Phyllotaxis
the spirals are counted, not admired
Series

Falloff exponent — the series

6 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Ten of the 24 orderings four exponents admit, from 15 readings of the wall. Every arrangement of the four falloff exponents is a cell, and a cell is filled when some level of some sampling puts the four walls in that order. 15 readings of runs that are shared cell for cell give ten of the 24, with the most common occurring three times. The marked cells are the orderings read at a half, the only level any round of this collection has published, and each of them occurs once. Exponent 5 is ranked first in 8 of the 15 readings and each of the others in two or three.

    The level was doing the ordering

    Four falloff exponents have been put in order by the rise at which half their runs stop settling, and a half is the only level that order has ever been read at. Read at nine levels the same runs give ten different orderings of the same four numbers, and the one comparison in the whole study that resolves runs the other way.

    part 3 · mechanism
  2. A rule too long-ranged makes no pattern; every shorter one makes the same pattern. Each dot is 4 runs from a coarse start at one exponent, separated by 0.2° of placement noise. Below p ≈ 1.1 the divergence scatters by tens of degrees, which is what an arbitrary sequence gives. From p = 1.25 to p = 8 — a sixfold range — every run ends on 8/13 with the scatter between 0.75° and 1.06°.

    The exponent that barely matters

    An unchecked claim, repeated since the first essays, held that the repulsion's falloff exponent hardly changes the answer. It could not be tested, because the function that would have taken it never passed one down. Tested at last, it is true on a disc — by three and a half degrees across a sixteenfold range — and on a stem it decides whether there is a pattern at all.

    part 4 · emergence
  3. Every bracket on the wall at a half, before the refinement and after. One pair of bars per falloff exponent: above, the rises consistent with that exponent's crossing on forty starting angles and the 5 published rises; below, the same on eighty angles and 9 rises. Two of the four have an open end before — at a half that is the coarse end, where no published rise has a share confidently above the level — and none is open after, the widest closing at 1.57 times. The four walls sit inside a factor of 1.111 of one another, against 1.624 on the reading that could not locate them, so they came closer together rather than further apart.

    Four walls closer than they looked

    Two of the four falloff exponents had a wall with no upper end at all, and the other two were located to factors of two and a half and nearly four. Nine rises at eighty starting angles close every bracket — and the four walls turn out to sit inside a factor of 1.111 of one another, which is narrower than the narrowest bracket.

    part 4 · mechanism
  4. Two shapes, two ranges, one contrast. The exponential's lattice ends at 3.63 spacings and the gaussian's at 2.25 — ranges 47% apart — and at those two ranges the contrast is 5.74 and 6.09, 6% apart. The band is what the exponent route leaves: 3.98 at p = 1, where the uncut rule makes nothing, and 7.63 at p = 1.25, where it makes a lattice.

    Two shapes, one threshold

    Read in the same unit, an exponential falloff and a gaussian one disagree about where the lattice ends by half. The quantity they agree on turns out to be one the earlier work measured for an unrelated reason — and it agrees with a bracket left by a sweep of a completely different parameter.

    part 5 · emergence
  5. The bracket at each of the four corners of the refinement, at a half. The geometric mean bracket at each corner, and under it how much of the joint narrowing each half of the refinement accounts for. It is taken over the two exponents whose bracket is closed at every corner, with the two halves of the refinement crossed against each other. The rule across the bars is 1.54, the narrowest bracket the published rise list can express at all. On forty starting angles and the published rises it is 3.06 times with two of the four brackets closed; doubling the angles alone gives 1.96, halving the rise spacing alone gives 1.46, and both together give 1.44. In the logarithm, where the two effects add, the rises account for 98.8 per cent of the joint narrowing and the angles for 59.3 per cent — summing to 158.1 per cent, which is what overlapping causes look like and not two independent factors of two.

    Two refinements that do not multiply

    The design that located the wall did two things at once — doubled the starting angles and halved the rise spacing — and the arithmetic behind it assumed each would buy about a factor of two. The finer rises did ninety-nine per cent of the narrowing and the doubled angles added under one, because a bracket's ends are rises and no error bar can move them.

    part 5 · mechanism
  6. The settling share against the rise at exponent 5, on the refined rises. One line, over 80 starting angles at exponent 5. The rise runs coarse on the left and the level the wall is read at is the dashed rule. Between 0.02 and 0.013 the refined list holds two rises the published one does not, and at exponent 5 the share reaches 0.725 at 0.01732 against 0.550 at the ends of that stretch — 3.13 standard errors above the higher end. It turns in all three readings independently, which is what separates a feature of the curve from a bump in a sample. The ringed point is that maximum.

    A maximum in the gap

    Four falloff exponents have refused to separate on every quantity this thread has read off them, and the wall that was supposed to tell them apart cannot. Two of the four carry a maximum in the settling share at a rise the published list stepped straight over, and it is there in both halves of the sampling independently.

    part 6 · mechanism

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