Phyllotaxis
the spirals are counted, not admired
Series

Sixsides — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. How many sides the cells actually have. The mean is 5.908, which Euler's formula forces. The spread around it is not noise — a tiling of only hexagons cannot close up on a finite patch.

    Why the average cell has six sides

    Not because hexagons are efficient. Because Euler's formula leaves a tiling no choice — count the edges two ways and the mean comes out at six, whatever the cells would prefer. The efficiency argument is a different claim about a different thing.

    part 2 · tissue
  2. The statistic everybody reports is the one that cannot vary. Six arrangements of 900 points, from a whorled lattice to a set with no rule in it. The mean number of sides per cell is 5.97–6.04 on all six, because Euler's formula forces it. The mean squared departure from six runs from 0.023 to 1.83 — a factor of 79 — and the most hexagonal tissue in the set is the whorled one, at a rational angle.

    The second moment is the measurement

    The mean number of sides in a cellular tissue is six, and Euler's formula leaves it no choice — so it takes the same value on a golden-angle head, a whorled head and a set of random points. On heads of nine hundred organs the mean squared departure from six varies by a factor of eighty across the same three, and almost nobody reports it.

    part 3 · tissue
  3. The second moment of five arrangements' side counts against the number of organs on the head. μ₂ on logarithmic axes for heads of 300 to 10,000 organs. Golden: 0.455 at 300 and 0.101 at 10,000; Lucas: 0.362 at 300 and 0.086 at 10,000; 137.5°: 0.453 at 300 and 0.045 at 8,000; whorled: 0.070 at 300 and 0.003 at 8,000; Poisson: 1.727 at 300 and 1.749 at 8,000. The Poisson set is a mean over three seeds. The dashed line is 6.83 over the square root of the organ count, the level the golden head returns to just before each defect ring enters the cut.

    A second moment that goes to zero

    The mean squared departure of a cell's side count from six separates a random tissue from a whorled head by a factor of eighty, on heads of 900 organs. Read at thirty-three head sizes it is exactly the share of cells on the defect rings of a spiral head and falls as one over the square root of the organ count, it falls as one over the count on a whorled head, and the factor is 25 at 300 organs and 677 at 8,000.

    part 4 · tissue
  4. Four head sizes, one staircase. The same window swept at 539, 900, 1409, 3690 organs, each curve divided by its own median so that the overall fall with head size is out of the way and only the shape is left. The features line up. Across the three steps in size, 52 of the 53 features present at a smaller head are still present at the same angle at the next size up — nothing slides. What a bigger head does is resolve features between the ones already there, which is a statement about the instrument rather than about the arrangement.

    Nothing in the staircase moves

    Disorder swept across the divergence angle is a staircase, and every step of it had been read at one head size — which leaves open whether a step is the lattice changing or a ring of defects crossing the rim as the angle moves it. Read again at 539, 900, 1409 and 3690 organs, 52 of the 53 features present at a smaller head are still there at the same angle at the next size up. Not one slides. A bigger head adds steps between the ones already there — 4, 18, 31, 40 — so the staircase belongs to the angle and the head size decides only how much of it is resolved. The one size every other disorder figure here uses turns out to sit three per cent past a ring entry.

    part 5 · tissue

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