Phyllotaxis
the spirals are counted, not admired
Concept

The placement rule — where it appears

6 essays name this object, across one field. What follows is each of them, and the objects they name alongside it.
1/2 at the bottom, 5/8 at the top167 nodes · rise 0.2 → 0.004542 nodes per rung

A pattern with a rate

Every lattice on this site is a static object indexed by a parameter, and a plant is not. Put the rise on a clock, place each node where the repulsion from the ones below it is least, and the object that comes out has a history — which is the first thing here that could disagree with the ladder.

emergence · rising placement noisedegrees off the minimum00.250.50.7511.251.52field noisefraction of the barrier00.00250.0050.00750.010.01250.0150.02kept its branchchanged branchanother pairno latticeseeded 40 nodes of Lucas lattice · 10 runs per amplitude1 escape in 160 runs

Noise is not a slow rate

A stem seeded on the Lucas branch keeps it below ninety nodes per rung and abandons it above — which invites the objection that a real apex's fluctuations would knock it off regardless. Measured across a hundred and sixty runs of two independent kinds of noise, one escapes, at the amplitude where the pattern is already coming apart.

emergence · noise amplitude the rule: compute the energy round the circle, take its minimum, place the nodefield noiseperturbs the energy, before65intact runs kept the branch1changed branchplacement noisedisplaces the node, after44intact runs kept the branch0changed branch — none didthe one that moved: 8/13 at 137.8°, 1.31° of scatter110 intact runs of 1601 of 66 against 0 of 44

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 -8-6-4-20distance from the tip, in node spacingsvariation ÷ nearest shell, log₁₀0–22–44–88–1616–3232–64a golden-angle stem at a rise of 0.004 · shells in units of √hnearest shell dominates by 1.1× at p = 0.5, 9733× at p = 3

How far a primordium reaches

The placement rule's repulsion falls as an inverse cube because that is what two magnetised droplets do, and nothing about a plant supplies the exponent. Asking what it controls produced one tidy wrong answer and one measured right one — and the difference between them is the difference between a total and a variation.

emergence · interaction range 00.50011.50-0.25000.2500.5000.750falloff exponent, log₁₀scatter, log₁₀ degrees00.50011.50-0.25000.2500.5000.750falloff exponent, log₁₀scatter, log₁₀ degreesno latticethe same lattice, whatever pevery runsome runsno runneighbourhood 12/√h · 4 runs per exponenta lattice from p ≈ 1.25 upward

The exponent that barely matters

A code comment on this site claimed since its foundation phase that the repulsion's falloff exponent hardly changes the answer. It could not be tested, because the function that would have taken it never passed one down. Tested at last, it is true on a disc — by three and a half degrees across a sixteenfold range — and on a stem it decides whether there is a pattern at all.

emergence · falloff exponent cut at 3/√h8/13 at 137.62°0.58° of scattercut at 12/√hno divergence angle43.86° of scatterexponent 1 · identical but for the neighbourhood0.58° against 43.9°

A window that makes a pattern

A rule whose energy has no well-defined minimum produces a clean 8/13 lattice at 137.62°, with half a degree of scatter, when its neighbourhood is cut at three node spacings. Let it see twelve and the pattern is gone. Every simulation of this kind truncates something, and truncation manufactures exactly the result it is used to look for.

emergence · truncation

Named alongside it

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

RiseNoiseBranchDivergence angleRepulsionLadderMeristemThe neighbour graphRateCylinderEnsembleMetastability

All concepts