The gap that grows

Measured four ways, packing at a fixed head size splits in two: the criteria about distance favour the golden angle and the criteria about cells favour rational ones. That leaves an obvious question: why does the pattern collapse at rational angles, and what is the golden angle’s advantage made of?

The answer is about growth rather than about a contest.

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 2.9 at two hundred primordia to 8.4 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 0.841 and 0.844 across the same range, because a single triangle at the centre of the head decides it at every size. 137.3° stays between 0.839 and 0.863, and 106.5° between 0.843 and 0.845.

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 small in a head of a few hundred primordia — 0.862 of a spacing against 0.843 at 137.3°, set by one ring of the head rather than by the whole of it.

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.

Three distances, and the regularity underneath #

The claim that irrational angles keep filling has a sharper form than equidistribution, and it is worth having because it explains why the filling is so orderly.

The three-distance theorem says that for any irrational α, the first n multiples of α taken modulo one divide the circle into intervals of at most three distinct lengths — and the largest is at most the sum of the other two. That holds for every irrational, at every n, with no exceptions.

So a spiral lattice is never merely “not clumped”. At every stage of its growth, the angular gaps between successive primordia take at most three values, which is an extraordinary amount of regularity to get from a rule that does nothing but add a fixed angle repeatedly.

The theorem also explains the transitions. The three lengths change as n crosses the denominators of the convergents, and those are exactly the radii at which the parastichy numbers change. The counting transitions and the three-distance structure are the same phenomenon seen from two directions.

Why 137.0° looks poor and is not #

A case worth working through, because it separates the finite from the asymptotic.

137.0° is irrational as a real number, so by everything above its gaps stay bounded in the limit. And yet a head built at it shows visible rows.

The reason is that 137.0° is close to 137.142857°, which is 8/21 of a turn, a fraction with a modest denominator. For as long as the head has fewer than a few hundred primordia, the lattice behaves almost exactly like the twenty-one-row rational one, because the accumulated drift from the rational has not yet become visible.

Grow the head far enough and the drift accumulates, the rows shear apart, and the pattern fills. But “far enough” may be more primordia than any plant makes.

That is the general shape of the finite-versus-asymptotic distinction on this site. Every irrational works eventually; how long “eventually” takes depends on how well the number is approximated by simple fractions; and the golden angle is the one for which it takes no time at all, because it has no good approximations at any denominator.

What the measurement had to be careful about #

The largest gap is measured as the biggest circumradius among the triangles of the Delaunay triangulation, and two decisions in that had to be made rather than defaulted.

Scaling. A raw circumradius shrinks as the head fills simply because everything gets denser, so the measurement is expressed in units of the mean spacing. Without that, every angle would appear to improve with n and the comparison would say nothing.

Which triangles. Triangles touching the convex hull have circumcircles that reach out into empty space beyond the pattern, and including them measures the edge of the disc rather than the packing. Only triangles all of whose vertices have bounded Voronoi cells are counted.

Both are the sort of decision that silently determines the answer, which is why they are stated. A gap measurement that included the hull triangles would report a large gap for every angle and would rank them by the shape of the boundary.

Why the rational case is a cliff and not a slope #

One more feature that the growth curves show and that a fixed-size comparison cannot.

Approaching a rational angle from either side, the finite-size behaviour degrades smoothly — the near-rows get more visible, the gaps get bigger. But the limiting behaviour is discontinuous: every irrational angle, however close to the rational, has bounded gaps, and the rational itself does not.

So there is no sense in which a rational angle is the worst case of a continuum. It is a different case entirely, and the continuum of irrationals nearby all behave qualitatively alike in the limit while looking arbitrarily bad at finite size.

That discontinuity is the reason the packing measurements at fixed head size are so unhelpful. The quantity that separates the cases cleanly is defined only in the limit, and every measurement is necessarily at finite n.

What a plant would have to do to fail #

Turning the result around gives a useful sense of how much margin there is.

A plant using a rational divergence angle would produce visible radial rows and large empty wedges. Nothing about that is impossible — whorled phyllotaxis is exactly this, it is common, and plants that use it are not defective. Alternate leaves at 180° are a rational arrangement and they work perfectly well.

So the interesting question is not why plants avoid rationals — many do not — but why the ones packing many primordia into a disc avoid them. And there the answer is straightforward: a seed head with eight radial rows wastes most of its area, and a plant packing hundreds of seeds cannot afford it.

That is a functional argument and this site cannot test it, because it is about selection rather than geometry. What the geometry supplies is the magnitude: a rational angle’s largest gap grows by a factor of 2.9 between two hundred and sixteen hundred primordia, and by more beyond that, so the cost of a rational angle rises with the number of seeds. A plant with a dozen leaves is indifferent; a sunflower is not.

The measurement, restated #

Three numbers from the sweep are worth carrying.

