Phyllotaxis
the spirals are counted, not admired
Series

Hurwitz — the series

2 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The version of the claim that does survive measurement. The golden angle scores 0.4377, against 0.3462 for the best of 938 other angles sampled. The dashed line is Hurwitz's 1/√5, which no number can exceed.

    The claim that survives

    Of the three famous assertions about this subject, one is out by a factor of two, one is true of a branch rather than of plants, and one is right — in a sharper form than the version usually told, and about arithmetic rather than about packing.

    part 4 · wrong
  2. Every angle whose counts from 34 to 144 resist approximation within a per cent of the golden angle's. Over the counts a head shows from 34 to 144, an angle scores like the golden angle when those counts add up, each the sum of the two before, from a first pair near the golden ratio. 46 angles between 20° and 180° come within one per cent of its score of 0.44718, each drawn as a stem at its angle. The five nearest are 137.51° with counts 34, 55, 89, 144 at 100.000 per cent; 99.50° with counts 47, 76, 123 at 99.989 per cent; 106.45° with counts 44, 71, 115 at 99.931 per cent; 151.14° with counts 50, 81, 131 at 99.919 per cent; 132.18° with counts 49, 79, 128 at 99.907 per cent. The golden angle is the highest, and the Lucas angle at 99.50° is a ten-thousandth of the score behind it.

    What a head can mean by most irrational

    Hurwitz's bound, the one famous claim about this subject that survives, is a limit over every denominator, and a head shows only the counts between its innermost spirals and its rim. Over those counts an angle resists approximation like the golden angle exactly when the counts it shows add up, each the sum of the two before, from a pair near the golden ratio — and every such pair has an angle of its own. The golden angle still scores highest over every window measured, by a ten-thousandth: over counts from 34 to 144 the Lucas angle is 99.989 per cent of it and forty-six angles are within one per cent. What separates the golden angle from them is below the counts they share, at the centre of the head.

    part 5 · wrong

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