Phyllotaxis
the spirals are counted, not admired
Where the angle comes from

The pattern the cut leaves behind

A stem that never recovers from a removal is not disordered. Its divergences settle into a cycle of eight angles and repeat it exactly for the rest of the run, and a counter reading the positions calls the result 8/16 — a two-jugate lattice. The rule has a second attractor at the same growth parameter, and an ablation is how you get to it.
21 min read 17 figures One parameter, one familyForm is not mechanismCounted, not admired

Worth reading first: The organ that was taken away · Two at a time · The bifurcation diagram.

The previous essay left five stems that never return to their lattice, and described them by their statistics: a mean divergence near 185° and a spread of fifty to eighty degrees. Those are the numbers a disordered sequence would give, and reporting them and stopping would have been a mistake, because the sequences are not disordered at all.

They repeat.

Eight angles, exactly

Take the stem whose organ eight places back was removed. Three hundred organs later its divergence sequence is

46.6° · 96.1° · 136.9° · 272.6° · 137.6° · 270.9° · 229.9° · 272.1°

and then those eight again, and again, to the resolution of the azimuth grid, for as long as the run is continued. Not approximately: the check looks for an exact repeat, with a tolerance of half a degree against a grid whose step is a quarter of one, and refuses to report a period that only nearly repeats.

A cut eight back is never undoneThe divergences of a stem whose organ eight places back was removed, against the same stem uncut. It never returns. What it settles into repeats exactly every 8 organs — 47°, 96°, 137°, 273°, 138°, 271°, 230°, 272° — and holds that cycle for the whole 300-organ run, with a mean of 186° and a spread of 83°. A rule that corrects a displacement does not correct a deletion.100200300050100organs placed after the removaldivergence, in degreescycle of 8rise 0.005 · cut 8 backgenerated from a stated rule, not drawn to look right
Fig. 1 The divergence sequence after the cut, against the same stem uncut. The faint trace is a horizontal line at 137.8°; the other one is a repeating block of eight angles that has been running since about the fortieth organ after the removal.

The cut four places back gives a different cycle of the same length —

138.5° · 136.6° · 137.8° · 138.3° · 138.1° · 270.7° · 231.3° · 271.2°

— and the cut six places back gives a cycle of four:

230.4° · 271.9° · 138.3° · 270.7°

Five of the eight angles in the first cycle and five in the second are within a degree or two of 137.8° or of 270.9°, which is 137.8° doubled and wrapped. The cycles are not arbitrary sequences; they are made mostly of the settled divergence and of two of it.

A cut six back is never undoneThe divergences of a stem whose organ six places back was removed, against the same stem uncut. It never returns. What it settles into repeats exactly every 4 organs — 230°, 272°, 138°, 271° — and holds that cycle for the whole 300-organ run, with a mean of 228° and a spread of 54°. A rule that corrects a displacement does not correct a deletion.100200300050100organs placed after the removaldivergence, in degreescycle of 4rise 0.005 · cut 6 backgenerated from a stated rule, not drawn to look right
Fig. 2 The shortest cycle in the table — four angles, from a cut six places back. Its mean is 228°, its spread is 54°, and by any summary statistic it is the most disordered stem in this collection. It is also the most exactly periodic.

Every one of them is the same arrangement

Five stems never healed, with cycles of eight, eight, eight, eight and four angles and five different-looking motifs. They are one object.

Add each cycle up. The eight angles of the stem cut eight places back sum to 1462.7°, which is four full turns and 22.7° over. The cut four places back sums to 1462.5°, and the four-angle cycle of the cut six places back sums to 911.25°, which is two and a half turns — so over two of its cycles, eight organs again, it also advances 22.5°.

Measured directly off the finished stems rather than off the motifs, the azimuth advance from an organ to the one eight places later is 22.54°, 22.73°, 22.97°, 22.73° and 22.54° on the five arrangements. Every stem that fails to heal ends up as a block of eight organs that rotates by about 22.7° per block, and the one with a four-angle cycle is that same structure whose motif happens to repeat after half a block.

The effective divergence follows: 1462.7° over eight organs is 182.84° per organ. Which is to say each organ is placed nearly opposite the one before it, and the pair of ranks so formed precesses slowly — a twenty-second of a turn every eight organs.

Take away the organ four places back, and the next one goes into the holeThe last 34 organs of a stem at a rise of 0.005, unrolled. The open circle is the organ removed — four places before the tip. The ring at the top is where the rule puts the next organ with every organ present; the filled mark beside it is where the rule puts it with that one missing. The two are 167.6° apart, against a local spacing of 25°, and the vacancy itself is 168.5° from the undisturbed answer. Nothing else differs between the two runs: same rise, same history, same rule.moved 167.6°open ring: where the next organ was going · filled: where it goes · large open circle: the organ removedrise 0.005 · cut 4 back · height ×6generated from a stated rule, not drawn to look right
Fig. 3 The cut that starts it. Four places back is inside the unhealing band, the next organ moves 168°, and the arrangement that eventually establishes itself is not a variant of the golden spiral but a different lattice altogether.