At 135° — three-eighths of a turn, exactly rational — the largest gap in units of the mean spacing runs 2.9, 4.1, 5.9, 8.4 as the head grows from two hundred to four hundred, eight hundred and sixteen hundred primordia. Monotone, and growing.

At 137.508° it runs 0.841, 0.843, 0.844, 0.844 over the same sizes — the same to the third decimal, because it is one triangle at the centre measured four times.

At 137.3°, an ordinary irrational nearby, it runs 0.839, 0.862, 0.862, 0.862: bounded, and larger, because from a few hundred primordia outward one ring of that head holds a hole the golden head never has.

The distinction the whole-head reading supports is therefore between rational and irrational, and it is enormous. The distinction between the golden angle and other irrationals is real, is a theorem, and shows on a head of a few hundred organs once the reading is taken ring by ring rather than over the whole head.

What the asymptotic statement is, precisely #

The result deserves a careful statement, because “irrational angles pack better” is false as usually meant and true in this specific form.

Fix a divergence angle δ and generate n points. Measure the largest empty circle among them — the biggest circumradius in the triangulation — scaled by the mean spacing so the number does not shrink trivially as points are added.

For a rational δ = p/q, the points lie on q rays. Adding points makes each ray denser but adds no new rays, so the wedge between two adjacent rays keeps its angular width while the head grows. The largest gap therefore grows without bound: measured here it goes up 2.9× between 200 and 1,600 points.

For an irrational δ, every new point falls in a genuinely new direction, so the angular gaps keep subdividing. The scaled largest gap stays bounded: 1.003× over the same range, because it is decided at the centre every time.

That is a statement about limits, not about any particular head, and it is why packing measured at a fixed head size cannot settle the question. At 200 points a well-chosen rational can look fine. The difference is in what happens next.

Three sizes, one point set each #

The demonstration is arranged so the comparison is not confounded.

Each row is one divergence angle. Each column is one head size. The same generator produces all of them, so nothing differs between cells except the angle and the count, and the largest-gap statistic is computed by the same code in every cell.

Reading across a rational row, the wedges are visible and they get visibly worse. The head is not becoming less regular — the rays stay perfectly regular — it is becoming more anisotropic, all its regularity concentrated in one direction.

Reading across an irrational row, nothing much happens. The pattern at 1,600 points looks like the pattern at 200 with more points in it, which is the property being demonstrated and is easy to mistake for the figure not showing anything.

How near-rational is rational #

The interesting case is neither, and it is where the arithmetic gets its grip.

An angle very close to a simple rational behaves like the rational for a while. At p/q plus a small ε the points nearly land on q rays, and they stay nearly on them until the accumulated drift n·ε becomes comparable with the ray spacing. Before that crossover the gap grows; after it, the pattern breaks up and the gap stops growing.

So there is no clean division into good angles and bad ones. There is a crossover size for every angle, and it depends on how well the angle is approximated by simple fractions — which is exactly the quantity Hurwitz’s theorem is about.

An angle with a good approximation p/q behaves badly up to roughly q² points. The golden angle’s approximations are the worst available, so its crossovers come as early as they possibly can, at every scale at once. That is the connection between the arithmetic and the geometry, and it is why the arithmetic version of the claim implies the packing version while the packing version does not imply the arithmetic one.

Why this is the honest version of the packing claim #

Putting the three results together gives a statement that survives all of them.

At fixed size, the packing criteria about distance put the golden angle first among the angles near it, and the criteria about cells pick rationals.

Asymptotically, rational angles fail without bound and irrational ones do not, which divides the angles into two classes but does not pick a member of the good one.

Within the good class, the ordering is by resistance to rational approximation, the golden angle is the extreme, and there is a theorem saying nothing beats it.

Only the third statement singles out 137.5° — together with the other noble angles — and it is a statement about numbers. “Packs best” is a reasonable gloss on it in the same way that “is far from simple fractions” is a reasonable gloss on the arithmetic — which is to say, it points at the right thing and would never let anyone check it.

Why this is the measurement that generalises #

Of everything on this site, the largest-gap-against-head-size sweep is the one whose form is most reusable, and it is worth naming why.

It is a limit statement measured over a range rather than a value measured at a point. Almost every claim about pattern that turns out to be slippery — packing quality, regularity, evenness — is slippery because it was evaluated at one size, and almost every one becomes decidable when it is evaluated as a trend.

The same reframing fixes the other measurements here. Counting is radius-dependent, so the useful statement is about how the counts change. Packing criteria are size-dependent, so the useful statement is about which ones stay put. The model’s convergence is parameter-dependent, so the useful statement is where it stops.

A single number with an unstated size is the recurring failure. Sweeping the size is the recurring fix.

There is one refinement to that rule which this collection has since had to learn, and it belongs beside it. Sweeping a size only helps if the size is binding — if the answer would have moved had the parameter mattered. A sweep of a non-binding parameter returns the same number every time and reads as robustness, which is the most reassuring way a measurement can be wrong. So the rule has two clauses: sweep the size, and check that the sweep had something to find.