The second moment is the measurement

An earlier essay established why the average cell in a planar tiling has six sides: Euler’s formula for a planar graph, counted two ways, forces it as the tiling grows, and no preference of the cells enters. It is a theorem about graphs, not a fact about tissue, and the honeycomb efficiency argument is a different claim about a different thing.

That essay stopped there, which was the right place to stop for the question it was asking. This one asks the next question: if the mean is forced, what should a measurement of a tissue report instead?

Six arrangements, one mean #

Build six sets of nine hundred points. Two irrational divergence angles that produce spiral lattices, two rational angles a hair away from them, one rational angle that produces a whorled pattern, and one set with no rule in it at all. Tessellate each, cut the rim off, and count sides.

The mean number of sides per cell:

They agree to about one per cent. A set of points with no structure whatever averages the same number of sides as a lattice built to five decimal places, because both are planar tilings and a planar tiling cannot average anything else.

The same six, one statistic that varies #

Now measure $\mu_2$, the mean squared departure from six — the standard disorder measure for a cellular tissue, and the quantity Aboav–Weaire is a statement about — a quantity two cell laws make opposite demands on:

A factor of eighty between the ends, on arrangements whose means agree to a per cent — at nine hundred organs. The factor grows with the head: it is twenty-five at three hundred organs and nearly seven hundred at eight thousand, because a lattice’s exceptions sit on rings or at its centre while a random tissue’s are everywhere.

That is the essay in one comparison. The statistic a tissue paper reports is the one the topology has already decided; the statistic that distinguishes a lattice from a random scatter of points is one step further into the same distribution, and it costs nothing extra to compute because it is the same side counts.

Two disorder statistics, opposite verdicts #

There is a second measure of disorder already in this collection, and setting it beside $\mu_2$ shows that the two do not agree about which of these six arrangements is the most ordered — which is the strongest argument there is for choosing a statistic deliberately.

The other measure is metric rather than topological: how close the nearest pair of points come, as a fraction of the mean spacing. By that reading the golden angle is the best of the six and a rational divergence is poor, because a rational angle stacks points onto rays and leaves gaps between them.

By $\mu_2$ the order reverses. The whorled 144° arrangement scores 0.023 against the golden angle’s 0.253 — eleven times more regular — and 99 per cent of its cells are hexagons.

Both numbers are correct and they are measuring different things. A rational divergence gives every cell the same neighbours, because the pattern is a small number of identical rows repeated, so its topology is nearly perfect; what it does badly is distance, since the rows are far apart and the points within them are not evenly spaced. The golden angle does the opposite.

So a paper reporting $\mu_2$ alone would rank a five-rowed head as the most ordered tissue in the set, which is the reverse of every claim the packing literature makes about the golden angle — and neither side would be wrong. The statistic decides the question, and reporting one without the other is how two literatures come to disagree about a set of points they both have in front of them.

The practical form is short: a tissue measurement should carry a topological moment and a metric one, and they should be reported side by side rather than summarised.

Why the mean cannot help, said precisely #

It is worth being exact about what “forced” means, because it is not that the mean happens to be six.

The derivation is the earlier essay’s and is not repeated here. What matters for this one is its shape: counting edges two ways in a planar graph where three cells meet at each vertex gives a mean of $6 - 12/F$ over $F$ cells, which goes to six as the tiling grows and is six to a per cent by a few hundred cells.

Nothing in that derivation mentions where the points are. Move them anywhere — cluster them, spread them, put them on a lattice — and as long as three cells meet at each vertex the arithmetic is unchanged. The mean is a property of the plane and the number of cells.

What the arrangement controls is how the side counts are distributed about that mean, and $\mu_2$ is the first number that sees it.

Why the departure from six rather than the variance #

There are two second moments available and they are not quite the same, so the choice is worth a line.

The variance is the mean squared departure from the sample mean. $\mu_2$ is the mean squared departure from six. They differ by the square of the distance between the sample mean and six — which the theorem says is small, and which the table above confirms is at most 0.04, so the squared difference is under two thousandths.

On the numbers here the two are interchangeable to three decimal places, and the table would read the same either way. The reason to prefer $\mu_2$ is not numerical: six is the value the topology forces, so a departure from it is the physical quantity, and a departure from the sample mean is a departure from an estimate of a quantity already known exactly. Estimating something a theorem supplies, and then measuring deviations from that estimate, throws away a degree of freedom for nothing.

