Phyllotaxis
the spirals are counted, not admired
Where the angle comes from

Where the model stops

Below a growth parameter of about 0.18 this implementation does not converge — the settled angle wanders over a hundred degrees however long the run. That is the range where the literature says the interesting behaviour lives, and it is worth a figure rather than a quietly chosen axis.
5 min read 6 figures Form is not mechanismState the claim, state the test

Every model has a range where it works, and a site that plots only that range is hiding the more interesting half of its own honesty.

This one stops converging below a growth parameter of about 0.18.

Where this implementation stops convergingBelow about G = 0.18 the settled angle wanders over 113° however long the run. That is the model's limit, not a fact about plants.0501000.2500.5000.7501growth parameter Gspread of the last 30 steps (°) — 0 means settledfilled dark: convergedthe usable range is stated, not implied
Fig. 1 The spread of the last thirty steps against the growth parameter. Above about 0.18 it is zero — the model has settled. Below it, the angle wanders over a hundred degrees and does not stop.

What goes wrong

At slow growth the elements barely move outward between steps. After a dozen steps the newest dozen elements are all still within a few percent of the meristem boundary, effectively sitting on the same circle as the place where the next one has to go.

The problem then degenerates. Placing a point on a circle as far as possible from a set of points also on that circle is just “find the biggest gap”, and filling the biggest gap repeatedly is a sequence with no attractor in it — it depends on the entire history, and a model that keeps only a finite window loses exactly the information it needs.

Two things were tried and neither helped, which is how the limit was established rather than assumed.

Longer runs. Three hundred steps at a growth of 0.03 wander as much as eighty do. It is not a transient.

A bigger window. The window is already scaled to the growth rate — the number of elements within a fixed radius ratio is the logarithm of that ratio divided by the growth parameter, so it grows automatically as the parameter shrinks. Fixing the window at a constant was an earlier bug and fixing it improved the middle of the range and not the bottom.

Why the limit sits exactly where it is inconvenient

The regime this implementation cannot reach is the one the botanical literature cares about most.

High-order Fibonacci phyllotaxis — the 55-and-89 heads that get photographed — corresponds to slow growth relative to primordium size, which is the small-parameter end. Douady and Couder’s own treatment works there, using a continuous formulation and a much more careful treatment of the interaction, and it is where their Fibonacci ladder appears.

So the honest summary of what is here is narrow: this implementation demonstrates that the golden angle is an attractor of a repulsion rule over a broad range of growth rates, and that other regimes exist at higher rates. It does not reproduce the climb up the Fibonacci sequence at low rates, and a reader wanting that should go to the original papers rather than to this site.

Why it is drawn rather than mentioned

Because a range chosen to make a model look good is indistinguishable, from outside, from a range chosen because it is the right one.

Every figure on this site that sweeps the growth parameter starts at 0.2, and a reader has no way to know whether that is a considered choice or a convenient one. Plotting the convergence failure directly answers that: the axis starts there because below there the model does not answer, and the plot shows exactly how badly.

That is a small piece of work with a disproportionate effect on how much the rest of the figures can be trusted. A site that shows where its machinery breaks has said something about the parts where it does not.

The general form

The distinction to keep is between a limit of the model and a fact about the subject.

That the settled angle wanders below G = 0.18 is a fact about a sampled circle with an inverse-cube repulsion and a truncated window. It is not a fact about meristems, and if it were reported without that framing it would read as one — “slow-growing plants have no consistent divergence angle” is a sentence this data does not support and could easily be mistaken for.

The same distinction applies to the packing measurements, where several criteria disagree at finite head size, and it would be equally wrong to conclude that the golden angle is unremarkable. What both cases show is where the measurement runs out, which is a different thing from what the world does.

What the model settles on, against how fast the meristem growsA broad golden branch, a transition, and then the two-whorl regime at exactly half a turn. 8 of 27 converged settings land within 4° of the golden angle; 15 land more than 20° away.1001251501750.50011.50growth parameter Gangle the model settles on (°)goldentwo-whorlfilled: converged · hollow: still wanderingone rule, one knob
Fig. 2 The diagram the constant is one branch of. Sweeping the growth parameter gives a golden branch, a transition and a two-whorl regime at half a turn.
Two runs of the same rule from unrelated starting anglesBoth settle at 137.0°, within 0.5° of the golden angle, from seeds 166° apart.1201301401500255075100stepdivergence angle produced at that step (°)golden anglegrowth 0.40settled spread 0.00°
Fig. 3 Two runs of the same rule from unrelated starting angles, both settling on the same value.
The rule, 26 steps in, at a growth of 0.40The next primordium goes where the repulsion is least — the marked minimum at 216°. Nothing in the rule refers to any particular angle.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
Fig. 4 The rule itself, part way through a run. The next element goes to the minimum of the repulsion curve, and nothing in the rule names an angle.
One packing criterion across the angles, with the others' winners markedThe three criteria pick 137.5°, 138.0° and 135.0°. The golden angle is near the top of all three and the exact winner of none at this size.00.2500.5000.7501120130140150divergence angle (°)closest pair, as a fraction of the mean spacing (higher is better)137.508°400 points per anglethree criteria, three winners
Fig. 5 Packing quality against the divergence angle. Several reasonable criteria give several different winners.
Three settings of the one knob, and the heads they produceG=0.3 → 139.2° · G=0.62 → 143.0° · G=1.1 → 180.0°. The model was not told any of these angles.G = 0.30139.2° — goldenG = 0.62143.0° — otherG = 1.10180.0° — whorled (half)one rule, three growth ratesthe angle is an output
Fig. 6 Three settings of the one knob and the heads they produce. The model was told none of these angles.

Where this goes next

1001251501750.50011.50growth parameter Gangle the model settles on (°)goldentwo-whorlfilled: converged · hollow: still wanderingone rule, one knob 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. 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. 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.