A Lucas seed counts whorls

A stem seeded on the Lucas lattice is on a real branch of the placement rule, and not the one it prefers. Hurried, it holds the Lucas ladder through three more forks at 99.5°; given room, it abandons it and walks to the Fibonacci ladder at 137.5°. Bisected on the rate from a start at a rise of 0.12, the Lucas seed was kept up to 87 nodes a rung and lost from 91.

A stem that grows two organs a whorl walks the ordinary ladder at half the rise between whorls, because folded twice round it is an ordinary stem at twice the rise. That identity predicts the bijugate threshold, but only after saying what the threshold counts. If an ordinary stem loses the Lucas branch because it has made enough placements per rung to probe its way out, a bijugate stem should lose it at the same organ rate. If it loses it because the folded lattice has had enough nodes per rung, a bijugate stem should lose it at twice the organ rate — the same number of whorls a rung.

Fourteen configurations #

Each stem is grown in exactly the design the ordinary threshold was found with, carried into folded units: forty whorls of Lucas lattice as the seed, a folded rise falling from 0.12 to 0.004, and a counting window of 26 folded nodes striding four. A stem keeps the seed if its last window is still on the Lucas ladder.

The configurations vary what could matter. Jugacy: one, two and three organs a whorl. The grid: 256 to 2,048 azimuths, always divisible by the jugacy. The symmetry: whorls filled freely, or one member placed and the others copied round. A gap: a whorl rise of empty stem left between the seed and the first grown whorl, or none. Each is grown at every rate across a coarse range and at every rate, or every second one, across its edge.

Every record is one edge #

Along the rate, every one of the fourteen configurations keeps its Lucas seed in an unbroken run and loses it in an unbroken run. There is no rate that keeps it between two that lose it, and none that loses it between two that keep it. So each configuration has a single edge, and it can be read to within one rate step.

The ordinary stem keeps the seed to T = 90 and loses it from 91, on grids of 256, 512 and 1,024 azimuths alike. The bijugate stem keeps it to 182 and loses it from 183, or from 184 on the coarser and finer grids. The trijugate stem keeps it to 274 and loses it from 276, or keeps it to 270 and loses it from 275 on its coarser grid.

In whorls a rung #

A rate T places 2·T·ln φ = 0.9624·T placements between one transition and the next. For an ordinary stem those placements are organs; for a stem of k organs a whorl, the whorls between transitions number 0.9624·T/k.

Converted, the ordinary edge is 86.6 to 87.6 whorls a rung, the bijugate edge 87.6 to 88.1 and the trijugate edge 87.9 to 88.5. The three brackets touch end to end within two whorls, and the last whorls a rung that keeps the seed are 86.6, 87.6 and 87.9 — the same to within one and a half per cent.

The finer scan also narrows the ordinary bracket. The bisection that first found it read 87 to 91 nodes a rung; stepping the rate one organ at a time puts the ordinary edge between 86.6 and 87.6, inside the old bracket and a quarter of its width.

How much stem the question is asked over #

The design fixes the range of folded rise, 0.12 down to 0.004, a factor of 30. Measured in factors of $\varphi^2$ that is $\ln 30 / 0.9624 =$ 3.53 of them, and every stem, at every jugacy, has to hold or lose the Lucas branch across the same three and a half rungs.

What changes with the rate is how much stem it lays down doing so. A stem of k organs a whorl at rate T makes T·ln 30 organs over the range, and T·ln 30 / k whorls: the bijugate stem at T = 180 grows 612 organs in 307 whorls, at T = 160 in 273 whorls and at T = 220 in 375. The first forty whorls of each are seed. At the edge an ordinary stem at T = 90 lays down 307 organs, which are 307 whorls — the same length of lattice as the bijugate stem at twice the organ rate, which is the whole result said in whorls rather than in rungs.

Worked at the bijugate edge #

The conversion is worth doing once by hand. The bijugate stem keeps its seed at T = 182: 182 × 0.9624 is 175.2 placements between transitions, and with two organs to a whorl that is 87.6 whorls. It loses it at 183: 176.1 placements, 88.1 whorls. The ordinary stem’s last kept rate, 90, is 86.6 placements, and on an ordinary stem placements and whorls are the same.

