Proving Tile Aperiodicity

To determine whether a candidate shape is an aperiodic monotile, pursue two complementary, mutually-reinforcing strategies:

1. Existence of a tiling — construct a substitution/inflation system (via metatiles or clusters) that provably tiles arbitrarily large regions, hence the whole plane (Extension Theorem).
2. Non-existence of periodic tilings — either:
   • (a) show any periodic tiling forces a hierarchical supertile decomposition that contradicts translational symmetry (Berger-style), or
   • (b) find an independent geometric/algebraic obstruction (e.g., incommensurable lattice scale factors) that rules out periodicity directly, without needing the full hierarchical argument.

A shape is an aperiodic monotile iff (1) holds and ((2a) or (2b)) holds.

Progress:
- [ ] Step 1: Computationally screen the candidate (Heesch number / isohedral number bounds)
- [ ] Step 2: Search for natural clusters of the tile that suggest metatiles
- [ ] Step 3: Define metatiles + inherited matching rules from cluster boundaries
- [ ] Step 4: Build geometric substitution/inflation rule for metatiles; verify plane-filling
- [ ] Step 5: Prove every tiling by the base tile decomposes uniquely into these clusters
       (computer-assisted case analysis over small patches)
- [ ] Step 6: Prove metatile substitution forces non-periodicity (hierarchical uniqueness argument)
- [ ] Step 7 (optional but strengthens result): Find an independent algebraic/geometric
       incommensurability argument ruling out periodicity directly
- [ ] Step 8: Check edge cases — reflections, degenerate parameter values, alignment to a grid
- [ ] Step 9: Generalize if possible (continuous family, combinatorial equivalence)

Step 1: Computational screening

Before deep proof work, use SAT-based or brute-force search to get bounds:

• If the tile does not tile the plane, what's the minimum Heesch number required to exceed known records? A shape needing a Heesch number far beyond the current record (~6) is a strong aperiodicity candidate.
• If it does tile periodically, what's the minimum isohedral number? High lower bounds (e.g., ≥64) signal unusual structure worth deeper study.
• Generate large patches (10+ concentric rings) and visually inspect for recurring local structure — colour tiles by orientation/reflection class to reveal hidden regularity.

Step 2: Discover clusters

Look for:

• Rare/sparse features (e.g., reflected copies of the tile) that are always surrounded by a fixed congruent "shell" of neighbours.
• Chains or paths of similarly-oriented tiles hinting at direction fields analogous to Ammann bars/Conway worms.
• Leftover tiles organizing into a small number of additional cluster shapes (pairs, triskelia, singletons) that tile the whole patch when overlapping copies of the primary cluster are removed.

Step 3: Define metatiles and matching rules

• Simplify each cluster's boundary into a clean polygon (remove notches by symmetrizing/truncating).
• Where the simplification is ambiguous (e.g., two cluster types differing only in boundary shape), introduce distinguishing notches/labels so clusters don't overlap once simplified.
• Label boundary segments so adjacency in the original tiling becomes an edge-matching rule (A+/A-, etc.) on the metatiles.
• Mark orientation with arrows when a metatile has rotational symmetry, to disambiguate placements.

Step 4: Build and verify the substitution system

• Assemble a known-good patch of metatiles (by hand/construction), identify where supertile boundaries must fall (often at special vertices, e.g., centers of local rotational symmetry) using only combinatorial constraints (fixed interior angles).
• Note: supertiles need not be geometrically similar to the original metatiles at every finite stage — require only combinatorial equivalence (same adjacency/matching structure). Geometric similarity may emerge only in the limit (a fixed point of the renormalized system).
• Iterate the construction arbitrarily many times; invoke the Extension Theorem to conclude the metatiles (and hence base tile) tile the whole plane.
• Separately, compute the fixed-point ("converged") tile shapes and inflation factor via eigenvalue analysis of the substitution matrix if a clean geometric substitution system is desired.

