Phyllotaxis
the spirals are counted, not admired
Ladder

Sweep resolution — the ladder

3 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. 19 changes and 2 at the coarse step, 21 and 2 at half of it. The changes of surviving family each band makes, at the step the sweep has always used and at half of it. The golden 8/13 band goes 19 to 21 and the golden 5/8 band 2 to 2. Halving the step cannot lose a change, since the fine grid holds every coarse rise with the same answer; what it can do is find one, and it does so on one band and not on the other. So whether a tally is a floor or a total is a property of the band rather than of the sweep — a floor where there are islands narrower than the step, and exact where there are none.

    A count or a floor

    Nineteen changes of surviving family on the widest band have been quoted as a number since the band was cut, with nothing to say whether a finer grid would find more of them. Halving the step finds twenty-one there and nothing at all on the next band along.

    rung 1 · cylinder
  2. The rise at 0.005845 the coarse grid steps over, under offset 6 at both steps. Offset 6's cut drawn at every rise between 0.00596 and 0.00573, coarse above and fine below, with a cell per rise coloured by the family the cut leaves standing and pale where the cut recovers. It changes its answer 3 times at the coarse step and 5 times at half the step, and the gain is the single rise 0.005845, where the 4 family is kept with 8 on both sides. That island is 0.86 parts per thousand of the rise wide, against 1.47 for the smallest step the coarse grid takes anywhere on the band, so no coarse rise could have landed on it.

    New islands or old edges

    Halving a band sweep's step found two more changes of surviving family, and there are two quite different things they could have been. Every coarse change and every coarse island turns out to be carried by exactly one fine one, so the extra pair is a rise the coarse grid stepped over rather than a boundary it misplaced.

    rung 2 · cylinder
  3. The 125 steps of the golden 8/13 band, in units of the grid they are rounded to. The sweep's rises are held to five decimal places, so every step it takes is a whole number of units of that fifth place. Each bar is how many of this band's 125 steps are that many units: 99 of 1, 26 of 2. One unit is 1.653 parts per thousand of the rise at the handover of 0.00605, against a nominal step of 2.00, so a sweep asked for at one part in a thousand would land on the same rise twice and the finer grid has to be laid at 7 places instead.

    The last unmoved setting

    Halving a band sweep's step is only a halving if the sweep steps where it says it does, and this one does not: its rises are rounded to five decimal places, so at one handover the grain is 1.65 parts per thousand against a nominal step of two. Moving the last setting nobody had moved found the setting was never what it was called.

    rung 3 · cylinder

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