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

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.
6 min read 5 figures The angle is an outputForm is not mechanism

The golden angle is usually introduced as a number that plants use, which raises an obvious question that popular accounts step around: how would a plant know it?

The answer is that it does not have to. The angle is not an input to anything. It is where a very simple rule ends up.

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. 1 The rule, part way through a run. Elements drift outward as the meristem grows; the next one appears where the repulsion from everything already there is least. The curve on the right is that repulsion around the boundary, and the marked minimum is where the next element goes.

The rule

Douady and Couder wrote it down in 1992 and it takes one sentence.

New elements appear on the boundary of a growing meristem. Each one, once formed, drifts outward as the meristem grows. The next one appears at the angle where the repulsion from all the existing elements is least.

That is the whole model. It has one parameter — how fast the elements drift outward relative to how often they appear — and it contains no reference to the golden ratio, to Fibonacci numbers, or to spirals. A reader looking for where 137.5° enters will not find it, because it does not enter.

The repulsion falls off with distance, and the exponent barely matters; this implementation uses an inverse cube, which is what the magnetic droplets in the original experiment actually obey.

What it settles on

Run it and the divergence angle it produces settles down.

From a seed angle of 112°, the last thirty steps of a run give 137.00° with a spread of exactly zero. From 277° — a hundred and sixty-five degrees away — the same. Two runs with nothing in common but the rule arrive at the same place and stay there.

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. 2 Two runs of the same rule from unrelated starting angles. Both settle, and both settle on the same value.

That is what an attractor is: a value the system moves toward from a wide range of starting conditions, and stays at once it arrives. The golden angle is not stored anywhere in the rule; it is a property of the rule’s dynamics.

Why the rule finds it

The intuition is short and the precise version is a theorem.

A new element placed at angle δ from the last one is well away from its immediate predecessor. But it is also, after many steps, at angle 2δ from the one before that, 3δ from the one before that, and so on. If δ is a simple fraction of a turn, then after a few steps some multiple lands almost exactly on top of an earlier element, and the new one has nowhere good to go.

So the rule is under pressure to choose an angle whose multiples avoid whole turns for as long as possible — an angle badly approximated by fractions. And there is a unique number that is worst-approximable, which is the golden ratio, and its angle is 137.508°.

The rule does not know any of this. It has an energy landscape with a minimum, and the minimum happens to be there, because avoiding the neighbours over many generations is the same problem as avoiding rational approximations. That equivalence is where the arithmetic and the biology meet, and it is the only place they do.

Convergence is not exactness

The model here settles on 137.00°, not 137.508°, and the half-degree gap is worth accounting for honestly rather than rounding away.

Part of it is resolution: the boundary is sampled at 720 points, so the model cannot express an angle more finely than half a degree. Part of it is the finite window — only the nearest few dozen elements are counted, and the true problem involves all of them. Part of it is the discreteness of the model itself, which places elements one at a time on a boundary of fixed radius while a real meristem does something continuous.

The claim the model supports is therefore “settles near the golden angle”, and the site says that rather than the stronger thing. What makes it convincing is not the precision but the independence: an angle within half a degree of 137.508°, reached from unrelated starting conditions, by a rule with no reference to it.

What the model establishes

Three things, and it is worth separating them because they are usually run together.

That the angle need not be stored. A system with no memory of any particular angle produces one. That removes the mystery the popular framing creates, which is the mystery of how a plant knows a transcendental-looking number.

That the angle follows from crowding. The rule is nothing but “get away from the neighbours”, so any process with that character will land in the same place. That is a strong statement and it makes a prediction: systems with nothing biological in them should do it too, and they do.

That there are other answers. The parameter changes what the rule settles on, which is the next essay, and it is the part that turns a constant into a diagram.

What it does not establish

It does not establish that any plant works this way.

The rule is an abstraction of a physical situation — elements that repel, on a growing boundary — and the real mechanism in a plant is auxin: a hormone whose transporters concentrate it where it is already concentrated, so that an existing primordium depletes its neighbourhood and the next one forms where the depletion is least. That is a different process with the same character, and it was worked out decades after the geometry.

The relationship between the two is the honest one to state: the dynamical model shows what kind of process produces the pattern, and the biology says which process it is. Neither substitutes for the other, and a model reproducing the pattern is not the same as explaining it.

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. 3 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.
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. 4 Three settings of the one knob and the heads they produce. The model was told none of these angles.
How nearly each angle is a simple fraction of a turnA dip means a rational approximation and a lattice with visible rows. The golden angle's floor is 0.008; the rational angle reaches 0e+0.00.2000.400204060denominator qhow nearly q × angle is a whole turn (0 = exactly)golden 137.508°137.3°135° = 3/8distance to the nearest whole turnno dips means no rows
Fig. 5 The pattern itself, generated from a stated angle. Nothing is placed by hand, so every claim about it is a claim about the rule.

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. divergence 137.508°closest pair 1.60 × 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.