What lies between the steps

A seed head’s cells average six sides because Euler’s formula leaves them no choice, and how far they spread from six — the mean squared departure, $\mu_2$ — is the number that carries the information. Swept across the divergence angle, $\mu_2$ is not a smooth curve but a staircase, flat in stretches with abrupt steps and dips at the rationals. Nothing in the staircase moves read it again at four head sizes and found it one fixed object: fifty-two of the fifty-three features present at a smaller head are still there at the same angle at the next size up, and a bigger head adds steps between them — 4, 18, 31 and 40 at 539, 900, 1409 and 3690 organs.

That count gave no sign of flattening, and it raised the question the essay could not answer at its own resolution. If a larger head always resolves more steps, the staircase might have structure at every scale — something closer to a devil’s staircase, with a step at every rational — and the interesting quantity would be the law relating a step’s size to its denominator rather than any count. Or the steps might stop, and the staircase at each size be a finite thing. The essay named the test: take a gap between two steps at the largest head, sweep it at a hundredth of the grid, and see whether new steps appear inside it.

Where the gaps are #

At 3,690 organs the staircase was read every 0.005° from 137.0° to 138.2°, and a step is a place where $\mu_2$ jumps by more than a fifth of the curve’s own median between two samples. The forty steps that rule finds leave gaps between them of 0.03° at the median, and the two widest run from 137.1975° to 137.2725° and from 137.6725° to 137.7425°. Both were read again every 0.00005° — 1,500 and 1,400 samples — and so was one step, the one beside 21/55 at 137.4325°, across a window a hundredth of a degree wide.

The reading needs one fact about $\mu_2$ at a fixed head. It is a mean over cells of the squared departure of each cell’s sides from six, so it does not change at all until some cell gains or loses a side, and every change in it is a whole number of cells. The staircase at a fixed head size is therefore exactly a staircase — flat between flips — and the question is only how many flips there are and how large. Where the fine sweep’s samples fall on the coarse grid, fifteen of them in one gap and fourteen in the other, they give the coarse staircase’s values exactly: this is the same curve read more often, not a different reading.

No step inside either gap #

The first answer is the one the question asked for, and it is no. Of the 735 changes in the wider gap and the 489 in the other, not one moves $\mu_2$ by a fifth of its median; the largest in either is under a tenth. And a coarse grid can miss something without it being small: a dip that falls and recovers within 0.005° would sit between two coarse samples and never be seen. None does. At a hundredth of the grid, both gaps are as empty of steps as the coarse reading said.

So the staircase at 3,690 organs has a floor, in the sense the question meant. The forty steps are the steps; a finer grid does not add more between them. What grows with the head is the resolution of the steps, not an endless supply of them at every scale.

What the gaps hold instead #

Empty of steps is not the same as flat. The wider gap holds a sawtooth: $\mu_2$ falls from 1.0 of the median to 0.5 across it, and it gets there by climbing and falling in a regular pattern. It climbs a cell or two at a time — 216 climbs, averaging 2.4 cells, none larger than eight — and it falls in teeth: 178 falls averaging 4.0 cells, of which thirteen move ten cells or more, the largest twenty-three. Every change of ten cells or more in the wider gap is a fall. The other gap is the same shape with larger teeth: twenty falls of ten cells or more, the largest thirty-two, against two climbs that large, both at its very end.

The coarse grid saw the sawtooth and could not name it. Its samples, every 0.005°, land on the teeth’s slopes and floors and trace a curve that wanders downward, which is exactly what the gap looked like before; no two of them differ by a fifth of the median, because a tooth is a tenth or less.

A tooth close up #

Read in its own window, a tooth is as sharp as the fine grid can make it. Between 137.2635° and 137.2670° $\mu_2$ steps up by a cell or two at a time, falls twice by exactly twenty-one cells, and starts climbing again. The falls come at 137.26517° and 137.26528°, about a ten-thousandth of a degree apart, and together they take 0.106 of the median off $\mu_2$ within a fifth of a thousandth of a degree. Five of the wider gap’s thirteen teeth are falls of exactly twenty-one.

