Phyllotaxis
the spirals are counted, not admired
Series

Nautilus — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A golden spiral and a nautilus spiral over 2.5 turns, from the same start. After 2.5 turns the golden curve is 7× larger. The growth factors are 6.85 and 3.2, a factor of 2.14 apart.

    The nautilus question

    A golden spiral grows by 6.854 per turn. Measured nautilus sections give about 3.2. That is a factor of 2.14 — not a rounding error, not an artefact of where the centre is assumed to be, and not close.

    part 3 · shells
  2. A shell expanding by 3.2 a turn, divided by 13 septa to a whorl. A shell expanding by 3.2 a turn at axis distance 0.1, seen down its coiling axis, with 13 septa to a whorl, 27.7° apart; the last whorl's chambers are shaded. Each chamber is the one before it turned and scaled about the apex by 3.2^(1/13), so a length grows by ×1.0936 from one chamber to the next, an area by ×1.1960 and a volume by ×1.307896. Integrated over the tube's own rings, successive chamber volumes grow by 1.307896 to 1.307896, and a chamber and the one a whorl out differ by 32.7680, which is 3.2³ = 32.7680.

    What the septa count

    A nautilus's chambers are each a scaled copy of the last, and an earlier essay gave their ratio as about 1.3 — what a growth factor of 3.2 gives over a third of a turn. It does not: a third of a turn at 3.2 is 1.474 in length. A ratio of 1.3 is 4.43 septa a whorl as a length, 8.87 as an area and 13.30 as a volume, so the dimension decides the count threefold. And the count is an exponent in any reading of the growth factor taken from one chamber to the next: one septum miscounted at thirteen moves it by 9.14 per cent. A chamber and the one a whorl out give W³ with no count at all.

    part 4 · shells
  3. Every way a growth factor read off a section can be wrong, against the gap it has to clear. Each bar is a worst case computed by the library that measured it. dividers 278.2%, span 22.4%, septum 9.14%, ontogeny 8.99%, centre 6.88%, clock 4.06%, oblique 2.95%. The line is the gap the golden claim asks the measurement to resolve: 6.854 against 3.2 is 114.2 per cent. Added without cancellation the 7 sources come to 332.6%, which is outside that gap — so the claim is NOT refused by a section read carelessly.

    The error budget for a nautilus

    Every way a growth factor read off a shell section can be wrong has been priced here, one essay at a time. Added up they come to 332.6 per cent in the worst case and 279.6 in quadrature, against a golden-spiral claim that is 114.2 per cent away — so the budget does not refuse the claim at all. One entry decides it: the dividers, at 278.2 per cent on their own, and the dividers are the historical method and the only route measured that pushes a nautilus towards a golden spiral. Set them aside and the budget falls to 54.4 per cent and the claim is refused twice over. What the same budget cannot settle is anything smaller than half: 3.2 against 3.4 is inside it, and stays inside it until six of the seven sources are controlled.

    part 5 · shells

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