What a counter makes of it

A sequence of angles is one description. This site’s discipline is that a pattern’s claims are settled by machinery shown the positions and told nothing else, so the tops of the wrecked stems were handed to the counter.

It reads 8/16 off two of the five, and 8/14, 8/19 and 8/26 off the others. The 8 is unanimous and it is the block. The second number is what the counter makes of the precession: a block advancing 22.7° goes round in 360/22.7 = 15.9 blocks, so 16 is the arrangement’s own answer and 14, 19 and 26 are a counter struggling with a lattice whose two hop lengths are wildly unequal. That is the counter behaving as it should — this collection has spent three phases establishing that a counter which never refuses and never wobbles is a counter that is not measuring anything — and it is a reason to lean on the direct measurement of the block advance rather than on the reading.

A pair of the form (m, 2m) is the signature of a two-jugate lattice: a pattern built of two identical spirals half a turn apart rather than one. Two of these read that way, and the arrangement they are reading is genuinely two-ranked — 182.8° between consecutive organs is a hair off exactly opposite.

What the "whorled" bucket contains, at a rise of 0.0087 pairs, sharing 7 different factors, and every one of them is k and 2k. The bucket the previous census called whorled is the coarsest pattern the ladder has, repeated k times around the stem — not a residue of odd arrangements.5/107.9%5 × 1/23/67.9%3 × 1/22/46.1%2 × 1/24/85.5%4 × 1/27/145.2%7 × 1/26/122.8%6 × 1/28/160.1%8 × 1/2share of divergences at this riserise 0.008 · 3600 divergences35.4% share a factor
Fig. 4 Jugate lattices as a census rather than as a curiosity: how many arrangements at a given rise are k copies of one spiral. A pair (m, 2m) is what such a lattice returns to a counter that has never heard of jugacy.
three stems: 1, 2, 3 primordia at a time1-jugate at 137.51° counts 2 and 3; 2-jugate at 68.75° counts 2 and 4; 3-jugate at 45.84° counts 3 and 6. Every count shares the factor k, and dividing it out leaves an ordinary lattice: what a k-jugate stem is, exactly, is k copies of an ordinary one wrapped k times round.1 at a time2 and 3symmetry order 12 at a time2 and 4symmetry order 23 at a time3 and 6symmetry order 3divergences 137.51° · 68.75° · 45.84°counted 2/3 · 2/4 · 3/6
Fig. 5 The objects themselves. What the ablation produces is not quite one of these — a true two-jugate stem puts two organs at each height, and this one puts one organ at each height with consecutive organs nearly opposite — but it is what a counter shown the positions cannot distinguish from one.
A whorl and a spiral, from one lattice at two divergencesAt 120° the nodes fall on 3 rows and the two counts share a factor: the elements of each turn are level with one another. At 137.51° they never are.120° — a third of a turn3 and 6 — whorled137.51°2 and 3 — Fibonaccirise 0.055 in both panelsthe counts decide, not the eye
Fig. 6 The distinction being leaned on, from the phase that stopped calling every non-spiral arrangement a whorl: whether a pattern is k organs at a time or one at a time is a question about the heights, and the counted pair does not answer it.

Why this is a statement about the rule and not about a run

The site’s central figure is a bifurcation diagram: sweep the one parameter of the placement rule and watch which divergence it settles on. It gives a broad golden branch, a transition, and a two-whorl regime at exactly half a turn.

What the model settles on, against how fast the meristem growsA broad golden branch, a transition, and then the two-whorl regime at exactly half a turn. 8 of 27 converged settings land within 4° of the golden angle; 15 land more than 20° away.1001251501750.50011.50growth parameter Gangle the model settles on (°)goldentwo-whorlfilled: converged · hollow: still wanderingone rule, one knob
Fig. 7 The rule’s parameter sweep. Every point on it is the angle a run settles to from an arbitrary start, so the diagram shows what the rule does at each parameter and says nothing about how many different things it can do at one of them.

That diagram is built by running the model to convergence at each parameter from an arbitrary start. It is therefore a picture of which attractor a run happens to find, and by construction it cannot show a second one at the same parameter, because nothing in the procedure ever tries to reach one.