Twenty-one is a Fibonacci number, and so is fifty-five, which will come up at the step. A head’s spirals come in families of Fibonacci counts, and a set of cells that change together at one angle, one on each spiral of a family, would move by exactly that family’s count. The measurement here counts the cells and does not trace which cells they are, so the attribution is a reading of the numbers rather than something shown.

How large one flip can be #

A limit on the teeth comes from the tiling itself. Every corner of a Voronoi tiling of a head is shared by exactly three cells, and the only way its topology can change as the angle turns is an edge flip: two cells that shared an edge stop sharing it, and the two at its ends start to. Each of the four loses or gains exactly one side. At every one of the 2,900 samples in the two gaps every measured cell has five, six or seven sides, so a side gained or lost moves a cell’s squared departure by exactly one, and one flip moves the summed squared departure from six by at most four. A climb of one or two cells is a single flip, or a flip beside the rim; the wider gap’s largest climb, eight, is two flips at once. A fall of twenty-one is at least six flips inside one sample of 0.00005°.

So the teeth are not large single events of a kind the climbs are small versions of. They are coordinated: six or more edge flips, in different parts of the head, happening within a twenty-thousandth of a degree of each other, and the same count recurring from tooth to tooth. The climbs are flips happening one at a time.

Small flips go both ways, large ones fall #

Sorted by size, the flips divide cleanly. In the wider gap, changes of one or two cells go both ways, 178 climbs against 111 falls; changes of three to nine favour falls, 38 against 54; and of ten to twenty-nine cells there are no climbs at all and thirteen falls. In the other gap the pattern is the same and sharper: 57 climbs of one or two cells against 9 falls, and at ten to twenty-nine cells 2 climbs against 19 falls, with one fall of thirty-two.

That is why each gap slopes downward overall — the wider from 396 cells’ squared departure at its start to 206 at its end, the other from 508 to 280 — and why it does so in teeth. The head’s disorder is removed in coordinated falls and restored in single flips, and across a gap the falls win.

The two gaps compared #

The two gaps are not copies. The wider holds 394 interior flips, most of them single; the other only 122, but with teeth half again as large on average, falls averaging 10.3 cells against 4.0. It also sits nearer the rim’s influence: 367 of its 489 changes are cells crossing the rim cut, against 341 of 735 in the wider gap. Neither holds anything near a step. The difference between them is how their disorder is shed — in many small pieces or in fewer larger ones — and the shedding in both is in teeth that stop short of a tenth of the median.

What a step is made of #

The step at 137.4325° is a rise: between the staircase’s samples at 137.430° and 137.435°, $\mu_2$ climbs by 110 cells, 0.294 of the median, which is why the coarse rule counts it. Read at a hundredth of the grid it is not one event. It is two runs of flips a thousandth of a degree apart, each adding exactly fifty-five cells — the first in twenty-one changes, the second in thirteen — and within a run the cells arrive several at a sample, nine to eleven in the first and four to seven in the second. The largest single change anywhere in the step is eleven cells. No change in the window reaches a fifth of the median.

So a step and a tooth differ in kind rather than only in size. A tooth is a fall of a family’s worth of cells within one fine sample. A step is a steep run of flips — cells entering a defect ring one sample after another over a ten-thousandth of a degree or two — adding up to a family’s worth, and at a rational two of them. The coarse rule catches the runs because they are steep enough to put a fifth of the median between two samples 0.005° apart, and misses the teeth because they are not large enough, however sharp.

How many steps depends on the line #

The fifth of the median is a choice, made on the smallest head so that its nearly smooth curve had any features at all and the largest head’s sampling texture did not count as structure. Move the line and the count moves with it. In the wider gap, 394 interior flips exceed half a hundredth of the median, 104 a hundredth, 17 two hundredths, 6 a twentieth and none a tenth; in the other, 117, 53, 26, 4 and none. The count falls steadily as the line rises and runs out before a tenth, well short of the fifth.

