A seed measured in whorls

A Lucas seed counts whorls grew ordinary, bijugate and trijugate stems from forty whorls of Lucas lattice and found the rate at which each gives that ladder up: 86.6, 87.6 and 87.9 whorls a rung. One number to within a per cent in whorls, and one, two and three times as far apart in organs — so the edge is a property of the folded lattice and not a count of placements.

That settled what the edge counts and left the seed open, and the two questions are not the same one. Forty whorls is forty organs on an ordinary stem and eighty on a bijugate one, so a seed long enough to fix a branch in whorls may or may not be long enough in organs. What follows sweeps the seed across a fivefold range at all three jugacies and asks which.

The seed is the only thing moved #

Everything else is the design the earlier measurement used, unaltered: the folded rise falls from 0.12 to 0.004, the counting window is twenty-six folded nodes striding four, the seed is the Lucas lattice at the jugacy’s own angle, and the whorls are filled freely rather than by copying one placed member round. Only the seed’s length changes.

That matters because the point of the exercise is to discover what the earlier number was a property of, and a second change would make the answer ambiguous.

Why the question is not obviously settled by the identity #

The folding identity says a k-jugate stem is an ordinary stem at k times the organ rate, and it is tempting to treat that as answering the seed question too. It does not, on its own, because the identity is about the lattice and the seed is about the history: what a stem carries forward from its beginning could in principle be a number of placements — a memory of how many organs have been laid — rather than a property of the folded pattern.

Two at a time is the essay that separates those two readings in general, and the separation has come out on the lattice’s side every time it has been tested here. It was still worth testing, because a rule that was right for three quantities and wrong for the fourth is exactly what nobody would notice.

The edge is a threshold, and that is checked rather than assumed #

At each seed length and jugacy, stems are grown at twelve rates spanning the bracket, and each is asked at the end whether its counting window is still on the Lucas ladder. The rates that keep it are an initial run and the rates that lose it follow, at every one of the eighteen configurations grown — so there is an edge to bisect rather than a scatter to average.

Read as whorls, the three jugacies lie on top of each other #

At a seed of fifteen whorls the three edges are 30.80, 30.80 and 30.80. At thirty whorls, 63.52, 63.52 and 63.52. At twenty, 42.35, 42.35 and 40.42; at forty, 85.66, 87.58 and 85.66; at sixty, 130.89, 132.81 and 130.89; at eighty, 173.24, 177.09 and 177.09.

The departures are never more than five per cent, and every one of them is one or two steps of the grain the edge is bisected to. The bisection stops within two whorls of rate, which is 1.925 whorls a rung at every jugacy and every edge here — so a disagreement of 1.92 or 3.85 between two jugacies is the instrument’s last step and not a difference between the stems.

Read as organs, they do not #

The decisive comparison is a seed of the same length in organs at different jugacies, and it is available three ways over. Sixty organs is sixty whorls on an ordinary stem, thirty on a bijugate one and twenty on a trijugate one — and those lose the seed at 130.89, 63.52 and 40.42 whorls a rung, a factor of 3.24 between the ends.

So the seed counts whorls, and the answer is the one the identity predicts #

A k-jugate stem folded is an ordinary stem at k times the rate, and its seed folds with it: forty whorls of k-jugate Lucas lattice folds to forty whorls of ordinary Lucas lattice. Nothing in the folded picture knows how many organs were laid at each height.

That is the same reasoning a grown stem halves its ladder used for the transitions and a Lucas seed counts whorls used for the edge, arriving now at the third quantity in the same account. The three together say the folded lattice is the object and the organ count is a detail of how it is drawn.

One reading per jugacy would not have found this #

Each row of the table is three independent stems, grown separately and bisected separately, agreeing to the grain. That redundancy is what makes the whorls reading safe: a single jugacy swept across seed lengths would give the same straight line and say nothing at all about what the seed is measured in, and a single seed length across jugacies gives the earlier result and says nothing about whether it is a constant.

