In a paper submitted on May 28th, 2023, researchers introduced a newly discovered “hat” aperiodic monotile which mixes reflected and unreflected tiles in every tiling it allows. This discovery raised the question of whether a single shape could tile aperiodically using solely translations and rotations. However, the researchers found that a related shape, the equilateral member of the same continuum as the hat, is a weakly chiral aperiodic monotile that only allows non-periodic tilings if reflections are prohibited. By modifying the equilateral member, they also discovered a family of shapes called Spectres that are strictly chiral aperiodic monotiles allowing only chiral non-periodic tilings based on a hierarchical substitution system.
https://arxiv.org/abs/2305.17743