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

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.
6 min read 6 figures The angle is an outputOne parameter, one family

The rule has one parameter. It is the ratio of how fast the elements drift outward to how often a new one appears — how much room the meristem makes between primordia — and everything else about the model is fixed.

Sweeping it produces the diagram this site is arranged around.

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. 1 The angle the model settles on, against the growth parameter. Filled points converged; hollow ones were still wandering when the run ended. There is a broad golden branch, a transition, and then half a turn exactly.

What is on it

The golden branch. Over a wide range of growth rates the model settles within a few degrees of 137.5°. This is the regime that produces sunflowers, pine cones, pineapples and most of what gets photographed, and its breadth is the reason the golden angle is common rather than rare.

The transition. Above the golden branch the settled angle climbs, passing through values that belong to no named sequence.

The two-whorl regime. At high growth rates the model settles on exactly 180°, and stays there. Each new element goes directly opposite the last. That is distichous phyllotaxis — the alternating leaves of a grass, a maple, an elm — and it is not a degenerate case or a failure of the model. It is the commonest leaf arrangement there is.

So the diagram contains, in one sweep of one parameter, both the pattern that gets called mathematically remarkable and the ordinary alternating arrangement of a garden shrub. They are the same rule at different growth rates.

Why a diagram beats a constant

The popular telling gives a number and an air of mystery. The diagram gives a mechanism and a prediction.

It explains why the golden angle is common. Its branch is broad. A plant does not need to hit a precise growth rate to land on it; a wide range of rates gives the same answer, which is exactly what a robust developmental outcome should look like.

It explains why other angles occur. Lucas-number phyllotaxis is a real and repeatedly observed minority pattern, and a model that only produced the golden angle could not account for it. A model with branches can.

It predicts what changes the pattern. Anything that alters the ratio of primordium spacing to meristem size should move a plant along the diagram — and it does. Changing the growth conditions of a real plant can shift its phyllotaxis, and the transitions in a single plant’s life, as the meristem changes size, are a walk along this axis.

It says what to measure. A constant says what to look for; a diagram says what to vary.

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. 2 Three settings of the one knob and the heads they produce. The model was not told any of these angles — each is what the rule settled on.

Hysteresis, and the path taken

One feature of the real model that the sweep here does not show, and which is worth naming because it matters biologically.

In Douady and Couder’s full treatment the system shows hysteresis: which branch it lands on can depend on the path taken to get there, not just on the current parameter value. A meristem that grows gradually from a high growth rate down through the transition stays on one branch; one that starts in the middle may land on another.

That is the mechanism behind the observed sequences in real plants, where a shoot’s phyllotaxis rises through the Fibonacci sequence as it develops rather than jumping to a final value. The pattern is not chosen once; it is inherited from the pattern that preceded it.

This site’s sweep restarts the model from scratch at every parameter value, so it shows the branches without the history. That is a limitation and it is worth stating: the diagram here is the set of destinations, not the map of which one a given plant reaches.

The Lucas branch

The literature reports a Lucas-number branch — parastichy pairs of 4 and 7, 11 and 18, 29 and 47 — occurring in a small but consistent minority of real plants, and corresponding to a divergence angle near 99.5°.

The sweep here does not resolve it cleanly, and that is an honest limitation of a coarse implementation rather than evidence against it. A Lucas branch is narrower than the golden one, so a sweep at this resolution can step over it, and the discrete boundary sampling blurs exactly the distinctions that separate nearby branches.

What the site can show is that the Lucas angle produces Lucas counts — that a head built at 99.5° gives 47 and 76, neither of which is a Fibonacci number. That is enough to establish the point the diagram is making, which is that Fibonacci is a consequence of one branch and not a law of plants.

Where the diagram stops

The sweep starts at a growth parameter of 0.18 and that is not an aesthetic choice.

Below it this implementation stops converging: the elements barely move between steps, several sit on the meristem boundary at once, and the settled angle wanders over a hundred degrees however long the run. The low-growth regime is precisely where the literature says the interesting high-order Fibonacci behaviour lives, and this model cannot reach it.

That is a real limit and it gets a figure of its own rather than a quietly chosen axis range.

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.
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. 5 Where this implementation stops converging, drawn rather than avoided by a quietly chosen axis range.
The spiral counts four different divergence angles produceFibonacci counts come from one angle. The Lucas angle — which the same dynamical model reaches on a different branch — gives 47 and 76, and neither number is a Fibonacci number.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
Fig. 6 What the spiral counts come out as at four different divergence angles. Only one of them gives Fibonacci numbers.

Where this goes next

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. 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. 0501000.2500.5000.7501growth parameter Gspread of the last 30 steps (°) — 0 means settledfilled dark: convergedthe usable range is stated, not implied 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.