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.Comment: 23 pages, 12 figure