It also matters at the edges. On a small or heavily rim-cut sample the mean can sit appreciably away from six, and in that case the difference between the two moments is a measure of how badly the tessellation was extracted rather than of anything about the tissue. $\mu_2$ carries that information; the variance hides it.

How many cells it takes #

The statistic is worth a sample-size calculation, in the currency this collection prices everything else in.

$\mu_2$ is a mean of squared departures over $n$ cells, so its standard error is the standard deviation of those squared departures over $\sqrt{n}$. On a spiral lattice about a quarter of the cells are non-hexagonal and contribute 1 each, so the squared departures are mostly zeros and ones and their standard deviation is about 0.45 — giving a standard error near $0.45/\sqrt{n}$.

To separate a spiral lattice’s 0.25 from a Poisson set’s 1.83 needs essentially no cells at all: a dozen would do it. To separate the golden angle’s 0.253 from the 137.0° head’s 0.291 — a difference of 0.038 — needs about 1,100 cells, which is a head of fifteen hundred primordia measured to its rim.

That asymmetry is the useful output. The statistic settles “is this tissue ordered” on a fragment and cannot settle “which ordered angle is this” on a whole head. Since the second question is answered instantly by counting the spirals, the division of labour is clear enough: count for the angle, measure $\mu_2$ for the disorder, and do not expect either to do the other’s work.

The most hexagonal tissue is the least interesting one #

The row that should stop a reader is the first. At 144° — two fifths of a turn exactly — the tessellation is 99% hexagons and $\mu_2$ is 0.023, an order of magnitude tidier than the golden-angle head.

That angle produces a whorled pattern: five straight radial rows, with everything lined up. It is the arrangement the whole subject treats as the degenerate case, the one where the parastichy numbers share a factor and no divergence angle is recoverable from the counts.

So if “hexagonality” were a measure of how well a tissue is arranged, the best-arranged tissue in this set would be the one with the least structure in it.

That is the same inversion area evenness shows on a head’s interior, where the evenest cells belong to exact rational angles whose organs sit on rays while the gaps between the rays grow without bound. Two criteria, the same answer: a regular-looking local statistic is optimised by a pattern that is globally degenerate.

And it does not single out the golden angle #

The other row worth pausing on is the pair in the middle. The golden angle gives 0.253. A rational angle eight thousandths of a degree away gives 0.255.

No distinction. The head at 137.0°, half a degree away, gives 0.291 — a real difference, and not an optimum: it is simply a different arrangement with a slightly broader side distribution.

It belongs with the criteria that do not single the angle out, and the ones that do are of a different kind. Side counts and cell areas are readings of a tiling’s topology and of the radial law; the closest pair and the largest gap are readings of distance, and read on the head’s interior they put the golden angle above every rational neighbour on a half-degree grid and level with the other noble angles. So the general result is narrower than a run of negatives made it look: no criterion that counts sides or compares areas distinguishes the golden angle from its rational neighbours on a finite head.

The claim that survives is arithmetic — resistance to rational approximation — and the distance criteria are its geometric form: away from the centre, the closest pair a noble head keeps is Hurwitz’s constant under a square root. This measurement, being a count of sides, does not touch it either way.

Why the rim cut is not cosmetic #

One methodological point, because the site has been burned by it here before.

The measurements above use only cells inside 86% of the head’s radius. That cut was introduced in an earlier essay for a reason worth repeating: sixty boundary cells once inverted the sign of Lewis’s law on this site, giving a fitted slope of −0.63 where the true value is small and positive. The population responsible was a few dozen enormous five-sided cells one ring inside the convex hull.

Side counts have the same defect in a milder form. A cell near the boundary has fewer neighbours because the neighbours are missing, not because the arrangement put fewer there — so a rim cell’s side count is a measurement of the edge of the sample.

Including the rim would raise every $\mu_2$ in the table, and it would raise them unequally, because the arrangements differ in how much of the disc the outermost ring occupies. The comparison would then be partly a comparison of rim effects.

What a tissue paper should report #

Three numbers, and the first two are free once the cells are drawn.

The mean side count — as a check that the tessellation is well formed, not as a result. If it is not six, something is wrong with the tiling or the rim cut, and that is exactly what it is good for.

