A spread of cell areas is not a growth law

Packing, measured against the interior cleared away a long-standing error about how evenly a golden head divides its area, and in doing so it found a formula. With organ $i$ at radius $i^p$, a cell’s area is proportional to $i^{2p-1}$ whatever the divergence, so the coefficient of variation of cell area over a head is a closed form in the exponent,

$$\mathrm{cv} = \frac{|2p - 1|}{\sqrt{4p - 1}}.$$

It rests on nothing but the arithmetic of how area is shared out, and it was checked against every reading of that essay. The final section then ran it backward. If area evenness is that function of $p$, then a measured spread of cell areas is a measured radial exponent. A spread of 0.10 means $p = 0.455$ or $p = 0.555$, and whether cells grow or shrink outwards says which. It went on to describe the test that would decide whether this held, a specimen whose placement law is known independently and whose cell areas were measured on the same head, and it said what a disagreement would mean: that area evenness was reading something other than the growth law.

That test is cheap to run on a head whose exponent is written down, so it is run here. Three readings of the same exponent are taken side by side, and each is then made to work under conditions a real specimen would impose.

Two heads with one spread #

The picture is the whole of the problem in one figure. On the left, a golden head of five hundred organs with an exponent of 0.6. It has no disorder at all: every organ is where the rule put it, and the cells vary in area only because they grow steadily from centre to rim. The shaded cells, those larger than the mean, lie in a clean outer band. The spread of its areas is 0.164.

On the right is a head with an exponent of exactly a half. Left alone, all its cells would be essentially equal and its spread would be nearly nothing. Every organ has been moved a fifth of a spacing, and its spread is now 0.176. The shaded cells are scattered over the head with no pattern, and there is no gradient to be found. As a matter of arithmetic the two heads report about the same number, and the proposed reading would call them about the same growth law.

What a spread allows #

Taken on its own terms the formula is excellent. Exact 900-organ golden heads, with the cells beyond 86 per cent of the radius cut away, read spreads of 0.253, 0.115, 0.012, 0.088, 0.166 and 0.295 at exponents of 0.4, 0.45, 0.5, 0.55, 0.6 and 0.7. The formula gives 0.258, 0.112, nothing, 0.091, 0.169 and 0.298. Those differ by little more than the rim cut, and at an exponent of a half, the cells’ small remaining spread of 0.012 is the head’s own lattice.

It does have a shape that matters. The curve has a minimum at a half, with one branch either side, so every spread allows two exponents. Read from the dashed line at 0.10 they are 0.455 and 0.555, the two the earlier essay named. A statement about which way cells grow picks between them, and that requires looking at something other than the spread. So the reading is already, even in its proposed form, a two-part measurement: a spread and a direction.

Three readings that agree #

Two other routes to the same exponent are available from the same specimen, and comparing them with the spread is the test the predecessor asked for.

The first is the trend. If a cell’s area is proportional to $i^{2p-1}$ and its radius to $i^p$, then area goes as radius to the power $(2p-1)/p$. Fit log area against log radius across the head and the slope gives $p$ directly, with the direction built in. It needs the cell areas and it needs their positions, and no more.

The second needs no cell at all. Rank the organs by their radius, so that the innermost is first, and fit the logarithm of radius against the logarithm of rank. On any head where radius increases with index the rank is the index, and the slope is $p$. A cell is not involved. It does require being able to find every organ, since a missed organ shifts the rank of everything outside it, but this is the independent measurement the predecessor described, and it needs no parastichy followed by hand.

On exact heads the three agree. Built at 0.4 the readings are 0.401 by spread, 0.398 by trend and 0.400 by rank; at 0.55, 0.548, 0.548 and 0.550; at 0.7, 0.698, 0.695 and 0.700. Taken this far, the essay that proposed the reading was right, and the formula runs both ways.