Two sweeps crossed is what the question needed, and it is why the second half of this essay exists at all — the seed sweep was run to answer the first question and answered a second one nobody had asked.

And the edge is not a number #

That is the expected half. The unexpected half is what the sweep does to the edge itself.

A seed of fifteen whorls is lost at 30.80 whorls a rung. Forty is lost at 85.66 to 87.58. Eighty is lost at 173.24 to 177.09. The edge rises with the seed, over the whole range swept, and a straight line through all eighteen readings gives

$$\text{edge} = 2.2373 \times \text{seed} - 3.079$$

straight to within 2.67 whorls a rung over a fivefold range of seed lengths.

Where the straight line is least trustworthy #

Three things about the fit are worth saying before anything is built on it. It is a line through eighteen points spanning a factor of five and a bit in the seed, which is a short lever. Its intercept is −3.08 whorls a rung, which is not zero and is not far from zero, so the relation is proportional to within the reading’s own noise and is not demonstrably exactly proportional. And nothing outside the swept range is claimed: below about a dozen whorls a stem cannot be read at all, and above eighty the stems get expensive without getting more informative.

What the line does support is the one conclusion it is used for. Over the range every earlier measurement here could have been taken in, the edge moves by a factor of more than five, and a single number for it is therefore a number with a hidden argument.

What that does to the published number #

The eighty-six to eighty-eight whorls a rung reported earlier for the edge is not a property of the rule. It is the value of that line at a seed of forty whorls, and forty whorls was the seed in every stem ever grown for it — inherited from the design of an earlier measurement and never varied.

So the number is correct and its status was wrong. It was reported as where a Lucas seed is lost; what it is, is where a forty-whorl Lucas seed is lost.

What the earlier essay would have had to do to catch it #

Nothing exotic. The seed was forty whorls because the measurement it was copied from used forty whorls, and that measurement used forty because forty is a round number that is comfortably long. At no point did anybody decide that the edge was independent of it; the question was not asked, which is a different failure from getting an answer wrong and is harder to notice.

The general form is worth stating because it keeps turning up. A measurement has settings, and a setting is either varied and shown not to matter, or varied and shown to matter, or not varied — and the third case is indistinguishable from the first in the written result. Every number in a Lucas seed counts whorls was in the third case and read as though it were in the first.

The shape of the mistake, which is familiar #

The window nobody varied found the same shape on the counting side: every parastichy pair the settling table reports was counted over the top two hundred organs, and that number had never been moved. There the answer was that moving it changes nothing, which is the good outcome; here moving it changes the headline.

Both are the same failure of hygiene and neither is a failure of arithmetic. An unvaried setting is a free parameter reported as a result, and the only way to find out which it is, is to move it. A plateau the instrument should have had is the version where moving it finds two bounds rather than none, and it is the outcome to hope for: a setting with a plateau in the middle is a setting that can be chosen well.

Two rungs and a bit, per whorl of seed #

The relation is close to proportional and not exactly so. Dividing each edge by its own seed gives 2.05 at fifteen whorls, 2.09 at twenty, 2.12 at thirty, 2.16 at forty, 2.19 at sixty and 2.20 at eighty — rising steadily and appearing to settle a little under 2.2.

What a ratio of two means #

A stem carries its seed about twice its own length into the fall of its rise before losing it, measured in whorls a rung. That is a statement with a plain reading: a seed of forty whorls survives until the stem’s rise is falling slowly enough that a rung takes about ninety whorls to cross, which is about two and a quarter seeds’ worth. That is a statement about how long a history survives a changing rate, and it is the kind of quantity the rate decides the branch is about — where a stem’s fate turns on how fast its rise falls rather than on where it started.

Why two and a quarter rather than one, or five, is not answered here. The number is measured and is not derived, and the creep from 2.05 to 2.20 says it is not a clean constant either. A quantity that is nearly but not exactly proportional is the usual sign of a transient on top of a scaling, and a grown stem halves its ladder found one of those too — a lag of a hundredth of a rung behind the static ladder, present at every jugacy and the same size in all of them.

