A whorl that misses its share

The symmetry that is not there showed that jugacy is a property of the positions a counter cannot read and a rotation can: turn a bijugate pattern half a turn and it lands on itself, turn a wrecked stem counted 2/6 half a turn and it lands nowhere. The rotation is the test, and it assumes the pattern being tested has exact symmetry to find.

A grown multijugate stem might not. The rule that grows one walks the ordinary ladder at every jugacy, but it places a whorl’s members one after another, each at the least repulsion from everything already there, its own whorl-mates included. Nothing in the rule says the members must end up a k-th of a turn apart. Whether they do is a measurement, and whether it depends on k is the question.

A rule with no instruction to be symmetric #

Each whorl is placed in order. Its first member goes to the azimuth of least repulsion from the whorls below. Its second goes to the azimuth of least repulsion from the whorls below and the first member, at the same height, and so on to the k-th. A member placed exactly on its share of the turn is a coincidence of that minimisation, not a constraint on it.

So the measure is simple. Sort a whorl’s members round the stem, take each gap between neighbours, and record how far the worst gap is from 360/k. A whorl that sits exactly on its shares misses by nothing; a whorl whose second member was pushed aside misses by the push.

A reason to expect a difference #

There is an argument for which jugacies should come out exact, and it is worth stating before any number because the numbers can then refute it.

A member lands exactly on its share of the turn only if that share is a mirror point of the members already placed — a position about which their repulsion is symmetric, so that nothing pushes the new member either way. The point opposite a single member is such a point, so a whorl’s second member of two lands exactly. The two points a quarter round from a pair of opposite members are mirror points of both, so a whorl of four fills exactly; an eighth round mirrors four members a quarter apart. But no point a third of a turn from a single member mirrors it — the member is a third of a turn away on one side and two thirds on the other — so a whorl of three has its second member pushed.

Carried through, the argument predicts exact whorls when k is a power of two, and a miss for every other k.

Seven jugacies, four lattices #

Whorls of two to eight members were grown at a fixed rise, a hundred and sixty whorls each and the last forty read, on a grid of 1,680 azimuths that every one of the seven jugacies divides — so no member is forced off its share by the grid. Each jugacy was read at the same four lattices: folded rises of 0.27, 0.18, 0.09 and 0.036.

Whorls of two, four and eight members miss their shares by nothing at every lattice. Whorls of three miss by up to 0.64°, 1.93°, 0.21° and nothing; of five by 1.29°, 1.29°, 0.21° and 0.21°; of six by 0.86°, 0.86°, 0.43° and 0.21°; of seven by 0.86°, 2.14°, 0.21° and nothing. Every jugacy that is not a power of two misses at the coarsest lattice, and every power of two is exact at all four. The prediction survives a test it could have failed at four jugacies.

Six members, which is two and three #

The whorl of six is the sharpest check on the argument, because six is divisible by two and by three. At a folded rise of 0.27 its gaps, in order round the stem, run 60.86°, 60.00°, 59.14°, 60.86°, 60.00°, 59.14°: the same three gaps twice.

That is the argument’s prediction exactly. A member and the member three places round from it are opposite each other, and the opposite point mirrors a member, so those pairs are exact and the whorl is symmetric under a half-turn. A third of a turn is not a mirror point, so within each half the members are pushed, and the whorl is not symmetric under a sixth of a turn. It is a bijugate arrangement of two identical, slightly lopsided triads.

Seven, which is prime #

Seven has no divisor to lend it a mirror point after the first placement, and its whorl shows no structure at all. At a folded rise of 0.18 its gaps round the stem run 49.29°, 50.36°, 52.29°, 51.86°, 52.50°, 53.14° and 50.57°, against an exact 51.43°. The widest is 1.71° over and the narrowest 2.14° under, which is the whorl’s largest miss, and no two gaps repeat.

Five behaves the same way at a folded rise of 0.27 — gaps of 72.64°, 70.93°, 71.57°, 72.43° and 72.43° against 72° — with only a chance equality between two neighbours. The argument does not predict how large a miss will be, only whether one exists and what symmetry survives it, and at both primes the answer is none.

