Phyllotaxis
the spirals are counted, not admired
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The thread: Two routes, one number

The strongest thing a figure can do is arrive at a quantity twice by arithmetic that shares nothing. A lattice on a cylinder predicts where a disc's spiral counts change; a dispersion relation predicts how many peaks an integration will produce; a branching tree predicts the angles a dynamical model settles on. Where the two routes meet, the number is not an artefact of either.
-0.075-0.050-0.02500.025510152025number of peaks around the ringgrowth rate of that mode8 counted8 predictedDₐ = 0.000008, D_h = 0.00064predicted 8, counted 8 What a plant might be doing

Turing's last problem

Turing's final work was on phyllotaxis and it was unpublished when he died. Its core is a ring of cells and two diffusing substances, and the thing it does is select a number of peaks — which can be predicted from the equations before anything is integrated, and then counted from what the integration produces.

 stem, lower third2 and 3stem, middle third2 and 3stem, upper third2 and 3disc, r = 0.18–0.3221 and 34disc, r = 0.45–0.6234 and 55disc, r = 0.82–0.9955 and 89stem at rise 0.05, disc of 1600 points, both at 137.51°counted from coordinates onlyone answer against three Stems and cones

Counting up the stem

The same counting machinery, pointed at a stem instead of a seed head, returns one answer three times where the head returned three answers. That contrast is a measurement rather than a preference, and it is the one the whole cylindrical argument rests on.

 137.51°, rise 0.098.5e-14°counted 2/3137.51°, rise 0.032.0e-13°counted 3/5137.51°, rise 0.0122.8e-14°counted 5/899.50°, rise 0.083.0e-13°counted 1/3151.14°, rise 0.071.1e-13°counted 2/399.50°, rise 0.021.1e-13°counted 4/7error in the recovered divergence anglecounts and hop lengths onlyworst 3.0e-13° Stems and cones

Two numbers out of the points

A seed head's divergence angle can be recovered from its spiral counts only to within an interval, because a range of angles gives the same counts. On a stem the counts come with lengths attached, two measurements pin two unknowns, and the lattice comes back to the last digit it was built with.

 0.2500.5000.75011.25-2.50-2-1.50-1-0.500log₁₀ of the rise between nodes (falling to the right is the plant growing)log₁₀ of the larger parastichy number2/33/55/88/1313/21500 rises, shortest vectors recomputed at eachratio 0.3820 against 1/φ² = 0.3820 Stems and cones

The Fibonacci ladder

Lower the rise on a cylinder and the parastichy pair climbs — 1 and 2, then 2 and 3, then 3 and 5 — each rung the sum of the two before it. Nothing in the arithmetic mentions Fibonacci, the transitions sit at computable rises, and consecutive ones stand in the ratio 1/φ².

 501001501020304050radius in the disclarger parastichy number, measured on the disc21/3434/5555/8989/144prediction from the cylinder, counts from the disc15 of 16 bands agree Stems and cones

A disc is a cylinder

Vogel's seed head makes the rise fall as one over radius squared, so a disc is not one lattice but a family of them. Feed that into the cylinder's ladder and it predicts where a sunflower's spiral counts change — with nothing fitted, and against a counter that never sees either model.

 Lewis slope — golden head0.009the law says 0.25Lewis slope — random set0.231the law says 0.25Aboav a — golden head1.177the law says 1.2Aboav a — random set0.593the law says 1.2filled where the tiling obeys the law it is being judged by900 points in each tilingLewis explains 32% of the area spread at best Packing and tiling

Two laws that want opposite tissue

Lewis's law and Aboav's relation are quoted side by side as properties of cellular tissue. Measured on the same two tilings they point opposite ways — the ordered head satisfies Aboav's with the textbook value of 1.18 and fails Lewis's completely; the random set does exactly the reverse.

 -2-1.50-1-0.500100120140160180divergence angle (°)log₁₀ of the rise between nodes1,2,32,3,596 × 150 lattices, each solved827 runs drawn Stems and cones

The forks are exact

Where a stem's pattern has to choose between two futures, three spiral families are equally short and the lattice is exactly equilateral. A numerical solver found those points; the numbers it returned turned out to be rational, and chasing that gave a closed form — including the fact that every fork sits at a rational divergence, and the golden angle at none of them.

 100120140-3-2-1log₁₀ of the rise at the forkdivergence angle at the fork (°)137.508° — Fibonacci99.502° — Lucas13 forks, each solved for three equal families137.4730° and 99.5495° Where the angle comes from

The tree and the attractor

The dynamical model settles on the golden angle over a range of one parameter and on the Lucas angle outside it, and it cannot reach the low-growth end at all. The lattice tree reaches everywhere, has no dynamics in it, and produces the same two angles as limits of two paths. Two routes, one pair of numbers.

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