A head with no gradient #

The disagreement comes as soon as the organs move. Take a uniform head, with an exponent of a half, for which every reading should return a half, and displace every organ independently. By trend it returns 0.498 at every displacement up to a fifth of a spacing and 0.494 at two fifths. By rank it returns 0.500 throughout, with 0.502 at the largest. The spread returns 0.489 at two hundredths of a spacing, 0.477 at a twentieth, 0.457 at a tenth and 0.422 at a fifth, reading the lower branch.

So a tenth of a spacing is enough to report a growth law that is not there. The head has no trend from centre to rim, and on the question the formula is about, its correct answer is that its cells do not grow outwards at all. A spread cannot be read that way, because nothing in a spread says where its variation came from, and the spread of a tenth-displaced head is what the formula calls an exponent of 0.457 or 0.552.

This is not a small effect that better measurement would resolve. A displacement of a tenth of a spacing is much smaller than the disorder Lewis’s law needs to switch on at all, and the disorder that real tissues carry has no reason to be smaller than that. For any head with real variation in its cells, the spread is mostly reading that variation, and the exponent read from it would be wrong.

The same head with a real law #

A real law does not rescue the spread. On a head built with an exponent of 0.45, it reads 0.449 with nothing moved, 0.448 at two hundredths, 0.445 at a twentieth, 0.436 at a tenth, 0.413 at a fifth and 0.378 at two fifths. The law is there and the spread reports it, added to something else. The trend reads 0.448 throughout and 0.445 at the end, and the rank 0.450 throughout and 0.452 at the end.

Those numbers contain a useful regularity. Below about a fifth of a spacing, a head’s spread is close to what the law gives and what the displacement gives, added in quadrature. A uniform head displaced by $\sigma$ spacings has a spread of about $0.94\,\sigma$: 0.094 at a tenth and 0.188 at a fifth. Combined with the law’s 0.115 at an exponent of 0.45 that gives 0.149 and 0.220, and the head reads 0.147 and 0.217. At 0.55 it gives 0.129 and 0.208, and the head reads 0.130 and 0.211. By two fifths the rule has begun to fail, the displaced head’s own spread falling short of 0.94 times the displacement.

That is why this is not a defect in the spread so much as a mislabelling of it. The spread is a real measurement of how varied a head’s cells are, and that has two sources which add independently. To read either one from the total requires the other. A trend gives the law, and the law’s part of the spread can then be subtracted, leaving the displacement. So the two measurements together are worth more than the original proposal: fit the trend for $p$, compute what the spread should be, and what is left over is a measurement of disorder in units of a spacing. The spread is not a reading of the growth law at all, and it is a good reading of whatever is not the growth law.

How much of a spread is law #

The quadrature turns into a plain accounting of how much a measured spread owes to each source, and the accounts are not encouraging. A spread squared is a variance, so the law’s share is its own spread squared over the total. On a head with an exponent of 0.45, whose law alone gives 0.115, the head displaced by a tenth of a spacing reads 0.147, so 61 per cent of the variance is the law and 39 per cent is displacement. At a fifth it reads 0.217, and the law’s share is 28 per cent. On a head with an exponent of 0.55, whose law gives 0.088, the shares are 46 per cent at a tenth and 17 per cent at a fifth.

The size of displacement that matches the law is easily stated: the two contribute equally when $0.94\,\sigma$ equals the law’s own spread, which is at 0.12 of a spacing for an exponent of 0.45 and 0.09 for an exponent of 0.55. Closer to a half, where the law’s spread is small, it takes even less. An exponent of 0.48, which is a real and measurable departure from uniform density, has a law spread of 0.04 and is matched by a displacement of about a twentieth of a spacing. In other words, the spread is at its least informative exactly where a head is near constant density, and that is where most of the discussion of heads has been.

Where the trend fails #

The trend is not safe everywhere either, and this is the most surprising row in the table. Every reading of cell area depends on a cell having an area that means something, and on a rational head it can stop doing so. Built at $5/13$ of a turn with an exponent of a half and displaced by two hundredths of a spacing, the trend reads 0.467. At a twentieth, 0.414. At a tenth, 0.376. At $3/8$ of a turn it is worse: 0.365 at two hundredths and 0.303 at a tenth. Exact, both read within a thousandth of a half. A rational angle’s organs lie along rays, and on the exact head cells are all clipped by the same geometry. Displace them and each ray’s neighbours are separated by a sliver, while the gap between rays keeps growing outward, so the cell areas become long splinters whose size depends on the wedge they happen to fall in. The spread on those heads reaches 1.1 at a twentieth of a spacing and 1.6 at a fifth, far beyond anything the formula allows.

