A tissue that was never shaken

Two numbers for a tissue put sixty-nine tissues on a plane with one axis for each of the two laws cellular tilings are quoted for — Lewis’s law, that a cell’s area rises with its number of sides, and Aboav’s relation, that a many-sided cell is ringed by few-sided ones. No single number placed a tissue on both. The variance of the side counts with either law’s own statistic did, to within one and a half times the scatter of ten heads, and so a report of those two numbers fixed the third.

Every one of those tissues was made by moving points: a golden head’s organs carried by a smooth field, then displaced one by one, and the tiling read as the cells nearest each organ. That is one way disorder arrives. Growing tissue has another. A cell divides, a new wall crosses it, and the two cells the wall ends on each gain a side; nothing moves. One law counts sides, the other pairs noted that division makes its defects in bound pairs by construction, which is what the first independent steps did and the smooth field undid. The question left open was where division puts a tissue on the plane of both laws, and whether the variance with $a$ still fixes Lewis’s slope for a tissue that was never shaken at all.

A tiling no set of points makes #

A tiling read as the cells nearest a set of points cannot hold a division: the new wall runs through the parent cell’s centroid, not halfway between two points. So the tiling is held as a map. It starts from the same 900-organ golden head every tissue here has started from, centre left out. Each corner where three cells meet is identified once and shared by all three, and each cell is its ring of corners in order. A division cuts a cell by a straight wall through its centroid; the wall meets two of its edges, a new corner goes on each, and each new corner is also inserted into the neighbour across that edge, which therefore gains a side.

Undivided, the map reads exactly what the earlier reading of the ordered head read: 607 measured cells inside 0.86 of the radius, a variance of side counts of 0.2503, Aboav’s $a$ of 1.1771, a Lewis slope of 0.0092 and every five touching a seven, all to the seventh figure. The readings below are the same four numbers taken the same way, so a divided tissue and a moved one are measured by one instrument.

Three rules divide the head. A random cell may be cut on a random axis. A random cell may be cut by the shortest wall through its centroid — Errera’s rule, the oldest guess at where a plant cell puts its new wall. Or the largest cell may divide first, by its shortest wall, as a tissue would if cells divided on reaching a size. The first two are read on ten seeds; the third is deterministic. Cells within 0.86 of the radius are eligible, nothing grows, and every area is taken against the tissue’s mean.

One division, four defects #

A division on the ordered head is nearly always the same event. Tallied over all 664 cells the rules can divide, the shortest wall cuts a hexagon into two five-sided daughters and turns its two hexagonal neighbours into sevens 51 per cent of the time; a random axis does the same 44 per cent of the time. The daughters are nearly equal, their areas in the ratio 0.974 on average. So one division makes two cells of half the usual area with five sides and two cells of full area with seven, every five touching both sevens.

That single event carries both laws at once. Two small cells with few sides and two ordinary cells with many are a Lewis correlation in miniature — area rising with sides — and a five between two sevens is the strongest anticorrelation of neighbours a tiling can hold. A moved tissue builds each of those separately and gradually: displacement varies the sides a little and the areas a little and lets them come to track each other. A division makes them together, already coupled, in one wall.

A tenth of the way in #

At two per cent — twelve divisions across the whole head — the quartets are isolated: each is a short chain of five, five, seven, seven laid over a tiling of hexagons, with the golden head’s own defect pairs visible between them. At a tenth, sixty-one divisions, the quartets are a few cells apart and beginning to touch, and the tissue inside the rim has 665 cells where it had 607. Averaged over ten seeds, it has eleven cells with four sides or fewer and seventeen with eight or more; its variance of side counts is 0.57, its $a$ is 1.20, its Lewis slope 0.164, and 94 per cent of its fives touch a seven. At half, the quartets have merged into a tissue of chains and clusters with 901 cells, 69 of them four-sided or less and 91 eight-sided or more.

Through all of it the mean number of sides stays at six: 5.985 with a tenth divided, 5.957 at twice the cells, the small deficit being the rim, where a measured cell’s neighbours can lie outside the cut. Why six sides is a statement about counting edges, and dividing cells does not change how edges are counted.

