Crystallography & Symmetry
patterns derived from their groups
Series

Limit periodic — the series

2 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. An aperiodic chain made of periodic ones. The first 40 letters of the period-doubling chain, a → ab, b → aa, with b dark, and below it the same sites sorted into layers by the power of two in n + 1. Layer j repeats with period 2ʲ⁺¹ and carries a when j is even and b when j is odd. Each layer is perfectly periodic; the chain is their union, and since no single period contains every layer, the chain has no period at all.

    An aperiodic chain whose peaks are all at rational positions

    The period-doubling chain, a → ab and b → aa, never repeats, and it diffracts in sharp peaks. Where those peaks are is the surprise. Every one sits at a fraction m/2ʳ, every peak with the same denominator has the same height, (2/3)·2⁻ʳ, and together they carry all the intensity there is. The chain is a stack of periodic chains of period 2, 4, 8 and so on, and its diffraction is theirs. Unlike the Fibonacci chain, it is the slice of no lattice in any number of dimensions.

    part 1 · aperiodic
  2. The period-doubling chain with its deep layers randomised two ways. The first 48 letters of the period-doubling chain (b dark), and below it the same chain with every site at depth 2 or more redrawn at random: in the middle row each deep site independently, with b at the deep layers' own density, and in the bottom row one letter per deep layer, the same for every site of that layer. The deep sites, every 4th, are outlined; a dot marks a letter that differs from the perfect chain. The shallow layers are untouched in both. The middle row is no longer built of periodic layers; the bottom row still is.

    A fault in the deep layers costs exactly the fine peaks

    Redraw the period-doubling chain's deep layers at random, site by site, and every peak finer than the fault's depth vanishes while every coarser one stays exactly where and as high as it was — provided the random letters keep the layers' own density, which is a third below an even layer and two thirds below an odd one. The lost intensity, (2/9)·2⁻ʳ, comes back as a flat floor to the last digit. Redraw them once per layer instead and nothing is lost at all: every peak survives, with a height drawn by lottery.

    part 2 · aperiodic

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