A grown stem halves its ladder

Half the golden angle ended on a derivation of what a bijugate stem walks. An ordinary stem at the golden divergence changes its pair 1/2 → 2/3 → 3/5 → 5/8 → 8/13 at rises of 0.12472, 0.04767, 0.01822 and 0.00696; a bijugate stem at 68.754° should change 2/4 → 4/6 → 6/10 → 10/16 at half those rises, with the spacing between rungs unchanged. The reason is one line: a lattice that arrives two organs at a time, wrapped twice round, is the ordinary lattice at twice the divergence and twice the rise.

That is a statement about lattices. A stem is not a lattice; it is grown, one placement against everything below it, and a grown ordinary stem lands on its ladder only because that was measured. No bijugate stem had been grown through a single transition to see whether the same is true of it.

Growing two at a time #

The rule is the one that grows every ordinary stem here. The rise falls exponentially with the organ count, at a rate T of 300 organs, and each organ is placed at the azimuth of least repulsion from the organs already there. For a stem of k organs a whorl the only change is that k organs share each height: the first member of a whorl is placed against everything below it, the second against everything below it and the first member, and so on.

Nothing in that says the members must end up a k-th of a turn apart, and nothing tells the rule which divergence to settle at or which ladder to walk. The stem is seeded with six whorls of ideal lattice at 137.5078/k and then left alone. The window of history each placement sees is capped at 220 organs a whorl member, so that a stem of k organs a whorl reaches as far back in whorls as an ordinary stem does in organs.

Folding a whorl to one node #

A counter that measures the distance from organ i to organ i + m needs an order of arrival, and two organs of one whorl have none. So each whorl is folded: its members’ azimuths are multiplied by k, averaged round the circle into one node, and its height and rise multiplied by $k$ and by $k^2$. On an ideal k-jugate lattice that is the ordinary lattice exactly, to seven parts in a million billion at k = 2, 3 and 4.

The folded stem is then counted exactly as an ordinary stem is: a window of 36 folded nodes slid up it two at a time, the pair read in each window and compared with the static ladder at that window’s rise. With one organ a whorl the grower is the ordinary one node for node, so the ordinary stem here is not a second program but the same one at k = 1.

Four stems on one axis #

Read against the folded rise, the four stems walk one ladder. The ordinary stem counts 1/2, 2/3, 3/5, 5/8, 8/13 and 13/21. The bijugate stem counts 2/4, 4/6, 6/10, 10/16, 16/26 and 26/42; the trijugate one 3/6, 6/9, 9/15, 15/24, 24/39 and 39/63; the quadrijugate one 4/8, 8/12, 12/20, 20/32, 32/52 and 52/84. Every pair is the ordinary stem’s multiplied by the jugacy, in the same order, with no pair skipped and none added.

The agreement with each stem’s own static ladder is as good as the ordinary stem’s: the counted pair is the ladder’s in 99.1 per cent of windows for one organ a whorl, 99.0 for two, 98.5 for three and 99.0 for four.

Three rises for one transition #

A transition on a k-jugate stem has three rises and they are worth keeping apart. The folded rise is where it sits on the ordinary ladder. The whorl rise, the height from one whorl to the next, is the folded rise over k. The organ rise, that height shared among the whorl’s members, is the folded rise over $k^2$.

The bijugate stem changes from 2/4 to 4/6 at a whorl rise of 0.0623, from 4/6 to 6/10 at 0.0232, then at 0.00901, 0.00345 and 0.00130. The ordinary grown stem makes the matching transitions at 0.1250, 0.0466, 0.0180, 0.00683 and 0.00263. The ratio is a half to within one per cent at every one of the five.

Worked at the first transition #

The first bijugate transition can be checked by hand. Its whorl rise is 0.0622806, and twice that is a folded rise of 0.1245612. Its organ rise is 0.0311403, and four times that is 0.1245612 again. The ordinary grown stem makes the matching transition, 1/2 to 2/3, at a folded rise of 0.124977, so the bijugate stem makes it at 0.99667 of the ordinary stem’s rise. The static ladder puts it at 0.124995.

The trijugate and quadrijugate stems give the same shape of answer with a third and a quarter: 0.0413823 times three is 0.1241469, and 0.0313486 times four is 0.1253944. Across all fifteen of their transitions the whorl rise times the jugacy is within 1.0 per cent of the ordinary stem’s own rise.

