Phyllotaxis
the spirals are counted, not admired
Concept

Attractor — where it appears

6 essays name this object, across 2 fields. What follows is each of them, and the objects they name alongside it.
golden 137.51°55 and 89FibonacciLucas 99.50°47 and 76Lucas151.14°50 and 81neither77.96°37 and 60neithercounted from the pointsone sequence per branch

Fibonacci is a branch, not a law

Fibonacci counts come from one branch of the model. The Lucas branch — which the same rule reaches at a different growth rate — gives 47 and 76, and neither is a Fibonacci number. The sequence is a consequence of an angle rather than a property of plants.

wrong · fibbranch 05e+51e+62e+60100200300angle around the boundary (°)repulsion from what is theregrowth 0.40 · 14 elements in playthe minimum is where the next one goes

The angle is an output

137.5° is not a constant of nature. It is where a rule settles — a rule that places each new element as far as it can from the ones already there, contains no reference to the golden ratio, and reaches the same answer from starting angles a hundred and sixty degrees apart.

emergence · attractor 1201301401500255075100stepdivergence angle produced at that step (°)golden anglegrowth 0.40settled spread 0.00°

Droplets with no biology in them

Douady and Couder dripped magnetised ferrofluid into a dish of silicone oil, and got spiral phyllotaxis with Fibonacci parastichy numbers out of a system containing no cells, no genes and no plant. That is the strongest evidence the pattern is physics — and the clearest warning about what a model can claim.

emergence · analogue 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°

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.

emergence · branch tree -2-1.50-1-0.500100120140160180divergence angle (°)log₁₀ of the rise between nodes1,2,32,3,522 × 150 lattices, each solved196 runs drawn

The angle is not the object

Every popular account of phyllotaxis is organised around a number. After a phase spent on stems, forks and frequencies, the number looks like the wrong thing to organise an account around — it is the limit of one path through a branching structure, it is at no fork, and a plant that has it got there by not jumping.

wrong · the tree not the angle grown from a coarse start100.0%divergence chosen at random10.8%share ending on a consecutive Fibonacci pair16 grown runs, starting divergences from 67° to 299°every one of them ended on 8/1367 nodes per rung · rise 0.4 → 4.8e-3100% against 10.8%

Continuity from a coarse start

At a fine rise, one divergence in seven gives a Fibonacci pair. Grow a stem from a coarse start at a divergence nobody chose, down to that same rise, and sixteen runs out of sixteen end on 8/13. The expansion phase's interpretation was that continuity does the work; this is the measurement it never had.

emergence · continuity

Named alongside it

The objects these essays reach for when they reach for this one.

BranchFibonacciRepulsionBifurcationDivergence angleMeristemRiseContinuityConvergenceφ, the golden ratioLadderAuxin

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