Phyllotaxis
the spirals are counted, not admired
Concept

Primordium — where it appears

A new organ at the moment it is placed, before it grows into a leaf, scale or floret. Phyllotaxis is decided among primordia at the tip, so the object of every rule here is a primordium rather than the organ it becomes.

Named by 8 essays across 3 fields — each of them below, with the objects they name alongside it.

A head of 200 primordia at a divergence of 137.51°. Nothing is placed by hand: the nth point sits at n·137.51° and radius √n. The closest any two points come is 1.60 of the mean spacing.

A head is a set of points

The nth primordium at n times an angle, and a radius of root n. Two lines of arithmetic produce a sunflower head, which is either remarkable or suspicious depending on how carefully the claim is stated — and stating it carefully is most of the work.

lattices · Lattice What the model settles on, against how fast the meristem grows. A broad golden branch, a transition, and then the two-whorl regime at exactly half a turn. 9 of 27 converged settings land within 4° of the golden angle; 15 land more than 20° away.

The bifurcation diagram

Sweep the one parameter of the rule and the settled angle traces a diagram — a broad golden branch, a transition, and a two-whorl regime at exactly half a turn. The famous constant is one branch of it, which is a more useful thing to know than the constant.

emergence · Bifurcation Three disturbances, three places to get in. The rule reads its neighbours, builds a profile of the energy at every azimuth, takes the least of it, and records a position. field noise enters at the profile; jostle noise enters at the neighbours; placement noise enters at the record. Two of the three are upstream of the choice and can change which minimum is taken; the third is downstream and never can.

The noise that arrives through the neighbours

The two kinds of noise this site had were idealisations that bracket the rule's choice. The realistic disturbance is neither: a primordium is placed exactly, and then the organ grows, so by the time the next one forms its neighbours have moved. That is a third kind, and it is invisible in every measurement a plant offers.

mechanism · Noise entry Three disturbances, three places to get in. The rule reads its neighbours, builds a profile of the energy at every azimuth, takes the least of it, and records a position. jostle noise enters at the neighbours; placement noise enters at the record. Two of the three are upstream of the choice and can change which minimum is taken; the third is downstream and never can.

A growing organ is part of the rule

Every model in the earlier essays places primordia on a surface and then treats the surface as furniture. But the surface grows between one placement and the next, and that growth reaches the rule through the only channel it has — where the neighbours are. What looks like a boundary condition turns out to be a term in the model.

mechanism · Mechanism claims Only noise that arrives before the choice can change what is chosen. Intact runs only, from the whole amplitude sweep. Placement noise displaces the node after the rule has picked an azimuth: 44 runs, none of which changed branch at any amplitude that left a lattice. Field noise perturbs the energy profile the rule picks over, so it can move the minimum into a neighbouring gap: 1 of 66 did.

Where the noise gets in

Ninety runs of noise applied after the rule has chosen, and not one changes branch. Fifty-six of noise applied to the choice itself, and one does. Only a disturbance upstream of the decision can restructure which nodes are neighbours of which — which is what a branch is.

emergence · Noise entry Three cut-offs at the same nominal width of 3 spacings. The weight the interaction is multiplied by, against distance. They halve at 2.08 (exponential), 2.50 (gaussian), 3.00 (hard) spacings — so a rule described as "cut off at 3 spacings" is three different rules until the falloff is named. Every later figure is read in half-weight radii for that reason.

A neighbourhood is a hypothesis

Every simulation of this kind stops summing somewhere. The earlier work found that where it stops decides what pattern comes out — so the stopping place is not a detail of the program but a claim about how far a primordium's influence reaches, and it should be written down as one.

emergence · The range of the interaction The landscape the rule chooses over, at a cut-off of 3 spacings. One height of an ideal lattice, swept around the circle. The exponential cut-off hands the rule a smooth landscape; the hard one hands it a landscape with steps, because a neighbour enters the sum as the candidate slides past it. Halving the sample resolution multiplies the largest jump between neighbouring points by 1.99 on the smooth curve and by 1.19 on the hard one — which is the definition of the difference, since a smooth function's steepest step is bounded by its derivative and a discontinuity's is not. An argmin taken over steps is pinned to the steps.

A hard edge is not a falloff

The prediction was that cutting the neighbourhood at three spacings would reproduce the pattern truncation had manufactured. It does — if the cut is smooth. A hard cut at the same distance produces no pattern at any width, and the reason is that it is the only one of the three whose neighbour set depends on where the candidate is.

emergence · Truncation Placements that went to a different minimum, per thousand. The rule's own counterfactual, run beside it: where would this node have gone with the noise taken away and everything else left alone? Placement noise displaces the node after the argmin, so the answer is always "here" — 0.2°, 0.4°, 0.8° all give zero. A jostle and a field perturbation are upstream of the choice and change one or two placements in a thousand while the lattice is still intact.

Which minimum was chosen

The rule takes an argmin, so there are two completely different things noise can do to it: move the answer, or move the question. One of them can change what is chosen and the other cannot, ever — and the difference is exactly zero against one or two placements in a thousand, at amplitudes where every other measurement says the two are identical.

emergence · Noise entry

Named alongside it

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

Divergence angleThe placement ruleMeristemNoiseBasinRateRepulsionRiseArtefactAuxinBranchCut-off

All concepts