The disorder is a staircase

Euler’s relation forces the mean side count of a bounded tiling to six, and the earlier work measured it on six point sets from a whorled lattice to a Poisson scatter: 5.97 to 6.04 on all of them. The quantity that is not forced is the second moment about six — how far the individual cells depart from it — and on the same six sets it runs from 0.023 to 1.830, a factor of seventy-nine.

That work recorded two numbers side by side without an explanation, which is the shape this collection keeps finding. μ₂ is 0.253 at the golden angle and 0.255 at a rational angle a hundredth of a degree away, and 0.291 half a degree away. The reading offered was that the statistic is sensitive to the arithmetic of the angle — to its partial quotients — rather than to how far it sits from 137.5°.

Nobody had swept it. This essay sweeps it, and the reading is wrong in a way that is more interesting than being wrong.

The sweep #

Build a head at each of four hundred divergence angles across a degree and a half, take the Voronoi tessellation of each, cut the rim, and measure the mean squared departure of a cell’s side count from six.

Three things are visible and none of them is what a single measured value suggests.

It is flat in stretches. From 137.36° to 137.44° it is 0.207 to 0.213. From 137.47° to 137.54° it is 0.249 to 0.255. From 137.56° to 137.61° it is 0.364 to 0.374. Those are not trends with noise on them; they are plateaux, on a deterministic function of the angle with no random seed anywhere in it.

The steps between them are sharp. Between the second and third of those stretches, over five hundredths of a degree, μ₂ goes from 0.25 to 0.37 and stops.

And the dips are at rationals. At 138.4615°, which is 360 × 5/13, μ₂ is 0.019 at nine hundred points, against 0.4 to 0.5 a fiftieth of a degree away. At 137.1429° = 360 × 8/21 it is 0.025. At 135° = 360 × 3/8 it is 0.008. The dips at 137.4545° (21/55) and 137.6471° (13/34) are shallower — and the fractions are the ladder’s own, which is not a coincidence and is not what this essay is about.

The ordering is by denominator: the smaller the denominator, the deeper the dip.

Why a rational is ordered #

At an exact rational p/q every point lands on one of q rays. That is what “rational divergence” means — after q organs the angle has come back to where it started — and the arrangement is a set of radial rows rather than a spiral.

A head of q rays is a clean lattice — and it is one a spiral counter reads as q rows rather than as two families. Its Voronoi cells are almost all hexagons, and the ones that are not sit on a fixed number of seams where the rays meet themselves. So μ₂ is small, and it gets smaller as the head grows, because the seam cells are a fixed population being diluted by an increasing one.

This site has met that inversion before. Read on a head’s interior, the evenest cell areas belong to exact rational angles, whose organs sit on rays at the radial law’s own spacing while the gaps between the rays grow without bound. The earlier work found that the most hexagonal tissue in a set of six is the whorled one at 144°, at 99% hexagons. The staircase is the same fact swept rather than sampled, and swept it shows what sampling could not: the ordered angles are not a handful of special cases. They are dense, and every one of them has a dip.

What the plateaux are #

The stretches between the dips are the part that changes what a measured value means, and they have a straightforward reading.

μ₂ is a statistic of the contact graph — which cells touch which — and that graph is a discrete object. Two divergence angles that produce the same set of neighbours produce the same side counts and therefore the same μ₂ exactly. The neighbour set changes only when the angle crosses a value at which one pair of cells stops touching and another starts, and those crossings are what the steps are.

So the plateaux are ranges of angle over which the head has one arrangement, and the width of a plateau is how far the angle can move before the tessellation notices. At nine hundred points that is a few hundredths of a degree.

That claim is stronger than the figure shows and it is worth saying how much stronger, because it comes with a test the sweep has not run. If two angles produce the same contact graph then they produce the same side count for every cell, so their whole side-count distributions are identical — not close, not equal to three decimals, but the same multiset of integers. The second moment agreeing is a consequence of that and much weaker than it.

