Spirals in Periodic Tilings
Spiral tilings, as appealing as they are for their aesthetics,have not been studied well mathematically. One of the difficulties in this area of tiling theory is providing a mathematical definition of spiral tilings. A recently published attempt at providing a formal definition distinguishes a so-called L-spiral tiling (L-tiling) and an S-spiral tiling (S-tiling), with the two types being characterized by special properties of tile set partitions. Based on these existing definitions, we investigate the spiral structure in periodic tilings. Unlike spiral tilings, periodic tilings lend themselves easily to a definition and have been well studied. We first prove that it is not possible for periodic tilings to be S-tilings.We then study a subset of periodic tilings that can be L-tilings. In particular, we demonstrate that there exist examples for each type of isohedral tilings (a subset of periodic monohedral tilings) that are L-spirable.