A whorl moves a ring out and multiplies it

Two at a time showed what a bijugate plant is: not a new kind of pattern, but an ordinary lattice wrapped twice. Every counter in these essays that reads a stem assumed its elements arrive one at a time, and for teasel and for a real minority of plants they do not, and folding such a stem’s azimuth back through a half turn gives an ordinary stem. What a cut costs a whorl and the symmetry that is not there both used that folding to tell a real whorl from something that merely counts like one.

All of that was done on stems. The most striking structure a head has, and the one measured most carefully here, has only been read at one organ a time. The defects lie on rings found the cells of a golden head that are not hexagons sitting on rings a factor of $\varphi$ apart, and a defect ring is where a family begins found what those rings hold: as many fives and as many sevens as the family that is losing its place in the lattice. If a jugate head is an ordinary lattice wrapped $k$ times, it ought to have those rings too, and the same argument should say exactly where.

The prediction, in one line #

A head’s local lattice is fixed by two things: how much azimuth an organ’s successive neighbours are apart, and how large the circumference is compared with the spacing. On a head of $k$ organs a whorl the rule repeats every $1/k$ of a turn, so a sector of $1/k$ of the head has the geometry of a whole ordinary head. Its circumference at radius $r$ is $2\pi r/k$. An ordinary head has that circumference at radius $r/k$.

So the lattice an ordinary head has at radius $r$ is found on a $k$-jugate head at radius $k\,r$. A ring of an ordinary head at radius $r$, holding $F$ fives and $F$ sevens, should appear on the jugate head at radius $k\,r$, and since each of the $k$ sectors has a copy, it should hold $kF$ fives and $kF$ sevens. Nothing is free in that statement.

Ordinary rings sit at 11.27, 18.24, 29.51 and 47.76 holding 21, 34, 55 and 89 of each. So a bijugate head should have rings at 22.5, 36.5 and 59.0 holding 42, 68 and 110 of each. A three-organ whorl should have them at 33.8 and 54.7, holding 63 and 102.

The heads #

Each head has organ $w$’s whorl at a divergence of the golden angle over $k$, with the $k$ members a $k$-th of a turn apart. Two versions are read. In the first, every member of a whorl shares one radius, which makes a rotation by $1/k$ of a turn an exact symmetry of the point set. In the second, each member sits at the radius $\sqrt{i}$ that its place in the sequence would give it, so that a whorl’s members lie very slightly apart in radius. That is what a rule that places organs one by one would do, and a whorl that misses its share showed the rule does not in any case hold a whorl’s members exactly a $k$-th of a turn apart. Both are tessellated as every head here has been, and every interior cell is counted by its neighbours.

The picture does the first part of the job by itself. At one organ a whorl, the marked cells make three thin rings close together. At two, they have spread apart: the innermost now lies where the second would have, and its pattern repeats twice round the head, a rotation by a half turn carrying every marked cell onto another marked cell of the same kind. At three, it has moved further out, and it repeats three times, a visible three-fold pattern in the marks.

Every count #

Read at eight thousand organs, the counts come out exact on every ring the rim leaves clear. At two a whorl, 42 fives and 42 sevens on the ring where the family is 21, 68 and 68 where it is 34, and 110 and 110 where it is 55. At three a whorl, 63 and 63 on the first, 102 and 102 on the second. At four and at five, only the first ring is clear of the rim, with 84 of each and 105 of each.

These are integer predictions and they are met exactly, which is a stronger result than any agreement to a percentage. It also gives the head the property that every ring holds as many fives as sevens, now through $k$ independent copies.

Every radius #

The radii are as close. The bijugate head’s rings lie at a mean radius of 22.6, 36.5 and 59.0, against the predicted 22.5, 36.5 and 59.0. The three-organ whorl’s lie at 33.8 and 54.7, four at 45.1, five at 56.4 against 56.3. So the ring is where the arithmetic says, to about the width of a cell or less, and the ring-to-ring ratio of $\varphi$ is unchanged on every head, since multiplying every ring’s radius by $k$ does not change their ratios.