Worked at a coarse rise #

The trijugate whorl at a rise per organ of 0.06 is the largest miss grown from rest, and its arcs can be read in the order its members were placed. From the first member to the second is 126.56°, from the second to the third 114.61°, from the third back to the first 118.83°. They add to 360.00°, as a whorl’s arcs must.

The second member has been pushed 6.56° further round than a third of a turn, exactly the direction the argument names: away from the first member, whose repulsion at the third-turn point slopes down away from it. The third member is then placed into what is left, and the arc it leaves back to the first member is 1.17° short of a third. At a rise of 0.02 the arcs are 121.64°, 119.30° and 119.06°, the same pattern a quarter of the size.

The push, seen #

The argument is about the repulsion a second member meets, and that can be drawn. Take a steady trijugate stem at a folded rise of 0.27, place the first member of a new whorl, and compute the repulsion the second member sees at every azimuth round the stem.

Its least value is not a third of a turn from the first member. The second member lands at 120.64°, pushed 0.64° further round, because the first member’s repulsion at the third-turn point has a slope there. On a quadrijugate stem at the same lattice the second member lands at 180.00°, opposite the first, where the first member’s repulsion is level. The same computation that places every organ on the stem puts one whorl exactly and pushes the other.

A trijugate whorl at each rise #

The trijugate miss is largest at coarse rises and vanishes at fine ones. Grown from rest at a fixed rise per organ, a trijugate whorl misses its third by 6.47° at 0.06, 3.52° at 0.05, 2.11° at 0.044, 1.64° at 0.04, 1.05° at 0.035, 0.70° at 0.03 and 0.47° at 0.025. At 0.02 it rises again to 1.64°; then 0.70° at 0.015, 0.23° at 0.01, and from a rise of 0.004 down it misses by nothing.

The reason the miss shrinks is that a whorl’s members are further apart, in units of the local spacing, as the rise falls: at a fine rise a member sits among many neighbours at nearly equal distances and its whorl-mates are a small part of what it feels. The bump at 0.02 is not explained by that, and it is real at both grids measured. Whorls of two and four members grown at the same rises are exact at every one.

Grown under a falling rise #

A stem whose rise falls carries its history, and the question is whether the history reaches the symmetry. At most rises it does not. A trijugate stem falling at T = 300 carries a mean miss of 1.078° near a rise of 0.035 against the fixed stem’s 1.055°; 0.703° against 0.703° near 0.03; 0.469° against 0.469° near 0.025. The slower stem at T = 900 agrees just as closely. Away from one band, the rise sets the miss and the route to it does not.

Near 0.02 the route decides. There the faster stem’s whorls miss by 7.01° on average and 8.44° at worst, four times the fixed stem’s 1.64°, and the slower stem’s by 3.07°, twice it. The faster the fall, the more of the miss survives, which is what a whorl still relaxing from the coarser arrangement above it would do. At 0.01 the falling stems miss by 0.47° and 0.34° against 0.23°, the band’s echo.

A step that lingers #

Below a rise of 0.004 a trijugate stem grown from rest is exact. A stem that fell to that rise is not: it keeps missing by one grid step, 0.234°, whorl after whorl, until whorl 337 at a rise of 0.00153 on the faster stem and whorl 1,006 at 0.00155 on the slower one. Only below that do its whorls close.

A single grid step is the smallest miss the grid can express, and a whorl stuck one step off is a whorl the placement rule has no reason to move: at a fine rise the repulsion differs by almost nothing between the exact share and the cell beside it. The history leaves the stem one step from symmetric for more than a decade of rise below where a stem with no history is exact.

Counted in grid steps #

A miss that small could be the grid’s own. So the trijugate stems were grown at the same four lattices on two grids, 1,536 and 1,680 azimuths, and the miss counted in each grid’s steps. It is 3 steps on both at a folded rise of 0.27, 8 and 9 at 0.18, 1 on both at 0.09 and nothing on both at 0.036.

