Solutions to Seven and a Half Problems on Tilings

Abstract

Four problems about tilings, related to the so-called: Heesch number, isohedral number, $m$-morphic figures, and $\sigma$-morphic figures, can be asked in four variations of the notion of tiling: protosets with more elements, disconnected tiles, colored tiles and tessellations in larger-dimensional spaces. That makes $16$ combinations in total. Five among them have been previously solved in the literature, and one has been partially solved. We here solve seven of the remaining combinations, and additionally complete that partial solution.