Lewis’s law, a tenth of the way in #

The law switches on early. With a twentieth of the cells divided by the shortest wall the slope is 0.114, a whisker under the threshold of 0.1145 that the earlier essays took as half a random set’s slope; with a tenth divided it is 0.164, and by random axes 0.156. Dividing the largest cell first gets there at three twentieths, 0.143. After that every rule keeps climbing, to 0.38 by the shortest wall and 0.31 by random axes once twice the head’s cells have divided.

What matters is where on the variance it happens. A tenth of the cells divided by the shortest wall has a variance of side counts of 0.57; three twentieths divided largest-first, 0.51. Among the sixty-nine moved tissues the lowest variance at which Lewis’s law is on is 0.71 — a field of a third of a spacing and steps of a fifth. Lewis’s law needs the sides to vary, and on a moved tissue it needs them to vary a good deal before area follows. On a divided tissue area follows at once, because the cells whose sides a division lowers are exactly the ones it halved.

Aboav’s a, up and then down #

The two random rules draw the curve the independent steps drew, with a flatter top. By random axes $a$ rises from 1.18 to 1.23 at a twentieth to a tenth of the cells divided and falls to 0.42 at twice their number; by the shortest wall it rises to 1.21 and falls to 0.47. Both rise because early divisions make quartets, each a pair of bound five–seven pairs, and a tiling of bound pairs anticorrelates strongly. Both fall because later divisions land on cells that are already fives and sevens, or beside quartets, and the arrangement coarsens into longer chains and larger defects: by twice the head’s cells divided, the shortest wall has left 275 cells with four sides or fewer and 277 with eight or more.

Largest-first does something else. Its $a$ climbs to 1.41 with a tenth of the cells divided, 1.56 at three twentieths and 1.67 at a fifth, above the band where Aboav’s relation holds, before falling back into it. The reason is which cells are largest. On the golden head the seven-sided cells are the largest, 3.18 in area against 3.14 for sixes and 3.12 for fives, and of the first thirty cells the rule divides, twenty-six are sevens. So it divides the head’s own defects — which lie on rings — one after another, and a seven cut by its shortest wall with its neighbours rearranged about it pulls the defect rings into tight, strongly anticorrelated clusters.

A random axis makes more variance and less law #

The two random rules differ only in the axis of the wall, and the difference is steady. By random axes the variance of side counts climbs faster — 1.55 at half the cells divided against 1.33 by the shortest wall, 3.09 at twice the cells against 2.40 — while Lewis’s slope climbs slower, 0.240 against 0.274 and 0.306 against 0.382. Per unit of variance the shortest wall makes more law.

The single division says why. A random axis cuts a hexagon into a four and a six in 13.9 per cent of single divisions; the shortest wall does it in 2.9 per cent. A four and a six from one hexagon differ by two sides and hardly at all in area, the daughters’ areas standing in a ratio of 0.980 on average by random axes. So a random axis adds variance to the side counts that area does not follow, and the shortest wall, which almost always makes two equal fives, adds variance that area follows exactly. A law about area and sides is strengthened by the rule that keeps the two tied and weakened by the one that lets them come apart.

Largest-first switches the law off again #

Dividing the largest cell first does not keep climbing. Its slope reaches 0.208 with half the cells divided and 0.214 at three quarters, and then, with every cell divided once, falls to 0.098 — below the threshold — at a variance of side counts of 0.49. A size rule that always halves the biggest cell ends a round of division with a tiling of halves that are all nearly the same size again, and a tissue whose cells are equal in area cannot have area tracking sides however its sides vary. The second round starts the climb again, to 0.131 at one and a half times the cells and 0.201 at twice.

That is the only rule here with anything like growth in it, since choosing the largest cell is what a tissue that divides on reaching a size would do, and it is the only one whose Lewis slope goes down. It is a hint of what a growing tissue does and not a measurement of it.

The pairing holds, then loosens #

A division makes its defects touching, and the pairing share says so. With a twentieth of the cells divided, 96 per cent of fives touch a seven by random axes and 96 by the shortest wall, and largest-first keeps 96 per cent. The share falls only as divisions accumulate, to 61 and 63 per cent at twice the head’s cells: by then there are enough small cells that many fives sit among other fives, and every five being bound to a seven, which is a property of the ordered head, is no longer a property of the tissue.

