Aperiodic Tilings V: the Refinable Frontier
2 days ago
- Substitution systems for aperiodic tilings can be ambiguous, preventing construction of deterministic transducers for coordinate-based navigation.
- The author developed an automated algorithm to refine ambiguous systems by splitting tile types based on their neighborhood patterns, removing ambiguity.
- The refinement works by analyzing all possible neighbor configurations and creating subtypes that encode enough information to determine child tiles.
- Postprocessing merges subtypes with identical expansions to reduce complexity while preserving transducer compatibility.
- The technique succeeded on the chair tiling, HTPF hat system, Spectre H7/H8, and Awkward Squares, often matching or improving upon known unambiguous systems.
- For the hat tiling, the refiner produced an 8-tile system (beating the previous 10-tile system) and for Spectre it replicated the 9-hex system.
- Zero-thickness spurs in expansions require iterative refinement passes; the algorithm may need multiple runs to achieve full unambiguity.
- The refiner also corrected a bug in the author's earlier Ammann-Beenker transducer, finally enabling deterministic coordinate generation for that tiling.
- This work enables automated disambiguation of new substitution systems without relying on luck or prior discovery of alternative systems.