Packing, measured four ways

“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.

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”. The first two are about distance — where the points are relative to one another. The last two are about cells — how the plane is divided among them. That difference turns out to be the whole of the answer.

An earlier reading, and what it was divided by #

This essay first reported that closest pair, largest gap and area evenness pick three different angles near 137.5°, that area evenness is won by rational angles, and that no finite measurement singles out the golden angle. The site’s own measurement confirmed exactly that: two criteria were required to pick two different angles.

Every criterion but six-sidedness is divided by a scale, the square root of the head’s mean cell area, and that mean was taken over every cell with a bounded polygon. The cells just inside the edge of a head are bounded and enormous, and on a 150-organ head they raise the mean from π to 38.8. The winners on largest gap wandered from one angle to another with the head size because the scale did.

Read on the interior’s own scale the result is different, and the rest of this essay is written to that reading. The earlier numbers are kept in two figures below, so that what was wrong can be seen rather than described.

What the measurement says #

On the interior’s scale at 400 organs, the golden angle’s closest pair is 0.9027 of a spacing, first of 72 angles; the best angle on the half-degree grid, 137.5°, reads 0.9026. Its largest gap is first as well. At 700 and 900 organs it stays first on closest pair and level to the fourth decimal on largest gap.

Area evenness goes the other way. It is won by 135°, three eighths of a turn, and the golden angle is fourth. Six-sidedness is won by the same rational angle, with the golden angle far down the list.

So the criteria about distance agree, and they agree on the golden angle. The criteria about cells agree too, and they agree on a rational angle. That is two answers rather than three, and the split between the two kinds is the finding.

The reading that was published #

This is the figure the earlier version of the essay was reasoning from. Counting the rim’s cells, area evenness at 400 organs runs from 0.0103 to 11.55 across the window, and the golden angle reads 1.8921, forty-fifth of 72.

Two angles half a degree apart cannot differ that much in how evenly they divide a disc among their organs. What they differ in is where the last ring of organs fell against the convex hull, and a criterion divided by that ranks the hull.

Why area evenness prefers rays #

The part of the old finding that survives is the part about rays, and it is worth dwelling on, because it is not a quirk of this criterion.

Set the divergence angle to a simple fraction of a turn and the head’s organs lie on a handful of radial rays. Along each ray they sit at the radial law’s own spacing, so every cell is a wedge of the same area — while between the rays are empty wedges that widen as the head grows. On the interior’s scale the evenest reading in the window belongs to three eighths of a turn, at 0.0074 on a 900-organ head.

The criterion is not badly implemented. It measures exactly what it says: how uniform the cells are. Uniformity of cells and uniformity of coverage are different properties, and a degenerate arrangement can maximise one while failing the other.

Why the criteria split in two #

The distance criteria ask where the organs are relative to each other. The cell criteria ask how the plane is divided among them. A rational angle’s organs are well separated along each ray and the division is perfectly even, but the rays leave holes, and only the distance criteria can see a hole.

So the two kinds were never going to agree, and nothing in the phrase “packs best” says which kind is meant.

What survives #

The claim is not empty, and its surviving form is narrower and sharper than the slogan.

On the distance criteria the golden angle is above every rational angle near it. Away from the centre of the head, its closest pair approaches a constant that every noble angle shares, and its largest gap is exactly 1/√2 at every ring where its lattice flips, which only the noble angles achieve. The property that picks those angles out is a statement about numbers: they are the hardest numbers to approximate by fractions, and a lattice built on one never develops the alignments that leave holes.

So packing singles out a class, and the class is the noble angles. Nothing about packing picks 137.508° from within it.

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 can change. And the class of alternatives matters, because the interesting comparison is not golden-versus-everything but noble-versus-the-rest — 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.

The exclusion that matters #

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

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. What is less obvious is that excluding them is not enough. A cell one ring further in is bounded and can still be hundreds of times the interior’s size, because the triangles along the hull are slivers whose circumcentres lie far away.

On a head of six hundred points, 578 cells are bounded and 516 of those lie inside the head’s own radius. The measurement here is over the 516. The rim of a Vogel head is also the part of its model least like a plant, so leaving it out of the scale costs nothing a plant would notice.

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 spread, obviously and visibly. Rational angles are terrible. And the intuition that an even arrangement is somehow optimal is sound as far as it goes — on the criteria about distance it is simply true.

What goes wrong is the compression into “the golden angle is optimal”, which converts a robust statement about a class of angles into a precise statement about one of them. The robust statement survives everything this site does to it; the precise version needs restating in terms of arithmetic, where it becomes exact and still names a class.

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.

How a single-number quality measure gets fooled #

The area-evenness result is worth generalising, because every plausible one-number measure has a degenerate case like it somewhere.

Closest pair is maximised by a configuration with a huge hole in it, as long as no two points are close. Largest gap is happy with points piled on top of one another, as long as the piles are well spread. Six-sidedness is forced to be high in any tiling at all, which makes it almost content-free. And area evenness is maximised by rays.

The defence is not a better statistic. It is reporting several and saying which kind each one is, which is what the figures here do, and what a claim of “most efficient” was always hiding.

Efficiency of what, for whom #

Behind the measurement problem is a definitional one, and it is fatal to the popular version of the claim.

A seed head is not packing seeds at the moment it is being laid out. The primordia are microscopic and the meristem is nearly empty; the packing everyone photographs is the result of later growth, by which time the arrangement is fixed. Whatever the angle is for, it is not for solving the problem visible in the photograph.

And “efficient” needs an objective. Maximum seed count? Then seed size and head size matter more than angle. Even light exposure, which is the argument for leaves rather than seeds? Then the criterion is angular coverage integrated over a season, which is a different computation entirely and depends on latitude. Mechanical stability? Different again. None of these is the criterion anyone means, because the claim is normally made without a criterion at all.