Half the rise, a quarter per organ #

The derivation said a bijugate stem reaches a given pair at half the rise, and so has to be twice as compressed to look the same. Measured, it depends which rise. Between whorls the bijugate stem is compressed twice as much. Per organ — the rise a person would measure by dividing a length of stem by the organs on it — it is compressed four times as much, and a trijugate stem nine times.

That is the number a comparison of two forms of one species would actually take. A bijugate shoot and an ordinary one counted at the same pair should carry organs four times as densely along their length, not twice, provided both are growing by the same rule and neither has changed branch.

The transition, on the stem #

On the grown stem the change from 6/10 to 10/16 comes at whorl 466, at a whorl rise of 0.00901, where the static bijugate ladder puts it at 0.00911. Nothing about the drawing announces it. The members of every whorl sit exactly half a turn apart before and after, and the change is visible only to a counter that follows the families — which is what a transition has always been: a change in which lattice vectors are shortest, not a change in the pattern’s look.

The lag is the ordinary stem’s #

A grown stem makes each transition a little after the static ladder puts it, because its arrangement carries its history. Measured on ordinary stems over a fifteenfold range of rate, that lag stayed within a tenth of a rung.

The four stems here are more exact than that. The ordinary stem’s mean lag is 0.0124 rungs, its worst 0.0223. The bijugate stem’s is 0.0131, the trijugate’s 0.0166 and the quadrijugate’s 0.0117, with worsts of 0.0258, 0.0223 and 0.0189. Every jugacy lags by a hundredth of a rung, within half a hundredth of the ordinary stem, and the quadrijugate stem’s first transition comes three thousandths of a rung early.

The spacing between rungs #

The derivation’s second consequence was that jugacy shifts the ladder without stretching it: the rises halve, so their ratios stay. The first two transitions of the bijugate stem are 0.0622806 and 0.0232193, a ratio of 2.682; the ordinary stem’s are 0.124977 and 0.0465937, a ratio of 2.682. The golden ratio squared is 2.618, and the difference between 2.682 and 2.618 is the ordinary stem’s own, carried into the bijugate one unchanged.

So the spacing is not merely unchanged in the lattice; it is unchanged in the grown stem, window for window, including the ways the grown stem departs from the lattice. Everything a cone or an ogive does to the ladder’s spacing on an ordinary shoot therefore applies to a multijugate one grown by this rule.

Where each stem settles #

The limit angle of a k-jugate ladder is 137.5078/k, and a grown stem that walks the ladder should settle near it. Over its last sixty whorls the ordinary stem settles at 137.512°, 0.004° from the golden angle. The bijugate stem settles at 68.734°, 0.020° from 68.7539°; the trijugate stem at 45.809°, 0.027° from 45.8359°; the quadrijugate at 34.348°, 0.029° from 34.3769°.

The azimuth grid’s step is 0.234°, and every stem settles well inside it. The slower-growing stems settle closer still: the bijugate stem at T = 600 lands at 68.754°, and the trijugate stem at T = 900 at 45.840°, 0.004° from its limit.

Six rungs in five transitions #

Every stem runs over the same folded rise, from 0.4 down to 0.0012, a factor of 333. Measured in factors of $\varphi^2$ that is $\ln 333 / 0.9624 = 6.04$ of them, and each stem passes five transitions inside it. The ordinary stem lays down 1,743 organs doing so; the bijugate stem 1,744 organs in 872 whorls; the trijugate stem 581 whorls, the quadrijugate 436.

At one organ rate a jugate stem therefore places k times fewer folded nodes a rung: 0.9624 × 300 is 289 for the ordinary stem, and 144, 96 and 72 whorls a rung for two, three and four organs a whorl. A quadrijugate stem walks its ladder with a quarter of the ordinary stem’s placements between transitions and still makes every one within two hundredths of a rung. The static bijugate ladder puts its transitions at whorl rises of 0.06250, 0.02380, 0.009108, 0.003469 and 0.001327; the grown stem’s 0.0623, 0.0232, 0.00901, 0.00345 and 0.00130 sit just below each, as a stem that lags a little should, and they reproduce the derivation’s quoted 0.06236, 0.02384 and 0.00911 to three figures.

Controls that move nothing #

Three things could have produced the agreement without the jugacy doing anything. Each was taken away.

The window. A bijugate stem sees 440 organs of history where an ordinary stem sees 220. Capped at 220, the bijugate stem’s transitions do not move by a single digit of the six its rises are stored to. The symmetry. A trijugate stem whose whorls are made exact by construction — one member placed, the others copied round — makes every transition at exactly the free stem’s rise. The number of placements a rung. A bijugate stem at T = 600 and a trijugate one at T = 900 place as many whorls a rung as the ordinary stem does organs; their transitions move by at most 1.3 per cent of rise, and so do the transitions of the same jugacy at two organ rates.

