Phyllotaxis
the spirals are counted, not admired
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Noise entry — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Only noise that arrives before the choice can change what is chosen. Intact runs only, from the whole amplitude sweep. Placement noise displaces the node after the rule has picked an azimuth: 44 runs, none of which changed branch at any amplitude that left a lattice. Field noise perturbs the energy profile the rule picks over, so it can move the minimum into a neighbouring gap: 1 of 66 did.

    Where the noise gets in

    Ninety runs of noise applied after the rule has chosen, and not one changes branch. Fifty-six of noise applied to the choice itself, and one does. Only a disturbance upstream of the decision can restructure which nodes are neighbours of which — which is what a branch is.

    part 5 · emergence
  2. Three disturbances, three places to get in. The rule reads its neighbours, builds a profile of the energy at every azimuth, takes the least of it, and records a position. field noise enters at the profile; jostle noise enters at the neighbours; placement noise enters at the record. Two of the three are upstream of the choice and can change which minimum is taken; the third is downstream and never can.

    The noise that arrives through the neighbours

    The two kinds of noise this site had were idealisations that bracket the rule's choice. The realistic disturbance is neither: a primordium is placed exactly, and then the organ grows, so by the time the next one forms its neighbours have moved. That is a third kind, and it is invisible in every measurement a plant offers.

    part 6 · mechanism
  3. Placements that went to a different minimum, per thousand. The rule's own counterfactual, run beside it: where would this node have gone with the noise taken away and everything else left alone? Placement noise displaces the node after the argmin, so the answer is always "here" — 0.2°, 0.4°, 0.8° all give zero. A jostle and a field perturbation are upstream of the choice and change one or two placements in a thousand while the lattice is still intact.

    Which minimum was chosen

    The rule takes an argmin, so there are two completely different things noise can do to it: move the answer, or move the question. One of them can change what is chosen and the other cannot, ever — and the difference is exactly zero against one or two placements in a thousand, at amplitudes where every other measurement says the two are identical.

    part 7 · emergence
  4. A sixfold neighbourhood, and nothing to dilute. The prediction was that a rule with fewer neighbours would convert a jostle into divergence scatter more efficiently. Across a sixfold widening the ratio sits between 0.97 and 1.00, and the one point that differs is the narrowest, at 0.82 — smaller, where the prediction wanted larger. The row underneath is why: past four spacings the rule builds the identical lattice, internode for internode, so there is no neighbourhood left to widen.

    The neighbourhood was already settled

    The earlier work explained a small difference between two kinds of noise by saying a jostle is diluted among some thirty neighbours. Sweep the neighbourhood sixfold and the difference does not move — because past four spacings the rule builds the identical lattice, internode for internode. There was nothing to dilute.

    part 8 · mechanism

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