Kept means the same thing at every jugacy: the stem’s last counting window, at a folded rise of 0.004, is still on the Lucas ladder — 7/11 on the ordinary stem, 14/22 on the bijugate, 21/33 on the trijugate. That is the criterion the ordinary threshold was bisected with, carried into folded units, so the three edges are one measurement made at three jugacies rather than three measurements that happen to agree.

The placements prediction, refused #

Read in organs a rung instead, the three edges are 86.6 to 87.6, 175.2 to 176.1 and 263.7 to 265.6. The midpoint of the bijugate edge is 2.02 times the ordinary one and the trijugate edge 3.04 times. A threshold that counted placements would put all three near 87.

So the prediction that the Lucas branch is lost after a fixed number of placements is refused by a factor of two and a factor of three, and the prediction that it is lost after a fixed number of whorls is confirmed to a per cent. What the rule is counting, as it decides whether to leave the branch, is the lattice it is building, not the organs it places to build it.

The grid does not move it #

A threshold that belonged to the grid would move with it. On the ordinary stem, grids of 256, 512 and 1,024 azimuths give the same edge to the organ: kept to 90, lost from 91. On the bijugate stem, 512, 1,024 and 2,048 azimuths all keep the seed to 182 and lose it from 183 or 184. On the trijugate stem 768 and 1,536 azimuths keep it to 270 and 274 and lose it from 275 and 276.

The largest movement anywhere is four in the rate on the trijugate stem’s coarser grid, less than two per cent. At a grid of 256 azimuths an ordinary stem places each organ to a precision of 1.4°, and it still loses the Lucas branch at the same rate as a stem placed to 0.35°.

Exact whorls change nothing #

A bijugate stem whose whorls are made exact by construction has a record identical to the free stem’s at all 42 rates it was grown at. So does a trijugate one at all 31. The symmetry of a whorl, which the rule is under no obligation to keep, has no say in whether the stem keeps its seed.

That is not because the trijugate whorls stay symmetric. At T = 276, the first rate that loses the seed, a free trijugate stem’s whorls miss a third of a turn by up to 182.8° as it leaves the Lucas ladder — members crowding onto one another — while the imposed stem’s whorls are exact throughout. Both lose the seed at the same rate.

What losing looks like #

At the edge the kept stems all walk the Lucas ladder cleanly. The ordinary stem counts 1/3, 3/4, 4/7, 7/11; the bijugate one 2/6, 6/8, 8/14, 14/22; the trijugate one 3/9, 9/12, 12/21, 21/33 — each the ordinary sequence multiplied by its jugacy.

Losing it is less tidy than keeping it, and the untidiness differs by jugacy. The bijugate stem one rate past its edge goes straight to the Fibonacci ladder: 2/6, 4/6, 6/10, 10/16, 16/26. The imposed trijugate stem does the same, 3/9, 6/9, 9/15, 15/24, 24/39. But the ordinary stem one organ past its edge wanders through 1/3, 2/4, 4/9, 4/11, 4/13, 4/17 and 4/6 to end on 6/10, a pair with a shared factor rather than a Fibonacci one. And the free trijugate stem, its whorls coming apart, passes through thirty-four pairs, among them 9/48 and 45/60, and ends on 21/36.

The edge and the destination are different questions #

That difference matters for what the threshold means. Whether a stem keeps the Lucas branch is decided at the same whorls a rung by every configuration measured. Where it goes when it does not keep it is decided by other things — the jugacy, whether its whorls hold together, how close to the edge it is — and one organ past the edge an ordinary stem has not yet reached the Fibonacci ladder by the end of the rise.

The earlier bisection read its losing stems further from the edge, at 92 and more nodes a rung, where they ended on 8/13. So “losing the Lucas branch” and “reaching the Fibonacci branch” coincide well past the edge and not at it, which is the same distinction the branch essays draw between which ladder a pattern is on and which pair it happens to be counted at.

A pair with a factor, on a stem with no whorls #

The ordinary stem one organ past its edge ends on 6/10, a pair whose two numbers share a factor — the signature a count takes for a whorled pattern. It is not one. The stem is grown one organ at a time and has no whorls to be symmetric; its pair has a factor because of where its divergence happens to sit when the rise runs out, still between two ladders.

That is the trap a shared factor sets, met by a different route, and the same half-turn the rotation test uses would find no symmetry in it. A stem caught leaving the Lucas branch is one more way for a count to land in the bucket once labelled whorled without anything having arrived two at a time.