So the check is: take two angles inside one plateau, tessellate both, and compare the histograms cell for cell. If they agree exactly the plateau is a plateau. If they agree in their second moment and differ somewhere in their tails, the plateau is a coincidence of one statistic and the reading above is wrong — which is precisely the failure mode this thread exists to catch, turned on its own explanation.

Nothing here has run it. It costs two tessellations and a comparison, and it is the cheapest outstanding check in the thread.

Which settles that earlier work’s puzzle. 0.253 at the golden angle and 0.255 at a rational a hundredth of a degree away are the same plateau. They are not two close measurements of a smoothly varying quantity; they are two samples of one value, and the near-agreement was inevitable rather than suggestive.

What the earlier work got right and what it got wrong #

Right: the statistic is arithmetic. The dips are at rationals, they are ordered by denominator, and there is no way to describe the curve without mentioning fractions.

Wrong: the difference it measured is not a gradient in that arithmetic. It is the height of one step against another, and a step’s height is not a smooth function of how badly approximable the angles on either side are. Reading 0.253 against 0.291 as evidence about partial quotients is reading the difference between two plateaux as though it were a derivative.

There is a second thing the sweep corrects, and it is about method rather than about tissue. A quantity that is flat in stretches with sharp steps between them cannot be characterised by two measurements, however carefully each is made. The earlier work measured six point sets very carefully. What it needed was four hundred measured roughly, and the sweep is cheap — a head of nine hundred points tessellates in thirty milliseconds.

The plateaux are flat at this resolution and not otherwise #

There is a consequence of the two findings taken together that neither states, and it is uncomfortable enough to be worth putting plainly.

Every rational has a dip, and the rationals are dense: between any two angles, however close, there is a fraction with a denominator small enough to matter at some head size. So no stretch of this curve is genuinely flat. What the figure shows as a plateau is a stretch containing only dips too narrow for the sweep’s grid to land in and too shallow to see at nine hundred points, and a finer sweep at a larger head would resolve some of them.

That is the same objection this essay makes to the two-point comparison it replaces, one level down. Two samples inside a plateau were called a plateau because they agreed; four hundred samples showing a flat stretch are called a plateau for the same reason, and the reason is the sampling in both cases.

What survives it is the part that does not depend on flatness. The dips are at rationals, they are ordered by denominator, and the difference between two widely separated angles is a difference between steps rather than a gradient — all three hold whatever finer structure sits inside a plateau. What does not survive is any statement of the form μ₂ is constant on this interval, and the honest version is that μ₂ is constant on this interval to the resolution of a four-hundred-sample sweep at nine hundred points, with the resolution named.

The dips are not a nuisance #

It would be easy to read the dips as contamination — the angles nobody means when they say “a spiral head” — and to sweep only the irrational ones. That would be a mistake for two reasons.

The first is that the rationals are dense. Between any two angles there is one with a small denominator, so a sweep restricted to “the irrational ones” is a sweep over a set with no interval in it, and every plateau in the figure is bounded on both sides by dips it would have to exclude.

The second is that the dips carry the content. What makes μ₂ an arithmetic quantity at all is that a rational divergence produces a lattice of rays; the plateaux are the tessellations between those lattices, and their heights depend on which rays the arrangement is nearly on. A description of the curve that omits the dips omits the mechanism.

The dips were found rather than looked for #

The claim “every dip is at a rational” is only worth something if the dips were located without reference to the rationals, so it is worth setting out how they were.

dipsIn is handed the sweep — a list of angles and μ₂ values — and nothing else. It takes the median of the values, marks every run of consecutive samples below half that median, and reports the lowest point of each run. Nothing in it knows what a fraction is. Only afterwards is each dip handed to nearestRational, which searches denominators up to sixty and returns the closest.

