The block is the count it was cut from

A stem cut in the middle of its front does not come back. What it does instead is not disorder: the divergences settle into an exactly repeating sequence and hold it for the rest of a three-hundred-organ run, and every wrecked stem at that rise ends up in the same place — a block of eight organs precessing by about 22.7°, an effective divergence of 182.8°, which a counter reads as 8/16.

Eight is also the smaller number of the pair that was cut. The lattice was 8/13, and the orbit’s block is eight.

That coincidence has two readings and the measurement that separates them costs one afternoon.

Two readings of one number #

The orbit remembers. The block is the smaller parastichy number of the lattice the stem was on, so a wrecked stem carries its old count into its new arrangement. On that reading the second attractor is not one attractor but a family of them, indexed by where the stem came from.

Eight is what this rule does. The block is a property of the rule and the rise — some resonance of the placement dynamics that happens to equal the smaller number here — and a stem cut anywhere would end up on a block of eight.

The two readings agree about every number measured so far, because every number was measured at one rise. They disagree about what happens at another one, and they disagree cleanly: the first predicts a block of five on the 5/8 rung, and the second predicts eight everywhere.

The measurement #

The 5/8 rung has an unhealing band of two offsets — four and five places back — so it has stems to read an orbit off. Each is cut, continued for three hundred organs, and searched for an exactly repeating motif: the smallest period at which every divergence in the last stretch equals the one a period earlier to within half a degree. Nothing is fitted and no tolerance is chosen to make a near-repeat count.

The block is five. At a rise of 0.013, on a lattice counted as 5/8, both unhealing offsets settle into a five-angle motif: 139.69°, 135.70°, 283.83°, 202.27°, 282.42° at one of them, and a rotation of the same shape at the other.

So the first reading is right and the second is refuted. The block is chosen at the rung. The second attractor carries the count of the pattern it replaced.

The precession says it twice #

The block is one number; the rate at which it turns is another, and the two are not independent.

Add up one motif at the coarse rung and it comes to 1,043.9°, which is two whole turns and 323.9° — or, taken the short way round, −36.1° per block of five. At the fine rung the eight-angle motif sums to 1,462.5°, four turns and 22.5°. Measured directly on five wrecked stems there, the azimuth advance over eight organs is 22.54°, 22.73°, 22.97°, 22.73° and 22.54°.

Divide a full turn by each precession: 360/36.1 = 9.97, and 360/22.7 = 15.9. Ten and sixteen — twice the block, at both rungs.

An arrangement of blocks of m organs, each block advancing by one part in 2m of a turn, is a two-jugate pattern with 2m rows: the organs come in pairs opposite each other, and successive pairs step round by half a row. So the wrecked stem is not merely periodic. It is a whorled arrangement whose jugacy is two and whose row count is twice the smaller number of the lattice it came from.

What the orbit is, as an arrangement rather than a sequence #

A sequence of divergences is a hard thing to picture, and the block is easier to understand as a shape.

Take the coarse rung’s five angles. Two of them are near the settled divergence — 139.7° and 135.7° — and three are large: 283.8°, 202.3° and 282.4°. Walking round the stem with those five steps and repeating, the organs fall into two rows on opposite sides, with each group of five stepping the whole pattern round by 36°. A botanist looking at it would see a two-ranked stem — leaves alternating left and right — that slowly twists.

The fine rung’s eight angles do the same thing with eight: five of them near the settled 137.8°, three of them large, and the block advancing 22.7°. Both are the same species of arrangement seen at two sizes.

That is why the effective divergence is the number to quote and not the mean of the angles. Averaging 208.8° over five organs is not a statement that any two consecutive organs are 208.8° apart — none of them are — but that the block behaves like an organ arriving every fifth time at that spacing.

The wrecked stems are therefore not a curiosity of the sequence. They are a recognisable phyllotactic arrangement — one this site has otherwise only produced by asking for it — reached from a spiral pattern by removing one organ.

What a counter makes of it #

The site’s blind counter — shown the positions and nothing else — agrees about the first number and struggles with the second.

At the fine rung it reads 8/16 off two of the five wrecked stems, and 8/14, 8/19 and 8/26 off the others. At the coarse rung it reads 5/12 and 5/13. The smaller number is right every time; the larger one is anywhere from correct to nonsense.

The reason is in the geometry rather than in the counter. A block of m organs precessing slowly has two hop families of wildly unequal length: the step within a block is tiny and the step between blocks is nearly a full turn, so the second family the counter finds is whichever combination happens to come out short. The arrangement’s own answer for its row count is the precession — 2m — and the counter is not being shown the precession.

That is a limit worth recording rather than a defect. A counter reads what is short, and on a lattice whose two hops differ by a factor of ten it has one number it can trust.