That leads to a useful observation that bears on any reading of a jugate plant. The spacing of the defects along a ring does not change at all. An ordinary ring has $F$ pairs of defects on a circumference of $2\pi r$. A $k$-jugate ring has $kF$ pairs on a circumference of $2\pi k r$. So the spacing of 1.81 cells measured on a golden head should be the spacing on every jugate head too. Reading how densely a ring is populated tells nothing about the whorl. What tells is the count itself: the counts on a jugate head’s rings share a common factor, and it is $k$. The numbers 42, 68 and 110 have a greatest common divisor of 2, which no run of consecutive Fibonacci numbers has. So reading three rings from a head names its jugacy with no reference to any divergence.

A ring has one thickness #

The rings share one more thing, and it is the measure of how little the wrapping changes. A ring’s defects occupy a band of radius, and the band has a thickness that can be read directly: from its innermost defect to its outermost. On the ordinary head it is 2.40 units on the ring of 21 fives, 2.41 on the ring of 34, 2.42 on the ring of 55 and 2.43 on the ring of 89. Multiply the whorl by two or three and the rings are read at 2.39, 2.41 and 2.42, and at 2.40 and 2.41.

So a ring’s thickness does not change with its radius, its family or the number of organs in a whorl, and it is about one and a quarter spacings, one layer of cells. That is the same statement as the count from the other direction: if a ring were to get relatively thicker as a whorl wrapped the head, its cells could not stay at one count per sector. On every one of these heads the band of defects is the same width, and only how far round the head it runs has changed.

One scale for all of them #

The cleanest test of the claim that a sector is an ordinary head is not at the rings alone, since the whole staircase of defects should collapse, including what is between rings and what is close to the centre. Plot how many defects lie inside a radius, with both numbers divided by $k$, and the heads should become one curve.

They do. Each has a step at 11.27, at 18.24, at 29.51 and, where the rim allows it, at 47.76, of twice the ring’s family, the fives plus the sevens. Between steps each is flat, since the head between rings is all hexagons. The curves for two and three organs a whorl stop earlier, when the rim cuts them off. Each lies on the single-organ head’s curve to the width of a line, so no feature of an ordinary head’s defects is missing from a jugate head’s, and no new one has been added by the wrapping.

A head grown $k$ organs at a time has no defect structure of its own. It has $k$ copies of an ordinary head’s, each one sector wide, each scaled outward by $k$. That is the wrapped-lattice account of two at a time carried from a statement about pairs to a statement about every cell.

What the rotation requires #

The exact symmetry of the first head makes a clean test that a ring must pass. If each whorl’s members share a radius, a rotation by a $k$-th of a turn carries the head to itself, so every defect must have a partner of the same kind one sector on. At two and three organs a whorl every defect clear of the rim does, 100 per cent of them. At four and at five, 99.7 and 99.6 per cent, with the shortfall from cells near the cut.

Now take the same head with its members at their successive radii, $\sqrt{i}$. A member moves out by the radial step of a single organ, $1/2r$, which at radius 36 is a little over a hundredth of a unit, less than a hundredth of a spacing. It does not disturb the head in any way the eye could see. And that shift costs 18 per cent of the defects their partner at two a whorl, 21 per cent at three, 22 at four and 24 at five.

The counts do not change. On the staggered heads the rings hold 42 and 42, 68 and 68, 110 and 110, 63 and 63, 102 and 102, 84 and 84, 105 and 105, the same as on the exact heads. The mean radii do not change either. So a ring’s count survives a disturbance that breaks its rotation for a fifth of its cells, which is the same split that a ring’s count and its order showed under displacement. The count is about how many walls change hands in total, and the rotation is about which ones, and these have very different tolerances.

