The hole on the other branch

The clean version of the front result is a step: every offset out to the larger spiral count is felt, nothing past it is, and the transition between the two regimes spans two orders of magnitude in displacement. That is what most cells of most measurements look like.

Near a rung boundary it is not what any of them look like. There the response has a hole: the run of felt offsets ends where it should, then several offsets are quiet, and then a single isolated offset — well outside the front, sometimes half as far out again — moves the next organ by a hundred degrees or more.

That was found on the golden branch, and where the isolated offset sits was pinned down there: one inside the count coming in at the next rung. A 5/8 stem approaching 8/13 has its isolated offset at twelve; a 3/5 stem approaching 5/8 has it at seven.

The trouble with a rule of that form on the Fibonacci branch is that the numbers available to check it against are all Fibonacci numbers, and Fibonacci numbers have a great many arithmetic relations between them. Twelve is thirteen minus one; it is also five plus seven, eight plus four, and half of twenty-four. A rule fitted to two or three such cases has more explanations available than it has data.

The Lucas branch fixes that, because its counts are 3, 4, 7, 11, 18 — the same addition from a different pair — and the coincidences are not the same coincidences.

Four holes, two branches #

The design is seven rises with a golden-seeded stem and a Lucas-seeded stem at each. Four of its fourteen cells have a hole, and they are split evenly between the branches.

On the golden branch: a 3/5 stem at a rise of 0.020, whose incoming count is 8 and whose isolated offset is at 7, displacing the next organ by 133.6°; and a 5/8 stem at 0.008, incoming count 13, isolated offset 12, displacing by 139.0°.

On the Lucas branch: a 3/4 stem at a rise of 0.024, whose incoming count is 7 and whose isolated offset is at 6, displacing by 102.9°; and a 4/7 stem at 0.010, incoming count 11, isolated offset 10, displacing by 97.0°.

Six is not a Fibonacci number and neither is ten. Seven is not a Fibonacci number and neither is eleven. The rule one inside the incoming count survives on lattices where none of the numbers it is stated in appear in the sequence it was found on, and that is the whole of what this essay adds.

Reading one Lucas cell in full #

The 4/7 stem at a rise of 0.010 is the cleanest of the four, so it is worth setting out.

Grown from a seed near 99.5°, it settles at 99.20° with a spread of a tenth of a degree over its last sixty organs. A blind counter shown its positions returns 4 and 7. Ranked by step length, the families after those two are eleven, then three, then eighteen — so the count coming in at the next rung is eleven, read off the geometry rather than off a sequence.

Now remove one organ at a time and record the displacement of the next.

Offsets one through seven all move it, by between four and a hundred and seventy degrees. Offsets eight and nine move it by 1.41° and less. Offset ten moves it by 97.0°. Offset eleven moves it by 0.5°.

So the response of this stem is not an interval. It is an interval of seven, then two quiet offsets, then one loud one at ten, then quiet again. The loud one sits one inside eleven.

The same stem three rises higher — the 4/7 cell at 0.016, in the middle of the same rung — has no isolated offset anywhere: past its front of seven, the largest displacement out to offset eleven is 1.41°. Same lattice, same counts, same branch, same rule. The difference is where in the rung it sits.

Why an unfelt offset is felt at all #

The mechanism is the one the golden-branch work identified, and the Lucas cells let it be stated without reference to any particular sequence.

The rule places each organ at the lowest point of a profile summed over its neighbourhood. That profile has more than one low point: there is the winner, which is where the organ goes, and there is a runner-up, which is where it would have gone had the winner been a little higher. Near a rung boundary the two are close in value, because a boundary is precisely a rise at which two candidate families have equal length.

An organ removed from well outside the front cannot change the winner’s position — that is the front result. It can change the ordering of the two low points, if the organ removed happens to be the one holding the runner-up up. When that happens the next organ goes to the other slot entirely, and the displacement is not small: it is the angular distance between the two slots, which is most of a turn.

So the isolated offset is the organ that guards the second slot, and the reason it sits at the incoming count less one is that the second slot is the one belonging to the family that is about to take over — which is the incoming family, and whose own step is that many organs.

Two things the Lucas branch decides that the golden one could not #

The first is the arithmetic form of the rule. On the golden branch, the incoming count after 5/8 is 13, and the isolated offset is 12 — which is consistent with incoming minus one, and equally consistent with the two counts of the current pair added, minus one (5 + 8 − 1 = 12), and with the smaller count plus the larger, minus the smaller’s own predecessor, and with a handful of other formulas, because on a Fibonacci ladder the incoming count simply is the sum of the current pair.

On the Lucas branch it is too — 4 + 7 = 11 — so that particular ambiguity is not resolved by moving branches. But the 3/4 cell does resolve one: its incoming count is 7 and its isolated offset is 6, while its own counts sum to 7 and its larger count is 4. Any rule phrased in the current pair’s larger number alone, or in a fixed offset from it, gives the wrong answer at that cell and the right one at the 5/8 cells.

The second is that the effect is not about Fibonacci at all. It would be easy, looking only at the golden branch, to read the isolated offset as one more appearance of the sequence — the collection is full of essays correcting exactly that reflex. Two of the four holes sit at offsets six and ten on lattices counted 3/4 and 4/7, which no reading in terms of Fibonacci numbers reaches.

Why testing a rule on a second branch is worth the trouble #

There is a general point here that this collection keeps running into, and the Lucas branch is the cleanest illustration of it available.

Phyllotaxis is full of relations between small integers, and the integers involved are nearly always Fibonacci numbers. That makes numerical coincidences abundant: any two Fibonacci numbers are related by the recurrence, by ratios approaching the golden ratio, and by a stack of identities. A rule stated in those numbers and checked on those numbers has a very high prior probability of appearing to work.

