Phyllotaxis
the spirals are counted, not admired
Series

Packing — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Closest pair across 120–155° at 400 organs, read both ways. On the interior's scale the golden angle reads 0.9027 and ranks 1st of 72, against 0.9026 for the best grid angle at 137.5°, with the window running from 0.0668 to 0.9026; counting the rim's cells the golden angle reads 0.7076 and ranks 2nd of 72, against 0.7129 for the best grid angle at 137.5°, with the window running from 0.0331 to 0.7129. The dashed upright is the golden angle, which a grid of decimal degrees never lands on and which is therefore read separately.

    Packing, measured four ways

    The claim is that the golden angle packs best, and it is measurable. Read on the interior of a head, the two criteria about distance put the golden angle first among the angles near it and the two about cells are won by rational angles — which makes the claim half right, and makes the right half a statement about a class of angles. An earlier reading of the same four criteria, divided by the cells at the head's edge, said the opposite.

    part 3 · tissue
  2. The cell area a 137.508° golden head's packing is divided by, at 10 sizes. Counting every bounded cell the mean area runs from 3.607 to 38.828, the largest at 150 organs, because the cells just inside the edge of the head reach out to circumcentres far beyond it. Leaving out every cell whose polygon crosses the head's own radius, it stays between 3.1425 and 3.1634 at every size drawn, which is π, the area the square-root rule gives each organ.

    Packing, measured against the interior

    An earlier reading of these heads reported that no packing criterion singles out the golden angle and that three criteria give three winners. Every one of those readings was divided by a mean cell area that, on a head of 150 organs, was 38.8 where the interior's is π. Divided by the interior's own, the criteria about distance put the golden angle first of 72 angles and the criteria about cells go to rational ones.

    part 4 · tissue
  3. A head growing outward beside a uniform head shaken until its cells vary as much. Left: an exact golden head of 500 organs with radius growing as the organ's index to the power 0.6; its cell areas have a coefficient of variation of 0.164, all of it a trend from centre to rim. Right: a head with an exponent of a half, whose cells would all be equal, every organ moved by 0.2 of a spacing; its coefficient of variation is 0.176, none of it a trend. Cells larger than the mean are shaded. A reading of the spread gives the two heads about the same exponent.

    A spread of cell areas is not a growth law

    An earlier essay proposed that a head's spread of cell areas is a measured growth law: a spread of 0.10 means a radial exponent of 0.455 or 0.555. On an exact head that holds to two figures. A uniform head with every organ moved by a tenth of a spacing also has a spread of 0.094, and reads as 0.457 with no gradient in it at all. By a fifth of a spacing it reads 0.42. Fitting cell area against radius instead is untouched on a golden head and collapses on a rational one, and ranking the organs by radius is untouched on both. The spread is law and disorder added in quadrature, which makes it a good reading of disorder once a trend has told it how much law there is.

    part 5 · tissue

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