A second moment that goes to zero

The mean number of sides in a tessellated head is forced to six, so it cannot tell one arrangement from another. The second moment can: $\mu_2$, the mean squared departure of a cell’s side count from six, is 0.023 on a whorled head at 144° and 1.830 on a set of random points, a factor of eighty between arrangements whose means agree to a per cent.

Every number in that comparison was read on heads of nine hundred organs. The factor has been repeated as a property of the arrangements, and it would be one only if $\mu_2$ did not depend on how many organs a head holds. Read at thirty-three head sizes, it does, and in three different ways.

What a constant factor would require #

For a factor of eighty to belong to the arrangements, $\mu_2$ would have to settle to a value on each of them as the head grows. There are reasons to expect it not to, and they can be stated before anything is swept.

On a golden head the cells that are not hexagons lie on rings at radii computed from the angle, each ring about a cell thick and holding a fixed number of fives and sevens. If that holds at every size, the number of exceptions grows with the radius of the outermost ring the head contains while the number of cells grows with its area, and $\mu_2$ falls as one over the square root of the organ count.

A whorled head’s exceptions do not form rings. If their number is fixed, $\mu_2$ falls as one over the count. And a Poisson set has no length scale at all, so a cell far from the centre is as irregular as a cell near it and $\mu_2$ should not move. Three predictions, three different laws, and a ratio between any two of them that cannot stay constant.

The sweep #

Golden heads at twenty-one sizes from 300 organs to 10,000, packed closely enough to catch the steps the ring picture predicts. Lucas heads at twelve. The whorled head, a rational divergence of 137.5° and a head at 137.0° at five sizes each. And the Poisson set at the same five sizes with three seeds each, the first being the seed the nine-hundred-organ table used.

Every point set is built the way that table’s was and tessellated and cut at 0.86 of the radius the same way, so at 900 organs each row reproduces its line of the table. The rim cut is kept fixed because a rim cell’s side count is a measurement of the sample’s edge, and a comparison across sizes is exactly where an edge effect would masquerade as a trend.

Five curves, three laws #

The figure is the whole answer on logarithmic axes. The Poisson set does not move: its $\mu_2$, averaged over three seeds, is 1.727 at 300 organs and 1.749 at 8,000, and lies between 1.73 and 1.83 at every size read. The whorled head falls from 0.070 to 0.0026. The golden head falls from 0.455 at 300 organs to 0.101 at 10,000, and the Lucas head from 0.362 to 0.086, both in teeth rather than along a curve. The 137.5° head falls with the golden one to 2,400 organs and then parts from it, reaching 0.045 at 8,000.

The dashed line through the figure is 6.83 over the square root of the organ count. The golden head comes back down to it every time it is about to take in a ring, and jumps away from it when the ring arrives.

On a spiral head the second moment is a count #

The first thing the sweep shows makes the rest simple. On all thirty-three golden and Lucas heads read, every cell that is not a hexagon has five or seven sides. There is not one four, one eight or one three among them.

A five and a seven each depart from six by one, so each contributes exactly one to the sum of squared departures, and $\mu_2$ is the number of exceptions divided by the number of cells — not approximately, but exactly, on all thirty-three heads. On a spiral head the second moment is not a measure of the shape of a distribution. It is a share: the fraction of the head’s cells that sit on its rings.

That is worth stating plainly because it is what makes the size law readable. A share of cells on rings is, near enough, a length divided by an area, and a length divided by an area cannot be a property of an arrangement. It is a property of an arrangement at a size.

The count moves in steps of twice a family number #

The golden head’s exceptions do not grow smoothly. At 300 and 400 organs there are 86. At 500 there are 120; from 600 to 1,100, 154; at 1,300, 209; from 1,500 to 3,000, 264; at 3,300, 366; from 3,600 to 8,000, 442; and from 8,600 to 10,000, 730.

The completed steps add 68, 110, 178 and 288, which are twice 34, 55, 89 and 144. Each step is one ring entering the rim cut and bringing its fives and its sevens, since a ring holds the family it keeps in each kind. The counts between are rings caught half-entered, and every one is the same way round. At 500 organs the 34 sevens of the ring keeping 34 are inside the cut and its fives are not; at 1,300 the ring keeping 55 has brought its 55 sevens and none of its fives; at 3,300 the ring keeping 89 has brought all 89 sevens and 13 fives. The sevens always enter first, which is the inner end of each five–seven pair crossing the cut before the outer end.

