Phyllotaxis
the spirals are counted, not admired
Concept

Local minimum — where it appears

A point where a quantity is smaller than everywhere nearby, without being the smallest overall. A placement rule chooses the least of several, so which minimum wins can change by half a turn under a disturbance that barely moves the profile.

Named by 6 essays across 4 fields — each of them below, with the objects they name alongside it.

A stem unrolled: 28 nodes at 137.51° with a rise of 0.090 circumferences. The counter is shown these coordinates and the circumference, and finds 2 parastichies one way and 3 the other. The faint strips left and right are the same stem: a family leaving one edge re-enters at the other.

A stem is a cylinder

The sunflower is the photograph, and it is the hard case. Nearly all real phyllotaxis happens on a stem, where the geometry is a lattice on a cylinder with two parameters — and where the spiral counts, which on a disc change with radius, are the same the whole way up.

cylinder · Stem lattice The two spiral families a counter finds between 0.68 and 0.92 of the radius. 34 spirals one way and 55 the other, found from the point positions alone — the counter is never told the divergence angle.

Counting the spirals

Almost every claim about phyllotaxis is a claim about how many spirals run through a pattern, and the count is almost never done. It can be done from the points alone, by a count that is never told what angle built them — and then a count of 34 is evidence rather than a restatement.

lattices · Counting The same counter, on a stem and on a disc. The stem returns 2 and 3 in all three bands. The disc returns 21/34, 34/55, 55/89 — three answers to one question, which is why a published count needs to say where it was taken.

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.

cylinder · Stem count What insisting on a fork angle costs: 43.3 degrees for one per cent of the network. The three ends are held where the optimum wants them, the branch point is moved everywhere inside them, and the cheapest network at each total angle is kept. The optimum sits at 74.93°. Everything within one per cent of the least cost runs 55.9° to 99.2° — a span of 43.3°, which read back as exponents covers 2.44 to 5.34 — and within a tenth of a per cent it still runs 13.6°. The prediction is steep in the exponent and the cost is nearly flat in the angle; they are the same curve read along its two axes.

An optimum too flat to reach

One per cent of a branching network's cost buys forty-three degrees of fork angle, covering exponents from 2.44 to 5.34, while the angle the theory predicts moves only fourteen and a half degrees across every daughter ratio there is. The prediction is steep and the cost is flat, and those are the same curve read along its two axes.

branching · Fork angle 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 angleRiseDiscriminationParastichyCylinderFibonacciLadderLattice offsetNoiseThe placement rulePrimordiumTolerance

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