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

The hat is a polykite: its boundary is a union of edges of the kite-shaped tiles of the Laves $[3.4.6.4]$ tiling (the dual of the Archimedean $(3.4.6.4)$ tiling, in which every vertex is surrounded by a triangle, two squares and a hexagon). Eight of these kites assemble into the $13$-sided, $60^\circ$-lattice-aligned shape $T_{\text{hat}} = K_1 \cup \cdots \cup K_8$; all its edge directions lie in the $12$ directions of the triangular lattice, which is what makes its tilings amenable to combinatorial — rather than metric — analysis.

The discovery was empirical before it was rigorous: David Smith generated large patches with Myers's polyform tiling software and Kaplan's Heesch-number program. A tile that fails to tile the plane has a finite Heesch number (the number of concentric coronas it can support); the hat's patches already exceeded the previous record of $6$ coronas, and forcing a periodic tiling would have required isohedral number at least $64$ — either observation alone signalled an extraordinary tile.

Everything proved about the hat is relative to the tiling convention used: tiles may be rotated, translated and reflected. The proof that follows splits into a geometric argument that no periodic tiling exists (Step 3), and a hierarchical-substitution argument that tilings exist and are forced into a unique non-repeating structure (Steps 4–6); the two halves are logically independent.