Phyllotaxis
the spirals are counted, not admired
Packing and tiling

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.
7 min read 6 figures State the claim, state the testCounted, not admired

“The golden angle is the most efficient packing” is the commonest explanation of phyllotaxis in popular writing, and it has the form of a testable claim. Efficiency of packing is a computable property of a point set.

So compute it. Several ways, because there is no single number called packing quality.

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. 1 One criterion across the divergence angles, with the golden angle marked. The other criteria pick different winners, all nearby.

Four criteria, all reasonable

Closest pair. How near do the two nearest points come, in units of the mean spacing? A high value means no point crowds another. This is the criterion that best matches “evenly spread”.

Largest gap. How big is the biggest empty patch, measured as the largest circumradius in the triangulation? A low value means no wasted space.

Area evenness. The coefficient of variation of the Voronoi cell areas. Low means every point has the same amount of room.

Six-sidedness. The fraction of cells with exactly six sides.

All four are defensible readings of “packs well”. They do not agree.

What the measurement says

Swept across the angles at a fixed head size, the closest-pair criterion, the largest-gap criterion and the area-evenness criterion pick three different angles, all within a few degrees of 137.5° and none of them equal to it.

Worse for the naive claim: the area-evenness criterion is won by rational angles. A lattice at 120° puts all its points on three radial rays, so the cells are near-identical slivers and their areas are extremely uniform — while the pattern has enormous empty wedges between the rays. Area evenness on its own rewards exactly the arrangement everyone agrees is badly packed.

That is a useful reminder about single-number quality measures. Each criterion is a legitimate reading of the phrase, and one of them is comprehensively fooled.

Why the criteria disagree at finite size

Because at any particular head size, which angle wins is partly an accident of how that angle’s lattice happens to fit that number of points.

An angle whose rational approximations have denominators near the number of primordia present will look either very good or very bad at that size, and the effect moves as the size changes. Re-run the sweep at a different n and the winner moves.

So the honest statement is that no finite measurement singles out the golden angle, and this site’s gate asserts exactly that: two criteria are required to pick two different angles, and if one ever singles out 137.5° the check fails and the essay is wrong.

What survives

The claim is not empty. It is asymptotic, and the asymptotic version is clean.

The property that distinguishes the golden angle is about behaviour as the head grows without limit, not about a contest at one size — and it shows up as the growth of the largest gap, which stays bounded for an irrational angle and grows without bound for a rational one.

The sharpest version leaves packing behind entirely and is a statement about numbers: the golden angle resists rational approximation better than any other, which is why its lattice never develops the alignments that produce rows. The packing story is a loose retelling of that, and the retelling loses the part that makes it true.

The general lesson

A claim of the form “X is optimal” needs three things before it means anything: the criterion, the range over which the optimisation runs, and the class of alternatives.

The popular version of this claim supplies none of them. Given the criterion, the answer changes. Given the size, the answer changes. And the class of alternatives matters, because the interesting comparison is not golden-versus-everything but irrational-versus-rational — which is a comparison the phrase “most efficient packing” does not even suggest.

Supplying all three turns an impressive-sounding assertion into a modest and correct one, which is usually what happens.

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. 2 What the criteria are computed from. Every packing number here comes from the Voronoi cells of the point set, and cells touching the outside are excluded — an unbounded cell has no area, and averaging over one silently is how a packing measurement goes wrong.
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. 3 The criterion that does separate the cases, and it separates them by growth rather than by value.
How many sides the cells actually haveThe mean is 5.908, which Euler's formula forces. The spread around it is not noise — a tiling of only hexagons cannot close up on a finite patch.5 sides13420%6 sides45869%7 sides7311%665 bounded cellsmean 5.908 sidessix is forced, not chosen
Fig. 4 The fourth criterion, and the one that carries least information: the mean is forced to six whatever the angle, so only the spread says anything.

The exclusion that matters

One methodological point, because it is where a packing measurement most easily goes wrong.

Cells on the outside of the point set are unbounded — they extend to infinity, and they have no area. Any statistic over cell areas has to exclude them, and a routine that quietly assigned them the area of their clipped polygon would be measuring the clipping radius rather than the packing.

This site excludes them and reports how many survived. On a head of six hundred points, about five hundred and eighty cells are bounded and measurable; the rest are the rim.

That is also why the sweeps here use a few hundred points rather than a few thousand: the measurement is over the interior, and the interior has to be large enough for the statistics to mean anything while the whole thing stays fast enough to run at build time on every figure.

Why the naive claim is attractive anyway

It is worth saying what the claim gets right, because it is not nonsense.

A golden-angle head is evenly packed, obviously and visibly. Rational angles are terrible. And the intuition that an even arrangement is somehow optimal is sound as far as it goes.

What goes wrong is the compression into “the golden angle is optimal”, which converts a robust qualitative statement into a precise quantitative one that turns out to be false as stated. The qualitative statement survives everything this site does to it; the precise version needs restating in terms of arithmetic, where it becomes exact.

That is a common fate for popular versions of technical results, and the honest response is not to say the claim is wrong but to say which version is right.

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. 5 Resistance to rational approximation across the angles, with Hurwitz’s bound drawn. This is the version of the claim that is exact.
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. 6 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

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 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. 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. 5 sides13420%6 sides45869%7 sides7311%665 bounded cellsmean 5.908 sidessix is forced, not chosen Why the average cell has six sides Not because hexagons are efficient. Because Euler's formula leaves a tiling no choice — count the edges two ways and the mean comes out at six, whatever the cells would prefer. The efficiency argument is a different claim about a different thing.