Phyllotaxis
the spirals are counted, not admired
Series

Gapgrowth — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. How the largest gap behaves as the head fills, on the interior's scale, from 150 to 2000 organs. On the interior's scale, the rational angle's gap grows by a factor of 3.8 over this range, from 2.48 to 9.38 spacings; the golden angle's runs from 0.841 to 0.844, decided at radius 0.871 at every size, and 137.3° reaches 0.863. A rational angle's gap is unbounded and an irrational one's is not, which is a claim about growth rather than about a value at any one head.

    The gap that grows

    A rational divergence angle develops an empty wedge that grows without bound as the head fills — a factor of 2.9 between two hundred primordia and sixteen hundred, and 3.8 from a hundred and fifty to two thousand. An irrational one does not. That is the division between rational and irrational angles that survives measurement.

    part 3 · tissue
  2. The largest empty circle of a 2400-organ 137.508° golden head, ring by ring. Each dot is the largest empty circle in one ring of the head, on the interior's scale. The whole head's is 0.8437 at radius 0.871, at the centre; the rings peak at the radii where the lattice flips from one pair of neighbours to the next, and at the 4 flips beyond radius six the head reads within 0.0020 of the closed form. The dashed line is 1/√2 = 0.7071.

    One over root two

    On the interior's scale a golden head's largest empty circle is 0.8435 of a spacing at every size from 150 organs to 2,000, because one triangle at its centre decides it. Everywhere else it is 1/√2 — a square cell at every ring where the lattice flips — and a closed form in the angle's continued fraction says that only noble angles hold it there.

    part 4 · tissue
  3. The holes of a 2400-organ head built at 137.3 degrees, and the lattice read from them with no angle given. Each dot is the largest empty circle in one ring a quarter of a unit wide, read from the positions of the organs alone. The line is the lattice of the one divergence that best makes those rings: 137.3003 degrees for a head built at 137.3, with quotients 2, 1, 1, 1, 1, 1, 4, 1 against the true 2, 1, 1, 1, 1, 1, 4, 1. Over 167 rings the fit misses by 0.0047 of a spacing, where the true angle's own lattice misses by 0.0048.

    A head's holes spell its angle

    A head's largest empty circles follow a closed form in its angle's continued fraction, and a closed form runs both ways. Given only the organs of a 2,400-organ head, the largest hole in each ring and no angle, a reading outward through the rings returns 28 of forty undisclosed angles to a thousandth of a degree, quotient by quotient, and a larger head tightens them a hundredfold. The other nine it finds are other angles, and they are not random: their expansions end as the truth's ends and start differently, because the start of a continued fraction lives at the centre of a head. And most of it is gone once each organ has moved by a hundredth of a spacing.

    part 5 · tissue

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