On Lucas heads the completed counts are 68, 126, 220, 372 and 618, and the steps add 58, 94, 152 and 246 — twice 29, 47, 76 and 123. They are Lucas numbers because the Lucas head’s families are, which is the control that says the steps count families and not Fibonacci numbers as such.

One more regularity holds on every completed count of both sequences: the fives outnumber the sevens by exactly six — 80 and 74, 135 and 129, 224 and 218, 368 and 362 on golden heads, and 37 and 31 up to 312 and 306 on Lucas ones. The rings are neutral, and the six is the handful of cells at the centre that Euler’s formula leaves over.

So the second moment falls as one over the radius, in teeth #

Multiply $\mu_2$ by the square root of the organ count and a law of one over the radius would come out flat. It comes out as a sawtooth with a flat floor. At 400, 1,100, 3,000 and 8,000 organs, the last sizes read before each ring enters, the rescaled value is 6.72, 6.84, 6.87 and 6.90: within three per cent across a twentyfold range of size. At 600, 1,500, 3,600 and 8,600 organs, the first sizes read after, it is 9.53, 9.90, 10.43 and 10.97.

Between entries $\mu_2$ falls as one over the number of cells, because the count is fixed and cells keep arriving; at each entry it jumps by the new ring’s share. The floor is the envelope, and the envelope is what the ring picture predicts. The rings sit a factor of φ apart and each holds a number of exceptions proportional to its radius — twice its family number, at a radius of about 0.54 times that number — so the total a head holds is a geometric sum set by its outermost ring.

The peaks drift from 9.5 to 11.0, and that drift should not be read as anything. A peak is sampled at the first size read after a ring completes, and how far past completion that size fell is a property of the sampling grid rather than of the head. The troughs were chosen as the last size read before each entry and happen to sit close to it, so their agreement is the tighter statement of the two.

Where the exceptions are, on the largest head #

On the 10,000-organ golden head the ring keeping 21, at a radius of 11.3, holds 42 exceptions; the ring keeping 34, at 18.2, holds 68; 55 at 29.5 holds 110; 89 at 47.8 holds 178; and 144 at 77.3 holds 288. The other 44 sit on the rings inside a radius of 9, where the rings run together, and no exception on the head is more than 1.23 units of length from its ring.

Between the rings there is nothing. The next ring out, keeping 233, lies at a radius of about 125, beyond this head’s cut at 86, so its 466 exceptions arrive only on a head of roughly twenty thousand organs, and until then the count stays at 730 while the cells keep coming. That is the whole mechanism of the sawtooth in one head.

Whorled and rational heads fall faster #

A whorled head at 144° has two four-sided cells and two five-sided cells inside the cut at every size read, and no other cell that is not a hexagon. The two fours contribute four each and the two fives one each, so the sum of squared departures is ten at every size, and $\mu_2$ is ten over the number of cells, exactly: 0.070 at 300 organs, 0.023 at 900, 0.0026 at 8,000.

The 137.5° head is a rational divergence, 55/144 of a turn, and it follows the golden head’s count — 86, then 154, then 264 — up to 2,400 organs, and then stops: 264 at 4,000 and 264 at 8,000. The closed form says why. The ring that would keep 89 sits at a rise that contains how far 144 turns of the divergence miss a whole turn, and 144 turns of 55/144 miss by nothing, which puts that ring at an infinite radius. With the count fixed, the rational head’s $\mu_2$ falls as one over the size from there: 0.155, 0.092, 0.045.

That is the source of the halving a rational’s floor shows when a head doubles. A rational head runs out of rings, and a fixed count over a growing head halves when the head doubles. A spiral head never runs out, and it falls at half the rate in the logarithm.

The factor of eighty is a head size #

Put the three laws together and the ratio moves with every doubling. The random tissue’s $\mu_2$ divided by the whorled head’s is 24.7 at 300 organs, 74.9 at 900, 212 at 2,400, 347 at 4,000 and 677 at 8,000. The eighty in the table was 79.3, read on a single seed at 900 organs; averaged over three seeds it is 74.9 at the same size. Divided by the golden head’s $\mu_2$ the ratio is 3.8 at 300 organs and 22.7 at 8,000, and divided by the rational head’s it is 3.8 and 38.7.

So a factor of eighty names a head of about nine hundred organs as much as it names two arrangements. At three hundred organs it is twenty-five, and at eight thousand it is nearly seven hundred. None of those is wrong and none is the factor; the question it answers does not have an answer until a size is attached to it.

