Phyllotaxis
the spirals are counted, not admired
Packing and tiling

The gap that grows

A rational divergence angle develops an empty wedge that grows without bound as the head fills — a factor of nearly four between two hundred primordia and sixteen hundred. An irrational one does not. That is the statement about the golden angle that survives measurement.
5 min read 5 figures State the claim, state the testCounted, not admired

No packing criterion singles out the golden angle at a fixed head size. That is a genuine result and it leaves an obvious question: what does distinguish it, and why does the pattern collapse at rational angles?

The answer is about growth rather than about a contest.

How the largest gap behaves as the head fillsThe rational angle's gap grows by a factor of 3.8 over this range; the golden angle's stays within 1.30. This is the claim about 137.5° that survives measurement.02.5057.50105001e+32e+32e+3number of primordialargest empty gap, in units of the mean spacing137.508° (golden)137.3°135° = 3/8 of a turngaps measured from the triangulationan asymptotic claim, not a contest
Fig. 1 The largest empty gap against the number of primordia, for three angles. One of them grows without bound.

What the measurement shows

Take the largest circumradius in the triangulation of a head — the biggest empty patch — and measure it in units of the mean spacing, so the answer does not simply shrink as the head fills.

For 135°, which is exactly three-eighths of a turn, the gap grows from 3.1 at two hundred primordia to 9.1 at sixteen hundred. It is not converging; it is growing, and it grows because the pattern has eight radial rows and the wedges between them get wider in proportion as the head gets larger.

For the golden angle it stays between about 0.7 and 0.9 across the same range. So does 137.3°, and so does 106.5°.

The distinction the measurement makes is therefore not golden-versus-everything. It is rational versus irrational, and it is sharp.

Why rational angles fail

If the divergence angle is p/q of a turn, then every qth primordium lands on exactly the same ray from the centre. The pattern has q radial rows and nothing between them.

As the head grows, the rows extend and the wedges between them widen in proportion. There is no mechanism by which anything fills them, because nothing is ever placed at an angle that is not one of the q rays.

That is complete collapse rather than degradation, and it happens at every rational angle, including ones very close to the golden angle. The failure is discontinuous in a way that a contest at fixed n cannot show.

Why irrational angles do not

The multiples of an irrational number, taken modulo one, are equidistributed — they fill the circle, and no gap survives forever.

There is a sharper statement, the three-distance theorem, which says that the first n multiples of any irrational divide the circle into intervals of at most three distinct lengths. That is a strong regularity, it holds for every irrational, and it is why any irrational angle produces a lattice that keeps filling.

How well it fills depends on how badly the number is approximated by rationals: a number with a very good rational approximation at some denominator behaves almost like that rational for a while, producing near-rows that persist until the head grows past them. That is what makes 137.0° look poor at moderate size while remaining bounded in the limit.

The claim in its correct form

Putting the three pieces together gives the version that survives.

Rational angles fail completely — gaps grow without bound, at every rational.

Every irrational angle works in the sense that gaps stay bounded.

How quickly it starts working depends on how well the angle is approximated by rationals, and the golden angle is the extreme case: it is the number for which those approximations are worst, so its lattice is the one that never has a bad phase at any scale.

That last clause is what “optimal” should mean here, and it is a statement about arithmetic rather than about geometry. It has a precise form, it is a theorem, and it is considerably more interesting than the packing story it usually gets flattened into.

What this means for a plant

Rather less than the claim implies, and it is worth being clear.

A plant does not need the extreme case. Any angle comfortably away from a simple fraction produces a pattern with bounded gaps, and the difference between the golden angle and a merely-irrational neighbour is invisible in a head of a few hundred primordia.

What a plant does need is to avoid the rationals, and the dynamical model explains why it does: a rule that places each element away from its neighbours is under continuous pressure away from any angle whose multiples pile up, and the golden angle is where that pressure has nowhere left to push.

So the biology does not require the theorem. The theorem explains why the attractor is where it is.

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. 2 Packing quality against the divergence angle. Several reasonable criteria give several different winners.
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. 3 How nearly each candidate angle is a simple fraction of a turn. A dip means a good rational approximation, and a good rational approximation means visible rows.
The version of the claim that does survive measurementThe golden angle scores 0.4377, against 0.3306 for the best of 1500 other angles sampled. The dashed line is Hurwitz's 1/√5, which no number can exceed.00.2000.400100120140160divergence angle (°)resistance to rational approximation (higher is more irrational)1/√5 — the bound137.508° — 0.4381500 angles on a 0.05° grid, plus the golden angle exactlythis claim is sharp
Fig. 4 Resistance to rational approximation across the angles, with Hurwitz’s bound drawn. This is the version of the claim that is exact.
Voronoi cells of a head at 137.51°171 bounded cells, averaging 5.87 sides. Cells with six sides are shaded; the others are what a tiling has to contain to close up.171 bounded cellsmean 5.87 sides
Fig. 5 The Voronoi cells the packing statistics are computed from, with six-sided cells shaded.

Where this goes next

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 Packing, measured four ways The claim is that the golden angle packs best. It is measurable, and the measurement gives three different winners on three criteria — all near 137.5° and none of them it. That does not make the claim wrong; it makes the usual statement of it wrong. 00.2000.400100120140160divergence angle (°)resistance to rational approximation (higher is more irrational)1/√5 — the bound137.508° — 0.4381500 angles on a 0.05° grid, plus the golden angle exactlythis claim is sharp The claim that survives Of the three famous assertions about this subject, one is out by a factor of two, one is true of a branch rather than of plants, and one is right — in a sharper form than the version usually told, and about arithmetic rather than about packing. 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.