Phyllotaxis
the spirals are counted, not admired
Series

Truncation — the series

2 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The same rule at p = 1, cut off at two distances. The top 220 nodes of two stems grown by an identical rule whose energy does not converge. Allowed to see 3/√h neighbours it produces 8/13 at 137.62° with 0.58° of scatter — a lattice no test here would question. Allowed 12/√h it produces 44° of scatter and no pattern. The truncation was doing the work.

    A window that makes a pattern

    A rule whose energy has no well-defined minimum produces a clean 8/13 lattice at 137.62°, with half a degree of scatter, when its neighbourhood is cut at three node spacings. Let it see twelve and the pattern is gone. Every simulation of this kind truncates something, and truncation manufactures exactly the result it is used to look for.

    part 5 · emergence
  2. The landscape the rule chooses over, at a cut-off of 3 spacings. One height of an ideal lattice, swept around the circle. The exponential cut-off hands the rule a smooth landscape; the hard one hands it a landscape with steps, because a neighbour enters the sum as the candidate slides past it. Halving the sample resolution multiplies the largest jump between neighbouring points by 1.99 on the smooth curve and by 1.19 on the hard one — which is the definition of the difference, since a smooth function's steepest step is bounded by its derivative and a discontinuity's is not. An argmin taken over steps is pinned to the steps.

    A hard edge is not a falloff

    The prediction was that cutting the neighbourhood at three spacings would reproduce the pattern truncation had manufactured. It does — if the cut is smooth. A hard cut at the same distance produces no pattern at any width, and the reason is that it is the only one of the three whose neighbour set depends on where the candidate is.

    part 6 · emergence

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