Why the creep is probably the seed’s own transient #

The likeliest explanation is that a short seed is not the same object as a long one. A stem seeded with fifteen whorls has a counting window of twenty-six folded nodes, which is most of its seed, so the first readings are taken while the seed is still most of what there is. A stem seeded with eighty has room for the seed to be a history rather than a presence.

That would make the low ratios at short seeds an artefact of the reading rather than a property of the stem, and it is checkable: the same sweep with a narrower counting window should lift the short-seed ratios towards 2.2 and leave the long-seed ones alone. Nothing here does that check.

What a stem at either end of the line looks like #

A trijugate stem seeded with eighty whorls is a long object: two hundred and forty organs of imposed lattice before anything is grown, and an edge at 177 whorls a rung.

An ordinary stem seeded with twenty is the other end — twenty organs of lattice, an edge at 42 — and the two scans have the same shape, which is the point of drawing both.

What is now known about this edge #

Three things, and the third is new. It is measured in whorls a rung, not in organs — that is the earlier result. The seed that sets it is measured in whorls too, so the whole account is a statement about the folded lattice and the jugacy enters nowhere. And the edge is proportional to the seed rather than being a constant, so every number previously attached to it carries a seed length as a hidden argument.

What has to be reported alongside the edge from now on #

The seed’s length, in whorls. That is the whole of the fix, and it costs nothing: every stem grown here already has a stated seed, and stating it beside the edge turns a number that was silently conditional into one that is openly so.

The same applies backwards. Every earlier statement here about where a Lucas seed is lost is true at a forty-whorl seed and undetermined elsewhere, and the essays that carry those numbers say so now. That is a correction rather than a retraction: the measurements were right and their scope was not stated, which is the kind of error that turns up most often here and elsewhere.

What is claimed, in one line #

A Lucas seed’s length counts whorls, so stems of one, two and three organs a whorl seeded with the same number of whorls lose the seed at the same rate; and the rate is 2.24 times the seed less three rather than a constant, which makes the eighty-seven whorls a rung reported earlier a property of a forty-whorl seed.

What it does not establish #

That a meristem has a seed at all. The seed here is an exact lattice of stated length imposed on a model stem, and a plant’s early growth is neither exact nor of stated length. Nor does anything here explain the factor of 2.24; it is measured over a fivefold range and is not derived from the rule.

Nor is the relation shown to be linear outside that range. Below about a dozen whorls a stem is too short to slide the counting window along at all, and above eighty the stems become expensive without obviously becoming more informative.

What would withdraw it #

A configuration whose kept rates are not an initial run. Two jugacies at the same seed length in whorls differing by more than the bisection grain. Two jugacies at the same seed length in organs agreeing. A reading more than three whorls a rung off the straight line. An edge at a forty-whorl seed outside the eighty-four to eighty-nine the earlier measurement reported. Each is checked every time the measurement runs.

What this does not touch #

The transitions. A grown stem halves its ladder established that a jugate stem passes the ordinary transitions at the ordinary rises divided by the jugacy, and nothing about the seed’s length moves any of them — a stem that has lost its seed is on the Fibonacci ladder and passes that ladder’s transitions wherever they are. The seed decides which ladder, not where the rungs sit.

So the two halves of this account remain separate quantities: a ladder that the folded lattice fixes, and a branch that a history and a rate decide between. That separation is what makes the seed worth measuring at all.

Still open: whether the window is carrying part of the ratio #

The ratio of edge to seed rises from 2.05 to 2.20 across the range swept, and the obvious suspect is the counting window rather than the stem: at the shortest seeds the window of twenty-six folded nodes is a large share of the seed itself, and at the longest it is a small one. The next measurement sweeps the window as well as the seed — the same eighteen configurations at windows of thirteen, twenty-six and fifty-two nodes — and asks whether the creep flattens. A ratio that is 2.2 at every seed under a narrow window would put the creep in the instrument; one that creeps under every window would put it in the stem, and would then be worth deriving.