“A Chiral Aperiodic Monotile”, David Smith, Joseph Samuel Myers, Craig S. Kaplan, Chaim Goodman-Strauss2023-05-28 (math)⁠:

The recently discovered “hat” aperiodic monotile mixes unreflected and reflected tiles in every tiling it admits, leaving open the question of whether a single shape can tile aperiodically using translations and rotations alone.

We show that a close relative of the hat—the equilateral member of the continuum to which it belongs—is a weakly chiral aperiodic monotile: it admits only non-periodic tilings if we forbid reflections by fiat.

Furthermore, by modifying this polygon’s edges we obtain a family of shapes called Spectres that are strictly chiral aperiodic monotiles: they admit only chiral non-periodic tilings based on a hierarchical substitution system.