Quasicrystalline structure of the hat and spectre monotile tilings

Penrose showed roughly 50 years ago that there exists a set of two tile shapes, the now well-known Penrose tiles, that can tile the plane but cannot form a space-filling periodic (crystalline) pattern.  The Penrose tilings exhibit both a non-crystallographic point group symmetry and a diffraction spectrum consisting of a dense set of Bragg peaks signalling a quasiperiodic structure.  Since then, many materials exhibiting this type of quasicrystalline structure have been fabricated in the lab and found in nature.  A similar story may now be just beginning to unfold.  In the past year, two examples of a single tile shape that can tile only in a nonperiodic pattern were discovered: the hat [1] and the spectre [2].  We show that the tilings formed by these shapes are quasiperiodic and that although the point groups are crystalline (hexagonal), the incommensurate ratios of wavelengths in the diffraction spectrum are fixed.  We also show that activation of phason degrees of freedom in these novel tilings involve new types of coordination that may affect the elasticity and annealing properties of physical realizations of the tiling structures.

[1]  Smith et al, arXiv 2303.10798 (2023)

[2]  Smith et al. arXiv 2305.17743 (2023)