Where division puts a tissue #

On the plane the difference is plain. The moved tissues fill a region; the two random division rules run across it and out of it, reaching slopes the moved tissues reached only when $a$ had already fallen, and going on to slopes of 0.3 to 0.38 with $a$ below 0.5, a region no moved tissue came near. They cross into Lewis’s law with $a$ still at 1.2, where the moved tissues crossed it with $a$ at 1.3 on the way down from the rise.

Largest-first goes where nothing moved went. The earlier essay found the corner with Lewis’s law on and $a$ above its band empty — the highest $a$ among moved tissues with the law on was 1.32 — and concluded that a tissue whose sides vary enough to make area track them has defects too many and too varied to keep the strong anticorrelation a large $a$ needs. Divided largest-first, the head sits in that corner twice, at $a = 1.557$ with a slope of 0.143 and at 1.666 with 0.147. The conclusion was right about moved tissues and is not a property of tissues. A route that makes its variation in sides by halving cells, and chooses which cells to halve by their defects, keeps the anticorrelation while it switches Lewis on.

The two numbers do not carry #

The earlier essay’s practical finding was that the variance of the side counts and Aboav’s $a$, both read from side counts without measuring a single area, place Lewis’s slope to within one and a half times the scatter of ten heads. That was measured on moved tissues and fitted on them. The test here is to carry the fit, unchanged, to tissues it has never seen.

It fails badly and in one direction. With a tenth of the cells divided by the shortest wall, the fit predicts a slope of 0.065 and the tissue’s slope is 0.164 — seventeen times the noise of 0.0058. By random axes it predicts 0.072 against 0.156, fourteen times. From a twentieth to half of the cells divided, both random rules sit above the prediction by more than ten times the noise at every stage, and largest-first by 3.7 to 27 times at every stage but one — the fifth divided, where it happens to land on the prediction. The moved tissues, against the same fit, sit at 1.5 times the noise.

So the two numbers were never a law about tissue. They were a calibration of a route: on tissues disordered by moving points, how much the sides vary and how their defects are arranged together fix how closely area tracks sides, because moving points couples those three in one particular way. Division couples them another way. A divided tissue at the same variance and the same $a$ as a moved one has two to two and a half times its Lewis slope, because its variation in sides was made by halving cells, and the cells with fewer sides are the smaller ones by construction rather than by drift.

What a report of two numbers is worth, again #

The earlier warning was that one number leaves a direction on the plane unmeasured. The correction here is that two numbers leave something unmeasured too, and it is the route. A tissue reported by its variance and its $a$ can be placed on Lewis’s law if it is known to have been disordered by movement; if it grew by division, the same two numbers underestimate its slope several-fold. A section whose cells cannot have their areas measured cannot have its Lewis slope inferred without knowing how it came to be disordered — or without some statistic that tells the routes apart.

That is also the reason to be careful with the second moment as the single number for how disordered a tiling is. It measures how far the side counts depart from six, and the same departure means different things for area depending on whether it came from moving points or from walls.

What these tissues leave out #

Nothing grows. A real tissue’s cells grow between divisions, so daughters approach their parents’ size before they divide again, and a tissue in steady growth has a distribution of areas this model does not: the halving is never undone here. Walls do not relax either; a new wall meets its neighbours where the cut lands and stays, where real walls shift towards the angles surface tension favours. The shortest wall through the centroid is one rule among several proposed for plant cells, and a wall placed through the nucleus or across the long axis of an elongated cell would divide differently. And every divided tissue starts from the golden head’s own ordered tiling, whose largest cells happen to be its defects — which is what made largest-first division special, and would not be true of a tissue that started from a random one.

Measurements that would overturn it #

An undivided map whose four numbers differ from the ordered head’s. A single division on the ordered head whose commonest outcome is not two fives and two new sevens. A divided tissue with Lewis’s law on at a higher variance than every moved tissue. A fit in the variance and $a$, made on moved tissues, that places divided tissues’ Lewis slopes within a few times the noise. Each would mean the reading here is wrong rather than incomplete.

Still open: a tissue that grows between divisions #

Division without growth halves every cell it touches and never restores it, so it builds a Lewis correlation into the areas that a growing tissue would partly erase. The next measurement lets each cell grow back towards the mean area between divisions — at a stated rate, relative to how often cells divide — and asks where on the plane a tissue in steady growth settles, whether it still switches Lewis’s law on at a variance lower than any moved tissue, and whether, once growth has had time to even out the areas, the moved tissues’ fit in the variance and $a$ starts to carry after all.