The rank is unaffected. On both rational heads it returns 0.500 at every displacement up to a fifth and 0.502 at two fifths. It reads where the organs are and asks nothing about the room around them.

That gives a clean order of how far each reading can be trusted. The spread is good only on an exact head, and it has an ambiguous answer even there. The trend is good on any head at an angle where the cells are sensible, whatever its disorder. The rank is good everywhere that the organs can all be found, and the transition radii are a fourth reading that needs only the counted pair, with no organ located individually.

What this does to the original claim #

The earlier essay’s account of area evenness stands. A head’s spread of cell areas under a stated radial law is what the closed form says, and that is the correction that essay was written to make: an evenness once credited to the divergence angle belongs to the radial law. What fails is the last section’s inversion, which said that a spread could then be read as an exponent, and which itself named this as the thing to test. It said that if the two exponents disagreed by more than a near-rational shear could account for, area evenness was reading something other than the growth law. They do, and it is. It is reading the growth law and the disorder together, and the disorder is usually larger.

So the plain answer to whether a measured spread of 0.10 means an exponent of 0.455 or 0.555 is no. It could equally mean an exponent of a half and a tenth of a spacing of displacement, and a spread alone has no way to separate them. A published value of a spread has no direct bearing on growth.

A reading of how much disorder there is #

The quadrature gives the spread a job it can do. Suppose a tissue’s cell areas are fully measured, as a reading of Lewis’s law would need in any case. Fit log area against log radius to get the trend and subtract its part of the spread, and what remains corresponds to about $0.94\,\sigma$ for independent displacement of $\sigma$ spacings. That gives a size for the organ-by-organ part of a head’s disorder without knowing any organ’s intended place, using only the same data a tiling already gives.

This does not rest on any particular cause of the disorder. A smooth field that moves whole regions of a head together varies the areas far more than the sides, so its spread would not obey the same factor, and a tissue that has divided since its organs were placed carries a different area distribution altogether. So 0.94 is a statement about independent displacement. A measured residual spread converts to an equivalent displacement, not to a mechanism, and anyone using the conversion should say so.

What this does not establish #

Heads here follow one power law from centre to rim, displacements are independent gaussian steps, and the rim is cut at 86 per cent of the radius in every reading. The rank is measured without any missed organs; a real specimen with a damaged patch would shift the rank of everything outside it, and a damaged centre would shift it most, so the rank’s robustness in this table is a robustness to displacement, not to loss. A trend would suffer less from the same loss. None of the readings has been tried on a head whose exponent changes with radius. The factor of 0.94 is measured at 900 organs on the golden angle and has not been tested at other sizes, and near a rational angle it plainly does not apply.

How this could fail #

A reading of the spread that held to a stated exponent on a head with a stated displacement of a tenth of a spacing, or more, would withdraw the main claim. A trend reading on a golden head that drifted with displacement by as much as the spread does would withdraw the order of the readings, and a rank that moved at the size of displacement that moves a trend would withdraw the rest. A displaced head whose spread did not combine with the law’s in quadrature, at sizes where each is well measured, would withdraw the last section.

Still open: two readings on one tissue #

The conversion from a residual spread to a displacement has only been run in the direction it was made. The test that would make it useful is a tissue disordered by some other route, a smooth field or a round of divisions, read for its trend, its residual spread and its Lewis slope. It would ask whether the equivalent displacement that the spread gives predicts where such a tissue sits on the plane of the two cellular laws, or whether, as with the two-number fit, it holds only for the route it was calibrated on. The prediction from what is already known is that it should fail for a divided tissue by a stated amount, and the size of that failure would be a number for how differently division varies a cell.