The random tissue is the one arrangement that behaves as the table assumed. Its three seeds spread from 1.62 to 1.90 at 900 and 2,400 organs, which is wider than any trend across sizes, and they sit around the value computed for the planar Poisson–Voronoi tessellation, about 1.78. The spread between seeds is what a single-seed table cannot show, and at 900 organs it is worth a factor of 1.13 in the ratio on its own. A ratio whose denominator is one draw from a random process carries that draw’s luck in it, whatever size the head is.

What the nine-hundred-organ table still says #

The argument of that table survives every size. The mean is forced to six on every arrangement at every size read, so it separates nothing, and $\mu_2$ separates a lattice from a random tissue by a factor of at least 3.8 at every size read. The order at the ends survives too: the whorled head is the lowest and the random tissue the highest at every size from 300 organs to 8,000.

The middle of the order does not survive intact. The golden and rational heads agree to within two per cent up to 2,400 organs, as the table found; past the size at which the rational head stops adding rings, its $\mu_2$ falls to 59 per cent of the golden head’s at 4,000 organs and 59 per cent again at 8,000.

And no single $\mu_2$ can now be read as a property of an arrangement. The table’s own suggestion that a golden-angle capitulum should give $\mu_2$ near 0.25 is a prediction for a head of nine hundred organs. A head of 3,000 golden organs should give 0.125, and one of 8,000 should give 0.077. A measured 0.25 on a head of 8,000 organs would not be a golden head with average order; it would be a head more than three times as disordered as the lattice it resembles.

What a comparison of tissues now has to carry #

Three things follow for anyone putting two specimens’ $\mu_2$ side by side.

The organ count, always. Two values from heads of different sizes are not comparable without it, and the correction is not small: doubling the organs divides $\mu_2$ by about 1.4 on a spiral head and by two on a whorled one.

A size-free number where one exists. On a spiral head $\mu_2$ times the square root of the organ count lies between 6.7 and 11 at every size read, and that rescaled value is what two spiral heads of different sizes can be compared on — to within the sawtooth, which is itself worth a factor of 1.6.

And the steps as a check on the tessellation. A spiral head’s exceptions arrive in counts of twice a family number, so a count of exceptions that is not the few dozen central cells plus a sum of such counts is a tessellation with something wrong in it: a cell missed, a rim cell counted, a centre misplaced. A mean fixed by a theorem is a check on the apparatus rather than a result, and a count that must arrive in known steps is the same kind of check, one moment further into the distribution.

What this does not establish #

That a real tissue is a spiral head or a Poisson set, or that its cells are the Voronoi cells of its centres. The laws above are for model arrangements with every organ exactly where the divergence puts it, and positional disorder dissolves the rings before it undoes the pairing. A real head’s $\mu_2$ will fall more slowly than a lattice’s, by an amount this does not predict.

The rim cut is 0.86 of the radius throughout. A different cut takes in each ring at a different size, which should move the teeth along the size axis without moving the floor; that is argued here and not measured.

The head at 137.0° was read and not explained. Its counts, 89, 176, 218, 444 and 670, are not built from twice a family number in any way this reading can see, and it is the one arrangement in the table whose exceptions the ring picture does not yet account for.

And three Poisson seeds are enough to show that the spread between seeds exceeds any trend with size, but not enough to fix the random tissue’s value to better than a few per cent.

What would withdraw it #

A golden or Lucas head with a cell of four or eight sides inside the cut, or a $\mu_2$ that is not its exceptions over its cells. Thirty-three heads were read and neither occurred.

A completed step that adds anything but twice a family number. Eight steps across the two sequences were read, and each added exactly that.

A golden head read just before a ring enters whose $\mu_2$ times the square root of its organ count lies outside 6.5 to 7.1, or a Poisson set whose $\mu_2$ trends with size by more than its seeds spread.

Still open: which steps of the staircase are rings entering #

Swept across the divergence angle, $\mu_2$ on a head of nine hundred organs is a staircase with dips at the rationals, and every step of it was read at that one size. A spiral head’s $\mu_2$ also changes in steps as the head grows, and a step in angle at fixed size could be a ring crossing the rim cut as the angle moves it, rather than a change in the lattice.

The measurement that separates the two is the staircase read again at three or four head sizes chosen to fall between ring entries. A feature that stays where it is across those sizes belongs to the angle; one that moves with the head belongs to the rings. The same reading would test whether the 137.0° head’s unexplained counts are rings entering at a rate its own continued fraction sets.