Where the cap binds #

The window control would be empty if the cap were never reached, so it is worth checking that it is. A placement sees $6/\sqrt{h}$ organs of history, where $h$ is the rise per organ, up to the cap. At the end of the bijugate stem the organ rise is 0.0012/4 = 0.0003, and $6/\sqrt{0.0003}$ is 347 organs — above the ordinary cap of 220 and below the bijugate stem’s 440.

So the capped stem sees less history than the uncapped one wherever the organ rise is below $(6/220)^2 = 0.00074$, a folded rise of 0.0030. The last transition, from 16/26 to 26/42, is made at a folded rise of 0.0026, inside that regime, and the capped stem makes it at the same rise to every figure stored. The bijugate stem has extra reach it does not use to walk its ladder.

One step back #

Of the eight grown stems one does something the ordinary stem never did. The trijugate stem at T = 900 walks 3/6, 6/9, 9/15, 15/24 and reaches 24/39 at a whorl rise of 0.002308, returns to 15/24 at 0.002292, and reaches 24/39 again at 0.002262, a span of two per cent of rise, before going on to 39/63.

It is a single event inside a few counting windows, and it is at the one jugacy whose whorls do not sit exactly a k-th of a turn apart. Whether those two facts are connected is not settled by one event, and it is recorded here as what it is: a back-step the scaling identity does not predict, seen once.

Counted without folding #

Folding needs the jugacy known before the count is made. The counter that follows families into chains needs no order of arrival and no k, and it reads the same kind of stem without folding — but slid up a grown bijugate stem at this rate it reads nine bands in ten only in bands of 60 to 100 organs, returning another rung below that and a pair on no rung above it.

So the ladder a folded count finds on these stems is available to a person who does not know k only through a band about a third of a rung of rise wide. A specimen whose jugacy is the question can be counted, but not by sliding one width up it, and not by the counter whose agreement with the ladder is reported here.

A whorl, and a cut #

The bijugate whorls stay exactly half a turn apart through all 872 whorls, and the quadrijugate ones exactly a quarter through 436. That exactness is fragile in a way the ladder is not. A single organ cut from a whorled stem destroys the symmetry its counts cannot see, and a rotation is what finds it gone.

A grown bijugate stem whose whorls are exact is therefore a stem nothing has been taken from, and exactness is one of the few properties of these stems a photograph could check without a count. The trijugate stems are the exception again, and the reason is in how a whorl’s members are placed.

What a plant with both forms would show #

The derivation’s checkable claim was about two forms of one species, one ordinary and one bijugate, counted at the same pair. The grown stems sharpen it into three numbers a specimen could contradict: the bijugate form should carry its organs four times as densely per unit of stem length at the same pair; it should settle within a grid step of 68.754°; and the ratio between its successive transition rises should match the ordinary form’s.

The first of those is the useful one, because it needs no angle measured and no spiral counted — only organs and lengths on two shoots counted at the same pair. It rests on the two forms growing by one rule at similar rates, which is exactly what a real pair of forms may not do.

What this does not establish #

That real multijugate shoots are grown by this rule, that their rise falls exponentially, or that their whorl members are placed one after another against their whorl-mates. The rule here is a model, and the grown stems establish what the model does, which is to walk the ladder the lattice argument predicted.

It also does not establish that a whorl keeps its symmetry. The bijugate and quadrijugate whorls here stay exactly half and a quarter of a turn apart; the trijugate ones do not, by up to 8.4°, and how far a whorl misses its share and why is its own measurement.

What would withdraw it #

A k-jugate stem whose counted pairs are not the ordinary stem’s multiplied by k. A transition whose whorl rise is not its folded rise over k, or whose lag behind the static ladder differs from the ordinary stem’s by more than half a hundredth of a rung. A stem that settles further than a twentieth of a degree from 137.5078/k. A capped window or an imposed symmetry that moves any transition. Each is checked every time the measurement runs.

Still open: what the ladder does to a seed on the other branch #

These stems were seeded on the Fibonacci branch and kept it. An ordinary stem seeded on the Lucas lattice keeps that branch only when its rise falls fast, and the scaling identity predicts where a bijugate stem’s threshold should be only once it says what is being counted — placements or whorls. That threshold is the next measurement on grown jugate stems.