups: for instance, does there exist a prototile admitting both tilings of levels 2 and 3, yet not admitting any tilings of level 1? The answer is yes--in fact, we show that for any a, b ∈ N there is a prototile whose level semigroup is exactly the set of nonnegative integer linear combinations of a and b.