Hierarchical compression uses multiple levels of dictionary lookup: level 1 catches local patterns, level 2 catches patterns of patterns, and deeper levels capture progressively longer-range structure. Periodic hierarchies — where the lookup structure repeats at each level — collapse after O(log p) levels. Beyond that depth, every lookup points to the same dictionary entry, and deeper levels contribute nothing.
Fibonacci hierarchies don't collapse. The aperiodic structure of the Fibonacci tiling ensures that n-gram lookup positions remain usable at every level, no matter how deep. The irrationality of the golden ratio (which governs the Fibonacci tiling's spacing) prevents the periodic alignment that would make deeper levels redundant.
The Quasicryth compressor validates this: a ten-level Fibonacci hierarchy achieves 35.99% on the enwik9 benchmark, with increasing advantage over periodic baselines as file size grows. Deeper levels activate on larger files, and each activated level contributes because the aperiodic structure guarantees fresh lookup positions.
The structural point: periodicity is a symmetry, and symmetries reduce degrees of freedom. A periodic hierarchy has fewer effective levels than its nominal depth because the periodicity creates redundancy between levels. Aperiodicity breaks this symmetry, giving each level independent lookup structure. The same total hierarchy, but more effective depth. Disorder (aperiodicity) creates information (usable lookup positions) by preventing the collapse that order (periodicity) produces. The quasicrystal's aperiodic structure — the same structure that gives quasicrystals their unusual diffraction patterns — gives the compressor its unusual depth.