Phyllotaxis
the spirals are counted, not admired
Series

Lewis — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Cell area against side count, at 0% disorder. The dashed line is Lewis's law, (n−2)/4. The fitted slope here is 0.014 against his 0.25, and the side count accounts for 20% of the variation in area.

    Lewis's law wants disorder

    Cell area rises linearly with side count — measured on cucumber epidermis in 1928 and quoted ever since as a property of packed tissue. It holds beautifully on a random point set, with a fitted constant of 1.64 against Lewis's 2. On a phyllotactic head it does not hold at all: the slope is 0.009, and area and side count are almost independent.

    part 3 · tissue
  2. Cell area against side count on a 900-organ head, every organ displaced by 0.2 of a spacing. The joint distribution of cell area, as a multiple of the mean, and side count, over the 639 interior cells of a golden head with every organ displaced by 0.2 of a wall spacing, seed one. The dashed line is Lewis's law, a quarter of the mean area for each side; the solid line is the fit, at a slope of 0.113, and the side count explains 30 per cent of the variation in area. Classes: 4 sides, 16 cells, mean 0.75; 5 sides, 159 cells, mean 0.89; 6 sides, 295 cells, mean 1.00; 7 sides, 146 cells, mean 1.11; 8 sides, 22 cells, mean 1.22; 9 sides, 1 cells, mean 1.26.

    Lewis's law needs the sides to vary

    Lewis's law holds on a random set of points and fails on a golden-angle head. Walked from one to the other by displacing every organ independently, the head's Lewis slope reaches half a random set's at a fifth of a wall spacing and nine tenths by seven tenths, and in between it explains up to 41 per cent of the variation in cell area — more than the 31 per cent it explains in the random set. Moved instead by a smooth field correlated over eight spacings, the head's cell areas become nearly as varied as a random set's and its slope stays at nought, because its side counts stay the lattice's. The law is not about how varied the cells are. It is about how varied their sides are.

    part 4 · tissue
  3. A random tissue of 700 cells whose size grows outward as a Vogel head of exponent 0.8 does. Points placed at random with the density of a head of exponent 0.8. Cells above the median area are shaded, and they lie outside: 537 cells whose sizes differ far more by where they are than by how many sides they have. A cell's side count does not follow its radius at any setting.

    A gradient keeps Lewis's slope and takes its share

    Lewis's law is reported as two numbers, a slope and how much of the variation in cell area it explains, and they get read as one. Grow a tissue's cell size steadily across the field of view and the two part completely. Over a gradient that takes cells from one size to several times that, the slope of a random tissue stays at 0.22 and its explained share falls from 33 per cent to 15. Both are predicted by a line of arithmetic from the flat tissue's own numbers, to within a percentage point or two. Dividing the field into twelve bands of radius restores nearly all of it. So a weak Lewis correlation in a published field may be a statement about the field.

    part 5 · tissue

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