Worked solution: Smith–Myers–Kaplan–Goodman-Strauss's aperiodic monotiles: the hat and the Spectre (2023)
Now play the classification trick again, one level up: group the metatiles themselves into larger clusters called supertiles, using rules just like before. It turns out these supertiles are built from the metatiles in exactly the same combinatorial pattern that the metatiles were built from hats — so the same grouping trick can be applied again to the supertiles, and again, forever, like Russian nesting dolls that never stop nesting.
Every single tile, at every level, ends up belonging to one and only one supertile in this endless hierarchy. If a tiling ever repeated itself by sliding, a large enough patch would have to overlap a slid copy of itself — but then some tile would belong to two different supertile hierarchies at once, which the uniqueness of the hierarchy forbids. So the tiling can never repeat, and — as a bonus — this nested construction is itself a recipe for actually building a valid tiling by the hat, arbitrarily large.
Applying the same style of forced case analysis one level up, Smith, Myers, Kaplan and Goodman-Strauss (2023, §5, Theorem 5.1) show that in any tiling by the four metatiles, the metatiles combine (after a technical bisection of the and tiles that makes the boundary bookkeeping cleaner) into larger level- supertiles that are combinatorially equivalent to the original metatiles : same matching rules, same adjacency structure, just built at a larger scale.
Because the level- supertiles have the same combinatorial structure as the metatiles, the same substitution can be iterated: level- supertiles group uniquely into level- supertiles, level- into level-, and so on without end, in a style pioneered by Robert Berger's original 1966 aperiodic tile-set construction. Crucially, this hierarchy is unique — every metatile belongs to exactly one level- supertile, which belongs to exactly one level- supertile, and so on.
Uniqueness of the hierarchy is what forces non-periodicity: if a tiling had a translational symmetry, then for large enough a level- supertile would overlap its own translated image, and any metatile in the overlap would then belong to two different infinite hierarchies at once — contradicting uniqueness. The same construction, run forward rather than used for contradiction, also proves that clusters of metatiles (and hence of hats) of unboundedly large size exist, which is exactly what is needed to conclude that the hat does admit a tiling of the plane at all — completing, together with the previous section's impossibility-of-periodicity argument, the proof that the hat is a genuine aperiodic monotile.