The first three hundred organs

The one famous claim about this subject that survives measurement is Hurwitz’s: the golden angle is the divergence hardest to approximate by fractions. Read over the counts a head can actually show, from its inner spirals to its rim, the claim loses most of its uniqueness. Between counts of 34 and 144, forty-five other angles resist approximation within one per cent as well — every angle whose counts add up, each the sum of the two before, from a pair near the golden ratio — and the Lucas angle comes within a ten-thousandth. What still separates the golden angle from all of them was found below the window, in the smallest counts, at the centre of the head.

That essay ended on a question it could not answer with arithmetic: whether the difference shows up in the positions of real organs, rather than only in a continued fraction, and how many organs from the centre it extends. This one grows a head at each of the forty-six angles and measures it.

Forty-six heads, and two readings of each #

Each head is 1,600 organs placed by Vogel’s rule, the $k$-th organ at radius $\sqrt{k}$ and at $k$ times the divergence. Each is read by the two instruments used on golden heads before: the closest pair of organs, and the largest empty circle among them, both on the interior’s own scale so that the rim’s inflated cells do not set the unit. A square lattice cell reads a largest hole of $1/\sqrt{2}$, and a noble lattice far from the centre keeps a closest pair of $\sqrt{2/\sqrt{5}} = 0.9457$.

The readings are then repeated with the centre left out, a radius at a time. Leaving out radius $r$ leaves out the first $r^2$ organs, so the sequence of readings says, organ by organ outward, how long each angle’s head stays different from the golden angle’s.

Three heads at their centres #

At their centres the three are plainly different drawings. On the golden head the closest pair is organs 1 and 4, 1.602 of the head’s units apart; on the Lucas head organs 1 and 5, 1.574 apart; on the head at 104.67°, organs 12 and 19, 1.241 apart. The first two are three and four organs apart, and three and four are exactly the counts at which the two angles’ arithmetic scores below the window are set — the golden angle’s at 3, the Lucas angle’s at 4.

That is the first sign that the head and the arithmetic are measuring related things, and the next section says how related.

The closest pair crowns the golden angle #

Over a whole head the golden angle’s closest pair is 0.9036 of the interior’s spacing, the widest of all forty-six. The Lucas angle’s is 0.8876 and 77.96°'s is 0.8750, and those three are also the three highest arithmetic scores below the window, in the same order: 0.4377, 0.4223 and 0.4136. So the one measurement of organ positions that can crown an angle crowns the golden angle, as the arithmetic says it should.

Past those three, though, the two rankings have nothing to do with each other. Across all forty-six the rank correlation between the arithmetic score and the closest pair is 0.11. The fourth and fifth by arithmetic, 64.08° and 54.40°, have closest pairs of 0.749 and 0.656, far down the family; the narrowest closest pair of all, 0.456, belongs to the smallest angle, 33.90°.

Why they agree where they agree #

The reason the two can agree at all is a piece of geometry worth writing out. Away from the centre, two organs $q$ apart at index $a$ are separated by about $q/(2\sqrt{a})$ in radius and $2\pi\sqrt{a}\,\lVert q\,d\rVert$ along the circle, where $\lVert q\,d\rVert$ is how far $q$ turns of the divergence miss a whole number. The squared distance is least at $a = q/(4\pi\lVert q\,d\rVert)$, where it comes to $2\pi\,q\,\lVert q\,d\rVert$. On the interior’s scale that is a closest pair of $\sqrt{2\,q\,\lVert q\,d\rVert}$ — the arithmetic score, in geometric form.

For 28 of the forty-six heads the closest pair is exactly such a pair, two organs apart by the count at which the arithmetic score below the window is set, and among those 28 the arithmetic ranks the heads almost perfectly: a rank correlation of 0.97. For the other 18 the closest pair is organs 1 and 2. Every one of the 18 is below 80°, and at an angle that small the first two organs sit close together for a reason the arithmetic cannot see — the angle is small. 77.96° is one of them, and its third place in both rankings is a coincidence of two different reasons landing together.