The ablation does try. Here the growth parameter never changes: the rise is 0.005 from the first organ to the last, the exponent is inverse cube throughout, and the only difference between the stem that settles at 137.84° and the stem that runs an eight-cycle is that one organ was deleted from a shared history. So the rule has at least two attractors at this parameter, and the diagram has been showing one of them for four phases.

The second one is not an exotic arrangement, which is the part worth pausing on. A divergence of 182.8° is two-ranked phyllotaxis with a slow precession, and two-ranked — distichous — arrangements are, after spiral ones, the commonest thing a plant does. The bifurcation diagram already contains a regime at half a turn; it puts it at a different value of the growth parameter, reached by making the meristem large relative to the organs. What the ablation shows is that the half-turn arrangement is available at the golden parameter too, as a second stable state of the same rule, and that the way to it is not through the parameter at all.

That is a different kind of claim about the model from any this site has made before. Everything up to here has been of the form the rule produces X at parameter G, tested by sweeping G. This is of the form the rule produces X or Y at one G, and which one depends on the history — bistability rather than a branch — and bistability is not visible to any procedure that reports where a run from an arbitrary start ends up.

Two runs of the same rule from unrelated starting anglesBoth settle at 137.0°, within 0.5° of the golden angle, from seeds 166° apart.1201301401500255075100stepdivergence angle produced at that step (°)golden anglegrowth 0.40settled spread 0.00°
Fig. 8 What settling looks like from the inside, at one parameter. The run finds the golden branch from an arbitrary start — which is the claim the site was founded on and which is untouched. What is new is that finding it from an arbitrary start is not the same as its being the only place to find.
The heads three settings of the one knob produceG=0.3 → 139.2° · G=0.62 → 143.0° · G=1.1 → 180.0°. The model was not told any of these angles.G = 0.30139.2° — goldenG = 0.62143.0° — otherG = 1.10180.0° — whorled (half)one rule, three growth rates, repulsion as 1/d^3the angle is an output
Fig. 9 The three regimes the parameter sweep produces. The cut’s cycle is not any of them: it appears at a parameter whose regime is the golden spiral, and it appears without the parameter moving.

Why eight, and why 22.7°

Neither number is put in by hand, and it is worth asking where they come from, even though the honest answer is that only half of it is understood.

The eight is the smaller parastichy number of the lattice that was cut. That is suggestive and it is not yet an explanation: the cut stems’ own smaller parastichy number is also eight, so the block could be inherited from the pattern that was destroyed or could be a property of the arrangement that replaced it, and nothing here separates those. The test is obvious and is not run in this phase — ablate a stem on the 5/8 rung, where the smaller number is five, and see whether the block is five or eight.

The 22.7° is the residue. Eight organs at a divergence near 180° advance nearly four full turns, and what is left over after those four turns is the precession. Nothing chooses it directly; it is what the rule’s own minimisation leaves when it settles into the near-opposite arrangement, and a change of a hundredth of a degree in the effective divergence would change it by a tenth. That it comes out within half a degree of the same value on five independent cuts is the interesting part, and it is why the five arrangements can be called one object rather than five.

Every transition as the rise fallsThe pair climbs 1/2 → 2/3 → 3/5 → 5/8 → 8/13 → 13/21. Consecutive transitions are 0.382, 0.382, 0.383, 0.382 of the previous rise — 1/φ² is 0.3820.0.2500.5000.75011.25-2.50-2-1.50-1-0.500log₁₀ of the rise between nodes (falling to the right is the plant growing)log₁₀ of the larger parastichy number2/33/55/88/1313/21500 rises, shortest vectors recomputed at eachratio 0.3820 against 1/φ² = 0.3820
Fig. 10 The ladder the cut stem has left. Its new arrangement is not a rung of this: 182.8° is not near the golden angle, and the pair a counter reads off it — eight and sixteen — is not a Fibonacci pair.

How it was found, which was by accident

The orbit was not looked for. It turned up while checking whether the azimuth grid decides the ablation’s answer.

The rule takes its minimum over a finite number of sampled azimuths, and every flat run in this collection uses 384 of them. At 384, with no noise at all, a stem at this rise does not settle on the golden angle. It settles into a cycle, at a mean divergence of 185.8° — the same neighbourhood the cut stems end up in — before anything has been removed.

