Worked solution: Smith–Myers–Kaplan–Goodman-Strauss's aperiodic monotiles: the hat and the Spectre (2023)
The first proof showed periodicity is impossible, but not that a tiling exists at all. To settle existence — and give a second, independent handle on aperiodicity — the authors look at any tiling that does exist and show every single hat in it can be sorted, unambiguously, into one of just four larger clusters, named , , , , based only on how a small neighbourhood of tiles is arranged around it.
These four cluster-shapes behave like a new, coarser set of building blocks with their own edge-matching rules inherited from the hats inside them — the fine texture of individual hats is abstracted away, leaving a simpler structure to analyse.
Smith, Myers, Kaplan and Goodman-Strauss (2023, §4) give eight ordered classification rules: scanning outward from a given hat, the first rule whose required neighbouring pattern is present assigns that hat one of eight labels (–, , , , or the catch-all ). Grouping hats by these labels, together with within-cluster and between-cluster matching checks (verified by exhaustively checking all the small local neighbourhood patterns that can occur), shows the labels partition every tiling by the hat into four kinds of clusters called metatiles: (built from hats), ( hat), ( hats), and ( hats), each inheriting matching rules — edges marked with orientation signs — from the geometry of the hats making it up.
Crucially, this classification is forced: every hat in every possible hat-tiling falls into exactly one metatile, with no ambiguity and no leftover tiles, once the matching rules are respected. This converts the infinite, seemingly wild variety of hat-tilings into a manageable question about how just four metatile shapes can fit together — precisely the same style of reduction (infinite to finite) that discharging achieves for the four colour theorem, though the underlying combinatorics here is different.
The next step shows these four metatiles satisfy their own substitution rule, which is what finally establishes that a tiling exists and settles hierarchical non-periodicity by an independent, Berger-style argument.