The Lucas angle, one organ deep #

The member of the family that matters most is the Lucas angle, since it is the one a growing stem has been followed onto here, and it is the cleanest test of the geometry above. Its head’s closest pair is organs 1 and 5, at 0.8876, against the golden head’s 0.9036. The ratio of the two is 1.0180. The arithmetic below the window predicts $\sqrt{2 \times 0.4377}$ against $\sqrt{2 \times 0.4223}$, a ratio of 1.0181. The head and the continued fraction agree on how much wider the golden angle’s closest pair is to four figures.

And that difference is one organ deep. Leave out the organs within a radius of 1.25 — organs 0 and 1 — and the Lucas head’s closest pair is still 2.7 per cent narrower than the golden head’s. Leave out organ 2 as well and it is 0.4 per cent wider, and it stays within a per cent from there to the rim. The whole of what separates the two angles in organ positions, by the instrument that can see it, is in the first two organs after the one at the centre: organ 1’s distance to organ 4 against its distance to organ 5, and organ 2’s to its own nearest neighbours.

The largest hole does not crown it #

The other instrument disagrees. Over a whole head the golden angle’s largest hole is 0.8436, and nine of the family have a smaller one: the Lucas angle’s is the smallest at 0.8279, with 100.55° and 96.68° a hair behind it. One triangle at the centre decides a golden head’s largest hole at every size, and the triangles at other angles’ centres happen to be tidier.

Leave out the centre and the family’s holes close up quickly. The spread across all forty-six falls from 23.3 per cent of the golden angle’s value over the whole head to 1.4 per cent beyond a radius of six, and every head is within a per cent of the golden angle’s from a radius of seven outward — the first 49 organs left out. Far enough out every noble angle’s holes are the same $1/\sqrt{2}$, which is the closed form’s own result, and 49 organs is how far a real head has to be read before that limit takes over.

The holes do what the closed form says #

That limit has a closed form behind it, and the heads can be checked against it rival by rival. A lattice’s largest hole changes only where the lattice flips, and at each flip it is a function of the angle’s continued fraction: $1/\sqrt{2}$ exactly for a noble angle’s cell, more for any other. So for each rival there is a last flip whose hole is a per cent or more off $1/\sqrt{2}$, and beyond that radius the lattice says the rival’s holes are the golden angle’s.

For the 21 rivals whose flips the walk from the pair 1 and 2 can follow — the ones above about 90° — the radius at which the head’s largest hole stops differing from the golden head’s ranks with that last flip’s radius at 0.94. At 104.67° the lattice’s last deviating flip is at 6.10 and the head’s difference ends at 5; at 136.73°, 5.83 and 5; at 140.59°, 4.50 and 3.5. The head sits just inside the lattice, as it must, since a head read with the centre left out to some radius still holds every flip beyond it.

The exceptions are the rivals whose last deviating flip is inside a radius of one and a half — the Lucas angle’s at 0.80, 106.45°'s at 1.39, 151.14°'s at 1.41 — and each of those heads differs a little further out than its lattice. Inside a radius of one and a half there are three organs, the one at the centre included, and nothing there is a lattice for the closed form to describe. That is the same boundary the closest pair ran into, from the other side.

The closest pairs take longer to converge #

The closest pair keeps the family apart for longer. Over the whole head the forty-six span 0.456 to 0.904, a spread of half the golden angle’s value. Beyond a radius of six the spread is 10.6 per cent; beyond seven, 3.5 per cent, where it stalls, held open by a handful of angles until a radius of fourteen. Only from a radius of seventeen outward — the first 289 organs left out — is every one of the forty-six within a per cent of the golden angle, and there the spread is 0.65 per cent.

The handful that hold it open are not a random selection. With the first 49 organs left out, the five narrowest closest pairs belong to 140.59°, 147.58°, 104.67°, 68.23° and 156.17°, whose convergent denominators add up only from 5, 5, 7, 5 and 7. With the first 196 left out they belong to 104.67°, 156.17°, 46.61°, 78.60° and 81.45°, adding up from 7, 7, 8, 9 and 9. The last to converge are the ones whose continued fractions take longest to become noble, which the next section but one makes into a measurement.

So the answer to “how many organs from the centre does the difference extend” depends on the instrument, and differs by a factor of six in organ count: about fifty by the largest hole, and about three hundred by the closest pair.

