Phyllotaxis
the spirals are counted, not admired
Series

Recovery — the series

2 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A round trip on four heads of 400 primordia: the divergence angle recovered from each. The counter is shown the points and nothing else. The worst recovery across the four is 0.035°.

    Recovering the angle from the counts

    Build a head at a stated divergence angle, forget the angle, and get it back from the spiral counts alone. Four angles, worst error twelve thousandths of a degree — and the only thing that crossed between the two halves was a list of coordinates.

    part 3 · lattices
  2. What the round trip returns as the organs are displaced: golden angle, 300 organs. Thirty heads at each size of displacement, from none to two and a half spacings, each run through the round trip and sorted by what came back. The undisplaced head counts 21 and 34. At 0.25 spacings, 97% the undisplaced pair, 3% a neighbouring pair; at 0.5 spacings, 50% the undisplaced pair, 50% a neighbouring pair; at 0.75 spacings, 30% the undisplaced pair, 70% a neighbouring pair; at 1 spacing, 13% the undisplaced pair, 80% a neighbouring pair, 7% a straddling pair; at 1.25 spacings, 7% the undisplaced pair, 57% a neighbouring pair, 7% a straddling pair, 30% refused; at 1.5 spacings, 40% a neighbouring pair, 3% a straddling pair, 57% refused; at 2 spacings, 10% a neighbouring pair, 90% refused; at 2.5 spacings, 3% a neighbouring pair, 97% refused. 10 recovered intervals in all exclude the true angle.

    A head displaced before it is counted

    The round trip from a head's spiral counts back to its divergence angle was tested on heads whose every organ sat exactly where the rule put it. Displaced by a normal error of up to two and a half spacings, heads of 900 organs keep counting a pair from their own sequence and return intervals holding the true angle to a spacing and a half; heads of 300 organs move to the neighbouring pair by half a spacing and then refuse, nine in ten of them by two spacings. Every moved count brings in the family whose chord was third shortest. Of 898 heads recovered, 19 intervals miss the true angle and 17 of those by about a tenth of a degree — displacement makes the reading coarser and then silent, not confidently wrong.

    part 4 · lattices

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