That gives the step count its proper reading. There is no natural size at which a flip becomes a step: the teeth and the climbs form a continuum from one cell upward. But the continuum ends. Nothing in either gap reaches a tenth of the median, and what the staircase counts as a step lies beyond the end of it — steep runs, not large flips. The count of forty at this head is the number of places where $\mu_2$ moves steeply enough to clear a fifth of its median in 0.005°, and it is finite for that reason.

The nearest misses #

The gaps are gaps by a margin, and the margin is small. Of the wider gap’s fourteen coarse intervals, the one from 137.2225° to 137.2275° changes $\mu_2$ by −0.190 of the median, a hair inside the fifth that counts as a step; read finely it is thirty-nine flips, moving ninety-three cells’ departure one way and the other for a net fall of seventy-two. In the other gap the largest, from 137.6825° to 137.6875°, is −0.183, and it is five falls and nothing else, sixty-eight cells in teeth inside one coarse interval. A threshold of 0.18 would have called both of them steps, and the staircase at this head would have had forty-two.

Across the wider gap as a whole each coarse interval holds about twenty-eight flips, and their summed sizes are 3.4 times the net change they produce: the curve inside a gap is busy, climbing and falling many times for every step’s worth of net movement. In the other gap the ratio is 1.8, because more of its movement is in teeth that all go the same way. What the coarse grid reports as a gentle slope is, finely, a great deal of activity that mostly cancels.

Half the changes are the rim #

The readings above are of interior flips, and they had to be separated out. $\mu_2$ is measured over cells inside 0.86 of the head’s radius, and as the angle turns, cells cross that cut. Of the wider gap’s 735 changes, 341 are a cell crossing it — 227 of them changing no side count at all, only the number of cells the mean is taken over — and in the other gap 367 of 489. The staircase’s steps are not rim effects, which was the whole of the last essay’s result, and a reading of their fine structure has to hold the same line or it would be counting the cut instead of the head. Counted with the rim crossings, the wider gap would have 735 “flips” rather than 394.

What this says about the staircase #

It says the staircase at a fixed head is finite, with the finiteness located precisely. Between its steps it has a texture of single-cell climbs and family-sized falls that no step threshold should count, and its steps are the places where a family’s worth of cells enters a defect ring over a ten-thousandth of a degree. As the head grows it resolves more of those places, and none of them moves; that is where the growing count of steps comes from.

It does not say the staircase has no structure at smaller scales in the limit of an infinite head. A larger head has more spiral families, larger ones, and narrower dips at higher denominators, and a head of ten thousand organs read the same way might have steps where this one has teeth. What it establishes is narrower: at 3,690 organs, the gaps the staircase leaves are genuinely empty of what it calls a step, at a resolution a hundred times finer than it was read at.

What this reading does not establish #

Two gaps and one step are three windows out of 1.2 degrees, and the widest gaps were chosen because they were widest; a narrow gap near a crowded rational might hold something these do not. The grid is 0.00005°, so a run of flips that took place inside a single fine sample would read as one event, and the teeth could themselves be very steep runs rather than simultaneous flips — at this resolution the two cannot be told apart. And the attribution of twenty-one and fifty-five to spiral families is by count alone.

Readings that would overturn it #

A change inside either gap, read at this resolution, of a fifth of the median. A dip that falls by that much and recovers inside 0.005°. A climb of ten cells or more in the wider gap. A step that, read finely, is a single event of more than a fifth of the median. Each would mean the finite staircase described here is wrong.

Still open: the step’s size and the family #

The step beside 21/55 is two runs of fifty-five cells, and the wider gap’s teeth are mostly falls of twenty-one. If each run is one cell per spiral of a family, then a step’s height should be predictable from which family’s spirals cross the ring at that angle, and the steps at other rationals — beside 13/34, beside 8/21 — should be runs of thirty-four, or twenty-one, cells. The next measurement reads every step at 3,690 organs at this resolution, divides each into its runs, and asks whether every run’s size is a spiral count of the head, and whether the count is the one the rational it sits beside predicts.