First place does not survive leaving out the centre #

The golden angle’s crown by closest pair belongs to the centre. Leave out the organs within a radius of one and it is second; within three, four or twelve, first again; within eight, fourth; within seventeen, fourth; within twenty — four hundred organs — eighth. Out there the differences are a few parts in a thousand, and which angle comes first is decided by where each head’s rings happen to fall relative to the cut, not by anything about the angle.

By largest hole it is tenth over the whole head, first only with the centre left out to a radius of 1.75 or 2, and nineteenth with four hundred organs left out. The measurement that ranked packing criteria and found them crowning different angles has the same shape here, inside a family chosen for being indistinguishable by the arithmetic over the window.

Where a rival’s difference ends #

The rivals do not all fade at the same radius, and what orders them is visible in their continued fractions. Every angle in the family has convergent denominators that eventually add up like the Fibonacci numbers; what differs is where they start to. The golden angle’s add up from 1, the Lucas angle’s too, and 104.67°'s only from 7 — its partial quotients are 3, 2, 3 and then ones.

Against that count, how far out a rival’s closest pair stays a per cent from the golden angle’s has a rank correlation of 0.90. Angles whose denominators add up from 1 stay distinct to a radius of between 1.25 and 5; those adding up from 8 or more, to between 10 and 14. By largest hole none stays distinct past a radius of six. So the organs over which a family member can be told from the golden angle are, near enough, the organs its continued fraction spends before it becomes noble — which is the geometric form of the statement that the family differs from the golden angle only below the counts it shares.

What the arithmetic said, and what the heads say #

The arithmetic said the golden angle is distinguished from its family at the centre. The heads say three things about that.

It is visible, but in one instrument and not the other: the closest pair crowns the golden angle over a whole head and the largest hole does not. It is confined, to the first three hundred organs by the more sensitive instrument and the first fifty by the other. And it is in the right organs for the right reason — the golden head’s closest pair is organs 1 and 4, three apart, at the count its arithmetic score is set, and where the arithmetic can see the closest pair at all it ranks the heads almost exactly.

What the heads add is the eighteen members the arithmetic cannot rank. Below about 80° a head’s first two organs are its closest, and resistance to approximation has nothing to say about them. The arithmetic’s claim about the centre is a claim about lattices, and the first organ or two of a Vogel head is not yet a lattice.

What a person would have to measure #

To tell a real head at the golden angle from one at a rival by where its organs sit, the measurement has to reach inside the first few hundred organs, and by the closest pair, which is the instrument that can tell. That is where the round trip through the counts cannot help, since it reads the rim, and where a real capitulum is least regular: its first organs are the youngest and were placed before the pattern settled. The measurement the arithmetic asks for is therefore possible in principle and hard in practice, and it is not a count.

It is also a measurement of one head against a family, not of a head against a claim. The golden and Lucas angles are branches a growing stem can reach; whether any stem reaches 104.67° or 151.14° is not known, and a closest-pair reading that pointed at one would be evidence of something new rather than a refinement of this.

What this does not establish #

That real organs sit where Vogel’s rule puts them at the centre of a head; the whole of the difference measured here is inside the first three hundred, where the rule is least likely to hold. That the per-cent threshold is the right one; a finer measurement would stretch every reach outward, and the ordering by where the counts add up is what is claimed, not the radii.

Nor does it extend the claim about the window. The forty-six were chosen as the family over counts 34 to 144. A head read over a different window has a different family, growing by about $\varphi^2$ a step outward, and this measurement has not been repeated on any other.

What would withdraw it #

A family angle whose whole-head closest pair is wider than the golden angle’s.

A family angle whose closest pair differs from the golden angle’s by a per cent or more beyond a radius of seventeen, or whose largest hole does beyond seven.

A head whose closest pair is neither two organs the arithmetic’s count apart nor organs 1 and 2.

Still open: whether the centre survives a head grown rather than placed #

Every head here is placed by Vogel’s rule, which puts each organ at an exact radius and an exact multiple of the angle from the first. A head grown by a placement rule — each organ put where the organs already present leave the most room — settles on its angle only after its first few organs, and its first organs are exactly where the difference measured here lives. The measurement is the same two readings on grown heads at the golden and Lucas angles: whether the closest pair still sits at organs three and four apart on one and four and five apart on the other, or whether growth rearranges the centre before the angle is fixed.