Step 5: Prove forced decomposition (computer-assisted)

This is the most labor-intensive step. Structure it as:

1. Enumerate all possible local neighbourhoods (e.g., "surroundable" k-patches: patches that can be surrounded by at least one more ring) using independent software implementations for cross-validation.
2. Define ordered classification rules: a sequence of pattern-match conditions on a tile's neighbours; the first rule that matches assigns the tile's cluster label. Ties/ambiguities get a composite label resolved later by one more neighbour check.
3. Perform within-cluster checks: verify that whenever a tile gets label L1, the required neighbour with label L2 actually appears in the correct position/orientation (using a spanning tree of the cluster's internal adjacencies — don't check all pairs, just enough to pin down the whole cluster).
4. Perform between-cluster checks: verify that every labelled boundary segment of a cluster only ever meets the segment type required by the matching rules, for every possible neighbouring cluster/orientation.
5. These checks only need to be done over the small enumerated patches — the local nature of the classification rules means correctness on small patches implies correctness globally.
6. Validate the whole pipeline with at least two independently-written implementations.

Step 6: Prove non-periodicity from forced hierarchy

• Show every tile belongs to a unique level-1 supertile, every level-1 supertile to a unique level-2 supertile, etc. (uniqueness is the crux).
• Argue: if a translational symmetry existed, for large enough k a level-k supertile would overlap its own translate, forcing a tile to lie in two different infinite hierarchies — contradicting uniqueness.
• This simultaneously proves (a) no periodic tiling exists and (b) a legitimate tiling of the whole plane exists (arbitrarily large forced patches).

Step 7: Independent incommensurability argument (optional, strengthens/duals proof)

When the tile is part of a parametrized family (e.g., two independent edge lengths), exploit structure directly:

• Degenerate the tile by contracting one edge-length class to zero; if this produces two different, simpler tilings (e.g., by two different "polyiamond"-style shapes) whose combined behavior is constrained, compare their lattices.
• If a periodic tiling of the original tile existed, both degenerate tilings would also be periodic, related by an affine map forced by the area ratio.
• Show the required scale factor (often an irrational like √2) cannot relate translation lattices of the underlying regular grid (e.g., triangular lattice distances are always √(integer) with constrained quadratic form) — contradiction.
• This argument needs balance/equidistribution lemmas (e.g., orientation classes of sub-shapes must occur in equal proportion in any tiling) — typically proven via a worm/strip decomposition argument tracking how translations act on parallel families of sub-tiles.

Step 8: Handle edge cases

• Reflections: determine whether tilings require a mix of reflected/unreflected copies. This affects whether the tile is a true "monotile" under orientation-reversing congruence, and may motivate seeking reflection-free variants.
• Degenerate parameters: check whether special parameter values in a family collapse to shapes with simple periodic tilings (these are the "exceptions" to otherwise-aperiodic families).
• Alignment to an underlying grid: when the tile is a polyform (union of cells from a fixed grid), prove (don't assume) that all tilings are grid-aligned — unaligned tilings are possible in principle (e.g., sliding-strip tilings) and must be ruled out via a general alignment lemma before any grid-based case analysis is valid.

Step 9: Generalize

• Check if nearby parameter values in a continuous family give combinatorially equivalent tilings via an explicit edge-length-rescaling bijection.
• Look for the same cluster/substitution structure arising from an alternative, more minimal set of clusters (fewer rules, smaller generators) — useful for exposition even if less convenient for the core proof.

Example 1: Input: A 13-sided non-convex polygon built from kites of a known Laves tiling; initial computer search finds large (e.g., 391-tile) patches with no obvious periodicity, and shows reflected copies are rare and each is wrapped in a congruent 3-tile shell. Output: Define 4 metatiles from the shell cluster, a leftover single-tile cluster, and two parallelogram-type clusters (plugging into local 3-fold symmetry points vs. not). Build substitution rules; prove via ~200-case computer-assisted analysis that every tiling decomposes into these clusters obeying matching rules; prove hierarchical uniqueness forces non-periodicity. Separately exploit a two-parameter family containing this tile to get an independent √2-incommensurability proof using triangular-lattice distance constraints.

Example 2: Input: A hexagon with proposed markings/matching rules forcing aperiodicity, but the markings require non-adjacent tiles to "communicate." Output: Recognize this does NOT qualify as a pure geometric aperiodic monotile — the matching rules cannot be realized as the geometry of a single connected 2D closed topological disk without 3D lifting or disconnection. Flag this as a different category of result (symbolic/matching-rule aperiodicity) rather than attempt a disk-tile proof.