The Lucas branch has the same recurrence and different terms. Anything that depends only on the recurrence survives the move; anything that depended on the particular values does not. That makes it a cheap and unusually sharp control, and it is available here only because a placement rule has more than one settled state and this collection has learned how to put a stem on the second one and keep it there.

Two results have now been carried across. The front is the larger parastichy number, checked on lattices whose larger numbers are 4, 7 and 11 as well as 5, 8 and 13. The isolated offset is one inside the incoming count, checked on incoming counts of 7 and 11 as well as 8 and 13. Neither survived by arithmetic accident, because the arithmetic is different on the two ladders.

The one thing the Lucas branch cannot do is separate incoming count minus one from the current pair summed, minus one, because both ladders are built by the same addition and the incoming count is the sum of the pair on both. Separating those two would need a lattice whose ladder is not additive, and this rule does not produce one.

What four hits out of four is worth #

Four cells all landing one inside the incoming count is the result, and it is worth pricing rather than left as a run of agreements, because four is a small number and the reader’s first question should be how often that happens by chance.

Take the null that the isolated offset is somewhere in the quiet region — past the front and no further out than the incoming count — with no preference. The 3/5 cell has a front of five and an incoming count of eight, so three slots; the 3/4 cell likewise has three; the 4/7 cell has four; the 5/8 cell has five. Four independent hits at the same relative position come to one chance in a hundred and eighty.

That is a real number and it is not an overwhelming one, which is the honest way to hold it. It is enough that the rule is not an artefact of two Fibonacci coincidences, which is what the Lucas branch was brought in to settle; it is not enough to treat one inside as established against, say, one or two inside, for which these four cells give almost nothing.

The refutation condition follows from the same arithmetic and is worth stating in advance. A fifth cell whose isolated offset sits somewhere else would leave four of five and a price of about one in forty — weakened but standing. Two misses would end it, since three of six at these odds is the ordinary case. So the rule is one cell away from being much better supported and two from being withdrawn, and the design that decides it is the same seven rises run at a rung boundary on a third branch.

Where the holes are, and where they are not #

Four cells of fourteen have a hole, and which four is itself informative.

Each of the four sits near a rung boundary on its own branch: the golden 3/5 cell at 0.020 is near the bottom of the 3/5 rung, the golden 5/8 cell at 0.008 near the bottom of 5/8, the Lucas 4/7 cell at 0.010 near the bottom of 4/7, and the Lucas 3/4 cell at 0.024 is a stem holding 3/4 past the rise at which the ideal ladder says it should have changed.

The cells in the middle of their rungs have no hole at all: the response is an interval, ends at the larger count, and everything past it is under a degree and a half across the whole range measured. So the hole is not a permanent feature of the response that is sometimes too small to see. It appears and disappears with position in the rung, which is the same variable that decides how strongly the last offset of the front is felt.

That gives the two results a shared account. Near the top of a rung the newest member of the front is weakly felt because its family has only just become the shorter one. Near the bottom of a rung the incoming family’s slot is close enough to winning that its guard becomes visible. Both are consequences of two families being nearly tied, at opposite ends of the same rung — which is also where a resolution parameter stops rounding and starts deciding, so a cell with a hole is a cell whose settings deserve a second look.

What an experiment would see #

The practical consequence is unpleasant and worth stating plainly.

An ablation experiment that swept the offset and reported the largest offset at which a removal was felt would, on a specimen near the bottom of its rung, report the isolated offset rather than the front. That is not one off. At the golden 5/8 cell it is twelve against a front of eight; at the Lucas 4/7 cell it is ten against seven. The reported number would be the incoming count less one, which is a number the specimen’s own counts do not contain — and so a number no arithmetic check on the pair could refuse.

The repair is to report the run rather than the maximum, which is what this collection’s measurement does and which is why the hole is visible here at all. An experimenter who records every offset separately can see the gap; one who records only the boundary cannot, and will report a number that is neither of the specimen’s counts and is wrong in a way that looks like a discovery.

What this does not say #

It does not say every stem near a boundary has a hole. Four of fourteen cells do, and all four are near a boundary, but not every cell near a boundary has one. The design is too coarse in the rise to say what fraction of the rung the hole occupies, and no attempt is made here to bound it.

It does not say there is only ever one isolated offset. Each of the four cells has exactly one. Whether a stem can have two, and whether a second would sit at the count after the incoming one, is not measured.

It does not say the rule is derived. One inside the incoming count is a description that now holds at four cells across two branches with six different integers involved. The account of why — the guarded second slot — is a mechanism that predicts the right family and does not by itself predict the offset exactly.

And it does not say a plant would show it. The displacement at an isolated offset is a hundred degrees, so it is not a subtle signal; whether a meristem has a runner-up slot in the sense used here depends on whether it is doing anything like this rule, which is the standing question of the whole collection.

The check that would refuse it #

Three assertions carry this, and they are arranged so that the interesting one cannot pass by accident.

The first is that the design contains cells whose response has a hole — at least three of them. Without that the rest is a claim about an empty set, and it would fail silently if a change to the rule or the rises removed the effect.

The second is the rule itself: every isolated offset in the design sits at exactly the count coming in at the next rung of that branch’s own ladder, less one. The incoming count is read from the lattice’s own step lengths rather than from a sequence, so on the Lucas cells it is 7 and 11 without anything being told what a Lucas number is. A single offset landing anywhere else stops the collection being built.

The third is that at least one of the cells with a hole is on the Lucas branch. That is the assertion that would fail if the effect turned out to be about Fibonacci lattices after all, and it is the reason for building the design in the first place: a rule confirmed only where its numbers coincide with a famous sequence is a rule that has not been tested.