The reason a stagger that small does so much is now on record. Each wall near a ring lies within two over the ring’s count of changing hands, on a line from nothing up to that reach. At two a whorl, the ring with 68 fives has walls within about 0.03 of a spacing of flipping. A stagger of 0.007 of a spacing changes which side of the flip about a quarter of those walls fall on. The fifth of defects that lost a partner is about that size, and the two together have an arithmetic that could be followed further, though it has not been here. Each wall has its own comparison of two lengths and each comparison has shifted, so a head with even the smallest departure from a whorl’s exact symmetry does not have exact rotational symmetry in its triangulation at the rings, however faithfully its organs keep it.

So a test of rotation on a real jugate specimen is a demanding one in a surprising way. A stem that has been counted against its symmetry was tested by rotating organs, and that survives a small error. A test that asks whether the defect pattern of a head repeats would fail on a real head, at a tiny fraction of the stagger any real plant will have. Reading the rings’ counts and their common factor would not.

What jugacy does to the number of defects #

The second moment of a head’s side counts has been a focus of its own in these essays, and on a spiral head it is the share of cells on the defect rings. It would be natural to suppose that jugacy changes it in a simple way, more organs at a time making more defects or fewer. Read at eight thousand organs the share is 7.7 per cent at one organ a whorl, 9.8 at two, 8.8 at three, 12.3 at four and 9.3 at five, which is no pattern at all.

There is a reason for that, and it makes the absence of a pattern informative. The whole of what can change is which rings fit inside the head. An ordinary head of this size has rings up to 47.76, then a next at 77.3, right at the rim cut. A head at four a whorl has its ring of $4 \times 21$ at 45.1 and its next, of $4 \times 34$, at 73.0, which is inside the cut and counted, and since a ring is a large cohort of defect cells the share jumps at each entry. So a share is a statement about where the rim falls, between entries. The same is true of the staircase of the second moment against angle at fixed head size, and it is a reminder not to read much into a share of defect cells on a head of one size.

On the other hand the number of rings a head has at a given size does change with $k$, and in a known way. Since rings lie at $k$ times the ordinary radii, a head of a fixed number of organs has fewer rings as $k$ grows, one fewer for about each factor of $\varphi$ in $k$. A real plant has a head of a certain size and a jugacy of two or three, and that means one ring fewer to read than an ordinary head of the same number of organs would have.

What this does not establish #

These are exact heads, with an angle chosen rather than settled on. A real bijugate head grown by a placement rule has whorls that do keep a power of two exactly and others that miss, and nothing here measures a head grown that way. Its rings would need a disc grower, which does not exist here yet. The stagger tested here is the smallest there is, so nothing is known about how larger errors in a whorl spread through the rings, except by analogy with independent displacement. Only the golden seed has been read, at $k$ up to five, and the count law has been confirmed on noble angles only. And jugate heads are rarely as large as the eight thousand organs the outer rings needed, so a real specimen would have only a ring or two, and a common factor taken over two numbers is weak.

How this could be wrong #

A ring of a jugate head holding a count other than $k$ times its family, on a head with every organ placed exactly, would mean a sector was not an ordinary head. A radius off by more than a cell would mean the same. The curves of defects against radius failing to collapse anywhere, inside the rings or between them, would mean that something about the wrapping had a structure of its own. Any of these would be strange, since the wrapping is exact by construction, and that strangeness is why they were worth checking rather than assuming.

Still open: a head with a whorl that misses #

A whorl of three grown by the rule misses its third of a turn by up to 6.5° at a coarse rise and hardly at all at a fine one. That is a far larger disturbance to a whorl’s symmetry than the stagger here, and it has a particular structure: it is largest early, and it fades. The next measurement places a head with each whorl’s members offset by that measured miss, a stated function of radius, and asks where along the radius the defects begin to keep their partners, whether a ring’s count stays exactly $k$ times its family while its members disagree by several degrees, and whether the point at which partners return can be read as the radius at which the whorl has settled to its exact share. If so, the defects of a head would date the settling of its whorls, which is not something any count can do.