Graph Theoretic Analyses of Tessellations of Five Aperiodic Polykite Unitiles

Aperiodic tessellations of polykite unitiles, such as hats and turtles, and the recently introduced hares, red squirrels, and gray squirrels, have attracted significant interest due to their structural and combinatorial properties. Our primary objective here is to learn how we could build a self-assembling polyhedron that would have an aperiodic tessellation of its surface using only a single type of polykite unitile. Such a structure would be analogous to some viral capsids that have been reported to have a quasicrystal configuration of capsomeres. We report on our use of a graph–theoretic approach to examine the adjacency and symmetry constraints of these unitiles in tessellations because by using graph theory rather than the usual geometric description of polykite unitiles, we are able (1) to identify which particular vertices and/or edges join one another in aperiodic tessellations; (2) to take advantage of being scale invariant; and (3) to use the deformability of shapes in moving from the plane to the sphere. We systematically classify their connectivity patterns and structural characteristics by utilizing Hamiltonian cycles of vertex degrees along the perimeters of the unitiles. In addition, we applied Blumeyer’s 2 × 2 classification framework to investigate the influence of chirality and periodicity, while Heesch numbers of corona structures provide further insights into tiling patterns. Furthermore, we analyzed the distribution of polykite unitiles with Voronoi tessellations and their Delaunay triangulations. The results of this study contribute to a better understanding of self-assembling structures with potential applications in biomimetic materials, nanotechnology, and synthetic biology.