$\mu_2$ — the disorder, which is the quantity that varies and the quantity every relation in this field is implicitly about.

The distribution itself — because $\mu_2$ is a second moment and two very different distributions can share one. A golden-angle head is 13% pentagons, 75% hexagons, 12% heptagons; a Poisson set runs from three sides to eleven. Both are summarised by a single number and only one of them is well summarised by it.

That is a modest recommendation and it is not novel — the disorder measure is standard in the cellular-tissue literature and has been for decades. What is worth saying is the reason: not that $\mu_2$ is a better statistic in general, but that the mean is determined by a theorem and therefore cannot be evidence about anything, which is a stronger objection than “it is uninformative”.

The head half a degree away is the most disordered lattice #

One row has been passed over and it deserves its own paragraph, because it is the only lattice in the table that is worse than the golden angle rather than indistinguishable from it.

The head at 137.0° gives $\mu_2 = 0.291$ against the golden angle’s 0.253 — about 15% more disorder — and 71% hexagons against 75%. The difference is small and it is larger than the difference between the golden angle and its own rational neighbour, so it is the only comparison in the table that is doing anything.

The reason is visible in the contact network rather than in the side counts. At 137.0° the pattern’s families are 8, 13, 21, 29, 50, 71, 92, 113 — a sequence that adds 21 over and over rather than adding the two previous — and the essay two fields away sets out why. A lattice whose family orders grow slowly has many families of comparable strength at a given radius, so its cells are drawn from a broader population of neighbours, and a broader population of neighbours is a broader population of side counts.

So $\mu_2$ is not measuring “distance from the golden angle”. It is measuring something about the continued fraction of the divergence angle, and 137.0° scores worse than 137.5° for the same reason it produces a whorled-looking pattern: a large partial quotient. Whether that generalises — whether $\mu_2$ tracks the continued fraction across many angles — is a sweep this collection has not run, and it is the obvious next measurement on this statistic.

What this leaves open #

The measurement here is of arrangements the site builds. Whether a real tissue’s $\mu_2$ distinguishes anything is a question about specimens, and the honest answer is that this collection has none.

What can be said is what a specimen would have to show. A real capitulum, if it is a golden-angle lattice of nine hundred organs, should give $\mu_2$ near 0.25 and about three quarters hexagons, and a larger one proportionally less. Substantially more disorder than that is either a genuinely irregular pattern or a tessellation taken from a poor image, and the two are separable by remeasurement. Substantially less is a whorled arrangement, and the counts would show it.

That is a check with a number attached, which is more than the mean has ever been able to offer.

The general shape, which is not about tissue #

Strip the subject away and the argument has a form worth carrying elsewhere.

A quantity is reported everywhere in a field. It is easy to compute, it is stable across specimens, and its stability is read as evidence that the systems being measured have something in common. Then someone works out that a theorem fixes it, and the stability stops being evidence of anything: it was never a measurement of the specimens at all.

The diagnostic is cheap and it is the question this essay is really recommending: what would have to be true of the sample for this number to come out differently? If the answer is “nothing that could happen in this system”, the number is a check on the apparatus rather than a result, and it should be reported as one.

Applied here it takes a minute. Move the points anywhere; as long as three cells meet at each vertex, the mean is six. There is no arrangement of primordia that averages five. So the mean cannot distinguish arrangements, and anything a paper concludes from its constancy across species is a conclusion about planar graphs.

This collection has now found the same shape three times in different clothes — a statistic invariant to shuffling, a statistic fixed by a theorem, and a parameter that stopped mattering above a threshold. In each case the reported quantity was real, correctly computed, and incapable of varying with the thing it was being used to describe.

Swept, rather than sampled #

The six arrangements above are six points on a curve nobody had drawn. Swept across the divergence angle, μ₂ is a staircase — flat in stretches, with sharp steps and narrow deep dips wherever a rational falls.

Which means the near-agreement recorded here was inevitable rather than suggestive, and the reading offered for it — partial quotients — is corrected in The disorder is a staircase.

A later note on measuring a dip #

This page treats μ₂ at an angle as a quantity. A dip in it is not a quantity but a comparison, and the work two after this one found that the comparison’s other half — the background — stops meaning anything at a large denominator, where the neighbouring rationals crowd.

The repair and what it recovers are in The background is not one sample.