In degrees that is 0.70° and 0.64°, 1.64° and 1.93°, 0.23° and 0.21°. The miss is a count of steps the grid does not change by more than one, which is what a real push resolved on two grids looks like; a grid artefact would scale with the grid and vanish at the finer one.

None of it reaches the ladder #

A trijugate stem at T = 300 grown twice — once with free whorls that miss by up to 8.44°, once with one member placed and the other two copied round so that every whorl is exact — makes its transitions at folded rises of 0.124147, 0.0465937, 0.0178404, 0.00683096 and 0.00261552 both times, to all six figures stored, with the same lag of 0.0166 rungs.

That is the same indifference the Lucas threshold showed: grown across its edge, a free trijugate stem that loses its seed has whorls missing their third by 182.8° while an imposed one stays exact, and both lose it at the same rate. The ladder and the branch are properties of the folded lattice; a whorl’s internal arrangement is not, and the rule’s counting never sees it.

What a rotation test would find #

The practical consequence is for the test the symmetry essay relied on. A rotation by a k-th of a turn maps an exact k-jugate pattern onto itself to rounding; a trijugate stem grown by this rule at a coarse rise maps onto itself only to within its miss, six or eight degrees at worst. A test with a tolerance tighter than that would call such a stem not trijugate — the opposite of the error a count would make on a wrecked stem, and a false negative rather than a false positive.

So a rotation test needs its tolerance stated against the rise, and the rise tells what it should be. At fine rises, where the stems here are exact or one step off, a tight tolerance is right. At coarse ones a trijugate stem grown one member at a time can miss by degrees and still be as trijugate as its lattice, its counts and its ladder say.

What a census would miss #

A census of counted pairs would not see any of this. A trijugate stem counts 3/6, 6/9, 9/15 whether its whorls are exact or pushed by eight degrees, which is why the bucket once labelled whorled could be opened up by counts alone and why a pair with a factor in it says nothing about how its organs arrived.

The same is true of the angle. The trijugate stems settle within three hundredths of a degree of a third of the golden angle, 45.8359°, while their whorls miss by degrees at coarse rises. The divergence between successive whorls is read from each whorl’s folded position, and whatever the pushes do to the members, the folded positions settle where a symmetric whorl’s would, to that three hundredths. The symmetry of a whorl is a property no counter and no divergence carries, and a rotation of the positions is the only instrument that sees it.

Sequential or simultaneous #

Everything here is a property of placing a whorl’s members in sequence. A rule that placed them together — each at the least repulsion from the whorls below only, with its whorl-mates ignored — would put every member on its share by construction, at every k. So the measurement distinguishes two kinds of rule, and a real multijugate apex could in principle be tested against it: exact trijugate whorls at a coarse rise would say the members form together; whorls of three pushed by a degree or more, and whorls of four exact, would say they form in turn.

Whorled organs are usually described as arising simultaneously, and that description is where this account stops being about the model and starts needing a specimen.

What this does not establish #

That a real apex places a whorl’s members one after another, at one height, against its whorl-mates. The results are for one repulsion law, falling as the cube of distance, one window of history and a grid of azimuths; a softer or longer-range law could push members by different amounts, and the parity pattern is predicted only for a rule that places members in sequence against a symmetric history.

It also does not establish the cause of the bump near a rise of 0.02, which is measured at both grids and on stems grown from rest, and is not explained here.

What would withdraw it #

A whorl of two, four or eight members that misses its share at any lattice measured, or a whorl of three, five, six or seven that is exact at every coarse one. A falling stem whose miss departs from the fixed stem’s by more than a grid step at the rises away from the band near 0.02. A miss in grid steps that changes with the grid by more than one step. An imposed-symmetry stem whose transitions differ from the free stem’s. Each is checked every time the measurement runs.

Still open: whether an exact whorl is where the rule returns #

The stems here start symmetric and are pushed. The next test runs the other way: a trijugate stem at a fine rise, where a whorl grown from rest is exact, seeded with whorls deliberately pushed several degrees off their thirds. Whether the rule closes them again, how many whorls it takes, and whether the one-step residue that lingers below 0.004 is where it stops, would say whether exact symmetry is something the rule restores or only something it happens to start from.