Why the smaller number, and not the larger #

The measurement says which number the block is. It does not say why, and the honest position is that the account below is a hypothesis with one piece of evidence for it.

The deletion damages the front, and the front is n organs deep — the larger number. But what propagates the damage is contact: the organ that fills the hole is out of position for the organs placed against it, and the organs placed against it are its contact neighbours, which sit m and n places later. Of the two, the m-family comes round sooner, so a disturbance that is renewed every m organs is renewed before one that is renewed every n.

A cycle of length m is what a disturbance renewing itself along the shorter contact family would produce, and it is what is measured at both rungs. That is one prediction and two confirmations, which is not much.

What would settle it is a rung where m and the front’s other numbers are not related in the usual way — a bijugate stem, where the pair is 2m/2n and the smaller number is even, or a Lucas-branch stem, where the pair is 4/7 rather than a Fibonacci one. Both are buildable here and neither has been cut.

The alternative account — that the block length is set by the number of organs in the arrangement’s own repeat, which for a two-jugate lattice with 2m rows is m — has the same evidence and makes the same predictions on Fibonacci rungs. It is the sort of ambiguity that a third rung would not resolve and a differently shaped lattice would.

Is it the same orbit, moved? #

Two blocks of different lengths could still be one object seen at two rises — the same orbit, stretched. Three quantities say otherwise.

The effective divergences differ. 208.8° at the coarse rung and 182.8° at the fine one, which are 151.2° and 177.2° taken the short way. One is near two-ranked; the other is not particularly near anything.

The precessions have opposite signs. −36.1° at the coarse rung and +22.7° at the fine one. The blocks turn opposite ways.

The row counts differ. Ten and sixteen. Two arrangements with different numbers of rows are different arrangements, whatever else is true of them.

How the stem gets there #

The orbit is where a wrecked stem ends. Between the cut and the orbit there is a transient, and it is worth describing because it is what an experiment would actually watch.

Immediately after the cut the divergences are large and irregular. The organ that fills the hole is a whole divergence from where it was going; the organs after it are placed against an arrangement that has been rearranged rather than merely displaced; and for a few dozen organs the sequence has a spread of tens of degrees with no repeat in it. Then it locks, and once it has locked it does not drift: the last eighty divergences of a three-hundred-organ run repeat to within half a degree, at every unhealing offset, at both rungs.

The tail spreads are the sharpest way to say how different this is from a recovered stem. A recovered stem’s last sixty divergences have a standard deviation of about a fifth of a degree; a wrecked one’s have 54° to 83°. There is nothing between the two anywhere in the tables: no stem drifts slowly back, and no stem holds an intermediate arrangement.

That is what makes “recovered” and “wrecked” a real dichotomy rather than a threshold imposed on a continuum. The recovery test could have been set anywhere between a tenth of a degree and ten degrees and produced the same tables.

Noise does not undo it, at either rung #

The orbit was already known to survive a jostle at the finer rung. It survives one at the coarser rung too: at 0.1° and at 0.4° of displacement per organ the wrecked stem’s tail sits at 208.74° and 208.72°, against 208.80° with no noise at all. The exact repetition is lost — a motif that has to match to half a degree cannot survive four tenths of a degree of jostling — and the arrangement is unchanged.

So both attractors have basins, at both rungs, and the choice between them is made once, by which organ was removed.

What this does not say #

It does not say the rule is bistable everywhere. The coarsest rung has no unhealing offsets at all, so it has no second attractor to be bistable with — at least not one a single ablation can reach.

It does not say the block is always the smaller number. Two rungs is two data points. The prediction for the 3/5 rung would be a block of three, and there is no way to test it with a single ablation because nothing there fails to heal; it would need a larger cut, which is a different experiment with a different control.

And it does not say a plant does this. Everything here is one placement rule with one deletion, on a stem whose heights are prescribed. What it does say is that if a plant places its organs this way, an ablation in the middle of its front should leave it in a two-ranked arrangement with twice as many rows as its old smaller count — which is a strange enough prediction to be worth someone’s scalpel.

The check #

Three assertions, and the first is the one that could have gone the other way.

The block at each rung must equal the smaller parastichy number of the lattice that was cut — checked at both rungs, where the two numbers are five and eight, so a rule whose orbit was always eight would fail at the coarse rung and stop the build.

The two rungs must give different blocks. That is the claim that the number is chosen rather than inherited, stated so that it fails if a future rise gives the same answer as an existing one.

And the precession must be one part in twice the block, to within a fifth of a row, at both. That is the arithmetic behind the two-jugate reading, and it is asserted rather than described because a block that precessed by an arbitrary amount would still be a periodic orbit and would not be a whorled lattice.