That order matters and the first version of the function got it wrong in an instructive way. It looked for local minima — samples lower than both their neighbours — which is the obvious implementation and which fails at both ends of the head-size range. On a sweep at three hundred points, where a dip is a fiftieth of a degree wide and spans ten samples of a fine grid, no sample inside a dip is lower than its neighbours, so a local-minimum test finds nothing at all on a sweep with three obvious dips in it. On a sweep at nine hundred points with the same grid, the dips are too narrow for the grid to land in and the local minima it does find are the corners of the staircase — which is how a dip came to be attributed to 23/60 at a twelfth of a degree away, and how the error was caught.

The run-based test finds dips at both resolutions, and the attribution then holds: every dip is within its own width of a fraction with a denominator at most sixty.

Denominator ordering, and what it is not #

The dips are ordered by denominator: 3/8 is deeper than 5/13, which is deeper than 8/21, which is deeper than 13/34.

It is worth being careful about what that does and does not say. It says the number of seams in a q-ray arrangement grows with q, which is straightforward — more rays, more places where a ray’s territory meets its neighbour’s at an awkward angle. It does not say that the ordering is a clean function of q across all fractions: 3/8 and 4/11 have different numerators and similar denominators, and nothing here measures whether the numerator matters.

Nor does the ordering license the step everybody wants to take next, which is that “more irrational” means “more disordered”. That step has its own essay and it is refuted there. The ordering is a statement about the dips, which are a set of measure zero in the angles; between them, the staircase does what it does for reasons that are about the contact graph rather than about approximability.

Why nobody had swept it #

It is worth asking why a sweep this cheap had not been run, here or elsewhere, because the answer is not that anybody was careless.

The statistic comes from the cellular-tissue literature, where it is a property of a specimen — this epithelium, this foam, this grain structure — and there is no parameter to sweep: the tissue measured is the tissue to hand. Importing it into a phyllotaxis setting silently introduces a knob that the original use does not have, and the knob is the divergence angle, which is exactly the quantity this whole subject is about.

The earlier work imported it and measured six arrangements, which is what the original use suggests: a table of specimens. Six carefully chosen point sets is a good design for the question “does the mean stay at six while the second moment does not”, and it answered that question decisively — a factor of seventy-nine on the same cells with the same rim cut.

It is a bad design for the question “what does μ₂ depend on”, and the two questions look alike. That is the general shape worth carrying away: a statistic borrowed from a field where it has no free parameter arrives with the habit of being reported rather than swept, and the habit comes with it even when the new setting supplies a parameter that varies over exactly the range the subject cares about.

What a measured μ₂ is a measurement of #

The practical upshot for anybody who reports one.

A single value of μ₂ from a real tissue is a measurement of the plateau that tissue’s arrangement sits on, not of its divergence angle. Two specimens with the same μ₂ may have angles a tenth of a degree apart or a hundredth; two specimens with angles a hundredth apart may be a step apart in μ₂ if the step falls between them.

So μ₂ is a poor instrument for the divergence angle and a reasonable one for the arrangement, which is what it was always a statistic of. Where it has been used in the cellular-tissue literature — as a measure of how ordered an epithelium is — that is the right use, and the sweep says nothing against it. What the sweep says is that comparing two μ₂ values as though they located two angles on a smooth landscape is comparing two step heights, and the landscape has no slope to locate anything on.

The next essay measures the one property of the staircase that turns out not to belong to the angles at all: the width of a dip, which halves twice for every doubling of the head, and therefore says which rationals a head of a given size can see.

One line for the ledger #

The sweep costs thirty milliseconds a point and four hundred points is twelve seconds, which is less than a single one of this site’s grown-stem ensembles. It had not been run for a long time, and it corrects a reading in the work immediately before it.

That is worth recording as a habit rather than as an incident. Any statistic this collection imports arrives with a parameter the original use did not have, because the original use is about specimens and this one is about a family of arrangements indexed by an angle. Sweeping the parameter is the first thing to do with such a statistic and it is cheap in every case so far — the packing criteria, Lewis’s slope, the Aboav relation, and now the second moment. Three of those four turned out to be won by rational angles when swept, and the fourth is this one.