ABSTRACT
Aperiodicity is a fascinating phenomenon. Traditionally, an aperiodic protoset is a set of
tiles in euclidean d-space that admit a tiling of by congruent copies, but no such tiling with a non-trivial translational symmetry. In the light of the discovery of the Schmitt-Conway-Danzer tile and its tiling properties ([4]), the notion of aperiodicity has been revised to require the stronger condition that no tiling by the protoset have a euclidean symmetry of infinite order. The purpose of this short note is to introduce a combinatorial analogue of this stronger notion, called combinatorial aperiodicity. Congruence of the tiles is here replaced by combinatorial equivalence of the tiles, and the
interest is in locally finite face-to-face tilings of by convex polytopes each combinatorially equivalent to a polytope from a finite protoset of convex polytopes. At this point it is still open whether or not combinatorially aperiodic protosets actually exist in any dimension. We describe some results about combinatorially aperiodic protosets and their tilings, and also discuss some open problems and conjectures.
