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.

翻译：最近发现的"帽子"非周期单砖在其所有铺砌中均混合使用未反射和反射的砖块，这使得一个单一形状能否仅通过平移和旋转实现非周期铺砌的问题悬而未决。我们证明，"帽子"的近亲——其所属连续统中的等边成员——是弱手性非周期单砖：若禁止通过强制手段反射，它仅允许非周期铺砌。此外，通过修改该多边形的边，我们获得了一族称为"幽灵"的形状，它们是严格手性非周期单砖：基于层级替换系统，这些形状仅允许手性非周期铺砌。