Worked solution: Smith–Myers–Kaplan–Goodman-Strauss's aperiodic monotiles: the hat and the Spectre (2023)

Section 6 of Smith, Myers, Kaplan & Goodman-Strauss (2023) and the companion paper (Smith, Myers, Kaplan & Goodman-Strauss 2024, arXiv:2305.17743) address a caveat left open by the hat: every tiling by the hat mixes reflected and unreflected copies (Section 4's classification rules explicitly track reflection), so the hat only tiles aperiodically if mirror-image copies of the tile are allowed as 'congruent' — the traditional convention in tiling theory, but not the only possible one.

The authors first observe that the equilateral member $\mathrm{Tile}(1,1)$ of the continuum, which tiles periodically when reflections are permitted, becomes what they call weakly chiral: if reflections are forbidden purely by convention, it admits only non-periodic tilings. To make the shape aperiodic by its geometry alone rather than by a stipulated rule, they perturb its straight edges into edges built from a smooth curve, oriented consistently so that only unreflected copies of the tile can ever meet edge-to-edge. This family of shapes, called Spectres, includes both polygonal and smooth-edged versions and tiles $\mathbb{R}^2$ using only translations and rotations — never reflections — because reflected copies simply cannot physically fit against the curved edges.

Because the underlying combinatorics of the Spectre tiling is inherited from the same hierarchical substitution and coupled-tiling machinery used for the hat, the Spectre's tilings are still forced to be non-periodic; the only new ingredient is the geometric trick that rules out mirror images. This makes the Spectre the first strictly chiral aperiodic monotile, closing the loop on the 'einstein problem' in the strongest possible sense: a single shape, no reflections needed, no matching rules beyond its own outline.