A golden seed has no edge #

The threshold exists because the Lucas ladder is the rule’s second choice. Grown from a coarse start at a divergence nobody chose, a stem ends on the Fibonacci ladder sixteen times in sixteen; seeded on it, a stem walks it on schedule at every rate. There is nothing to lose and no rate at which to lose it.

So the edge is a statement about how long a stem can be held on the other branch before it falls to the preferred one, and what the jugate stems add is the unit that holding is measured in. It is measured in lattice laid down between transitions, and a whorl of any size is one node of that lattice.

A gap after the seed #

The configurations with a gap leave one whorl rise of empty stem between the seed and the first grown whorl, the kind of slip a growth loop that adds the rise before placing rather than after makes without anyone intending it. It lowers every jugacy’s edge by the same factor: the ordinary stem keeps the seed to 55 and loses it from 56, the bijugate stem 110 and 111, the trijugate stem 165 and 168. In whorls a rung those are 52.9 to 53.9, 52.9 to 53.4 and 52.9 to 53.9, and each edge’s midpoint is 0.61 of its value without the gap.

That is what an estimate made while this measurement was being planned had seen. It put the bijugate edge near 106 to 110 organs a rung — between the placements prediction of 87 and the whorls prediction of 175, and it was read as the first quantity the scaling identity does not carry. It was a gap on the bijugate side of the comparison and not on the ordinary side: with the gap on both, the bijugate edge is 105.9 to 106.8 organs a rung, twice the ordinary stem’s 52.9 to 53.9.

Two stems either side of the edge #

The bijugate stems at T = 160 and T = 220 sit well inside the two regimes and look like it only to a counter. At 160 the stem keeps the Lucas ladder and ends counted 14/22 along 2/6, 6/8, 8/14, 14/22. At 220 it ends counted 16/26 along 2/6, 4/6, 6/10, 10/16, 16/26. Their top sixty whorls, unrolled, are two lattices of paired organs whose difference a person would see only by following the families round.

That is the practical difficulty with the whole question on a plant. The branch a shoot is on is a property of the lattice, and the rate it grew at is a property of its history; neither is written on a single whorl.

Why whorls #

The scaling identity makes the result less surprising than the question made it look. Folded k times round, a k-jugate stem is an ordinary stem whose nodes are its whorls and whose rise is $k^2$ times its organ rise. If the placement rule’s choice of branch depends only on the geometry of the lattice it has built — how many nodes the folded pattern has laid down while its rise crossed a rung — then the folded stem must lose the seed exactly where an ordinary stem does, counted in its own nodes, which are whorls.

The measurement says the rule behaves that way to within a per cent, and the one per cent is itself in a consistent direction: 86.6, 87.6 and 87.9 whorls a rung, rising a little with the jugacy.

That drift is the size of the ordinary scan’s own step. One organ of rate is 0.96 whorls a rung on an ordinary stem, half that on a bijugate stem and a third on a trijugate one, which was scanned every second organ. So the whole difference between the ordinary edge and the others is one step of the ordinary scan, and a scan of the ordinary stem at fractional rates between 90 and 91 is what would say whether it is real. That scan has not been run.

What this does not establish #

That a real bijugate plant keeps or loses a Lucas arrangement, that its rise falls exponentially, or that its whorl members are placed one after another. The edge is read at one starting rise, one seed length, one counting window and one final rise, the ordinary design carried over; the earlier bisection found the ordinary edge lower from a coarser start, and nothing here repeats that for jugate stems.

What would withdraw it #

A keep-or-lose record along the rate that is not one unbroken run of each. A bijugate or trijugate edge, in whorls a rung, more than two per cent from the ordinary edge. An imposed-symmetry record that differs from its free one at any rate. A gap that lowers one jugacy’s edge by a different factor from another’s. Each is checked every time the measurement runs.

Still open: whether the seed’s length counts whorls too #

Every stem here was seeded with forty whorls of Lucas lattice, which is forty organs on an ordinary stem and eighty on a bijugate one. A seed long enough to fix a branch in whorls may or may not be long enough in organs, and the next test is the edge against the seed’s length at each jugacy: whether a bijugate stem seeded with twenty whorls — forty organs — keeps the seed like an ordinary stem seeded with forty, or like one seeded with twenty.