Why the distance criteria agree #

On the rim’s scale the largest gap was smallest at 135.5° at 700 organs, with the golden angle twenty-sixth. On the interior’s scale the golden angle is level with the best at 700 organs, 0.8436 against 0.8435, and the same at 900.

The agreement has a plain reason and a sharp one. The plain one is that angles near a simple fraction leave holes and are bad on both distance criteria, while the region between fractions is good on both. The sharp one is that on a golden head both criteria are decided by the first five organs, and read away from the centre both tend to constants that every noble angle shares.

What changes with head size #

A last dependency, since it is where the earlier reading went wrong in plain sight.

Counting the rim’s cells, the golden head’s largest gap runs 0.240, 0.733, 0.382, 0.735 and 0.923 at 150, 300, 600, 1,200 and 2,000 organs — up and down by a factor of four with no trend at all. The rational angle climbs from 2.66 to 10.14 on the same scale, and its growth is real on either scale.

On the interior’s scale the same five golden heads read between 0.841 and 0.844. A packing statistic that swings by a factor of four between head sizes while its interior does not is reporting the rim, and a comparison of angles at any single size inherits the swing.

Why cells and not circles #

A note on the choice of machinery, since “packing” has an established meaning in mathematics that is not the one used here.

Classical packing asks how many non-overlapping discs of a fixed radius fit in a region, and the answer for the plane is the hexagonal lattice at 90.7% coverage. That is a beautiful result and it is not applicable: the points in a seed head are not equal discs, their spacing grows with radius, and the region is a disc rather than the plane.

So the measurements here use the Voronoi tiling instead, which asks a different and better-posed question: given these points, how is the available area divided between them. Every criterion in the figures is a statistic of that division, and the Delaunay triangulation underneath it supplies the neighbour relations the closest-pair and largest-gap criteria need.

Edge cells, and the statistic that was wrong #

One implementation point that changed the answer.

Points on the convex hull have unbounded Voronoi cells, and including them in an area statistic is not merely inaccurate, it is undefined. That was always handled. What was not handled is the ring of cells just inside the hull, which are bounded and whose size is set by how thin the hull’s triangles are.

The earlier version of this section said that clipping cells to the head’s boundary gave a systematically different answer, and offered that as the reason for not clipping. The difference was real and the reasoning ran the wrong way: the unclipped reading was the one measuring something other than the point set. A statistic divided by the mean of those edge cells is a statement about the hull.

Rim effects are a large fraction of a small head, and a packing comparison at a fixed size is exactly where they bite — as the reading this essay first published shows, in the two figures that keep it.

Where the popular claim came from #

Worth a short history, because the claim is not arbitrary and its origin explains its shape.

The mathematics that does support something like it is about coverage of the circle: the sequence of fractional parts of n·δ, and how evenly it fills the interval. There is a body of results — three-distance theorems, discrepancy bounds — saying that the golden ratio’s sequence is the most evenly distributed available, and those results are correct.

The step that goes wrong is translating “most evenly distributed on the circle” into “packs best in the disc” without saying which kind of packing. The first is a statement about angles only; the second involves radii, cells and a finite head. The translation survives for the criteria about distance and fails for the criteria about cells.

So the popular claim is a compressed version of a real theorem, which is why it feels authoritative, and why the correction is awkward: the underlying mathematics is not wrong, and on the criteria about distance the packing turns out to carry exactly the same arithmetic.

What a better-posed version would look like #

If the packing question is to be asked properly rather than abandoned, the form it needs is worth writing down.

State the criterion as a function of the point set, with the head size as an explicit argument, and say whether it is about distance or about cells. State whether the claim is at fixed size or asymptotic — the two can have different answers. State which points are excluded, and from what: from the statistic, and from the scale it is divided by. And say what would count as refutation: which other angle, scoring how much better, would settle it.

Asked that way the question has answers, and this essay is those answers. At fixed size and on the interior’s scale, the distance criteria put the golden angle first among its neighbours and the cell criteria prefer rationals; asymptotically the rationals fail and the irrationals do not; and among the irrationals the distance criteria single out the noble angles, which is arithmetic.

Asked the usual way — “is the golden angle the most efficient packing” — it has no answer, because it has no criterion, no size, and nothing that could refute it.

The cylinder’s version #

Asked on a stem, where the lattice is genuinely periodic, the packing question has an answer that points the same way.

On a cylinder there is a clean optimum: the lattice is best packed when it is equilateral, every element having six equidistant neighbours. Those configurations turn out to be exactly the branch points of the van Iterson tree, they occur at a thin discrete set of rises, and they have a closed form — the rise is √3/2q with q = m² + mn + n².

And every one of them sits at a rational divergence angle, with denominator 2q. None is at 137.5°.

So the strongest optimality statement the cylinder supports is: at the discrete set of rises where an equilateral lattice fits, the divergence that makes it fit is optimal for packing. It says nothing about the golden angle, and nothing at all about the rises in between, which are most of them.

Two geometries, two quite different analyses, and the same conclusion: packing arguments about phyllotaxis single out special configurations — the forks on a stem, the noble angles on a disc — and never one angle by itself.

A fifth criterion, swept #

The disorder of the resulting tissue is a fifth criterion, and it is a criterion about cells. Sweeping it across the angle gives the answer the two cell criteria here give: the ordered arrangements are at rational angles, and the golden angle is nowhere special on it.

The scale the four were divided by #

The next measurement is the one that found the error, and it is not another criterion but the scale all of them share. Read against the interior, the mean cell area is π at every size, area evenness turns out to be an instrument for the radial law rather than the angle, and the distance criteria away from the centre become Hurwitz’s constant — a prediction that any noble angle’s head can be built to test.