What the finer grid does to the rises already publishedThe two rises this site has argued from and the one it published as having no answer, each read at both azimuth grids, five stems apiece. A filled mark agrees with the position counter, a half mark contradicts it, an open mark is a refusal. At 0.013 and 0.005 the readings are identical at both grids, so nothing already written depends on the sample count. At 0.008 they are not: 384 azimuths gives 5/8, 5/8 and 1152 gives 8/13, 8/13, against a counter that says 5/8. That rise was chosen in the previous phase because the three shortest lattice offsets there are within a fifth of each other, and a stem with no answer answering differently on a different grid is the object behaving as it was said to.risefive stemsthe position counter0.0133845/85/85/85/85/85/80.01311525/85/85/85/85/85/80.0053848/138/138/138/138/138/130.00511528/138/138/138/138/138/130.0083845/85/85/80.00811528/138/135/8the previous phase's settingsgenerated from a stated rule, not drawn to look right
Fig. 11 The sample grid deciding an answer, from the phase that first caught it doing so. This is the same failure in a new place, and it is the reason the ablation work was moved to 1,536 azimuths.

Every flat run on this site is a noisy one, and the noise is what keeps them off the cycle: a jostle of a quarter of a degree at 384 azimuths gives a clean 8/13 at 137.8°, which is why nobody had seen this. The quantisation is acting as a disturbance of about a quarter of a degree in its own right, and evidently a disturbance of the wrong colour.

That is worth recording as a hazard rather than as a result. A deterministic model on a coarse grid can find a periodic orbit that is a property of the grid; the reason this essay’s orbits are not that is that they survive refinement. At 1,536 azimuths and at 4,608 the same cuts give the same verdicts, and the cycles are still exact.

The 13/21 rung, at two azimuth gridsFive stems at each of five disturbances, all at a rise of 0.0019, read through a lag window of 45 from a seed of 40 nodes. A filled mark is a stem that returned 13/21, which is what the position counter finds in it; an open mark is a refusal. The only difference between the two rows is how many azimuths the placement rule samples when it takes its minimum: 384, which every run on this site has used since the foundation phase, against 1152. At the coarse grid the rule locks onto its own sample points and the readout refuses 22 of 25 stems; at the fine one it reads all 25. The ceiling was a parameter of the program.disturbance0.080.10.130.150.18384 azimuthsstep 0.94°1 of 25 read 13/21scatter 44.9°1152 azimuthsstep 0.31°25 of 25 read 13/21scatter 0.4°rise 0.0019 · seed 40 nodesgenerated from a stated rule, not drawn to look right
Fig. 12 Refinement as a check rather than as a reflex, from the phase that established the practice here: a result that changes when the grid does is a result about the grid.

Noise does not undo it either

The other way an orbit can be an artefact is by being a knife edge — a state the arithmetic balances on and that any real system would fall off. So the cut was repeated with a jostle in the run.

At 0.1° the wrecked stem’s tail is 186° ± 83°, which is the noiseless run to within the reporting precision. At 0.4°, close to the amplitude at which the lattice fails on its own, it is 149° ± 92° — still nowhere near 137.8°, with the disturbance now doing most of the spreading.

The orbit has a basin, and the basin is wide enough to hold a stem that is being jostled hard enough to be close to losing its pattern altogether.

Placements that went to a different minimum, per thousandThe rule's own counterfactual, run beside it: where would this node have gone with the noise taken away and everything else left alone? Placement noise displaces the node after the argmin, so the answer is always "here" — 0.4°, 0.8° all give zero. A jostle and a field perturbation are upstream of the choice and change one or two placements in a thousand while the lattice is still intact.field 0.0052.10.70° of scatterfield 0.00750.01.00° of scatterfield 0.010.01.12° of scatterjostle 0.41.10.87° of scatterjostle 0.81.11.12° of scatterplacement 0.40.00.94° of scatterplacement 0.80.01.42° of scatter3 runs each · a basin change is half a local spacingplacement noise: zero by construction
Fig. 13 The statistic for a disturbance changing what the rule chooses rather than where it puts the answer. Noise does this once or twice in a thousand placements and does not accumulate; a deletion does it once and the pattern stays moved.
What becomes of a seeded branch at 65 nodes per rungEach bar is 3 runs at one amplitude, divided by what the blind counter found at the top of the stem. With no noise this rate ends on the Lucas branch. Placement noise displaces the node after the rule has chosen; field noise perturbs the energy the rule chooses over. Across 48 runs, 1 reached the Fibonacci branch with the lattice intact.placement noisedegrees off the minimum00.250.50.7511.251.52field noisefraction of the barrier00.00250.0050.00750.010.01250.0150.02kept its branchchanged branchanother pairno latticeseeded 40 nodes of Lucas lattice · 3 runs per amplitude1 escape in 48 runs
Fig. 14 What noise does to a growing pattern, from the phase that mapped it. Nothing in that map is a permanent change of attractor at fixed parameter, which is what makes the ablation a different kind of disturbance rather than a larger one.

What this is worth, and what it is not

It is not a claim about plants. No plant has been ablated here, and the model’s response to a deletion is exactly the thing an experiment would be testing. If a real apex reorganises to a two-jugate arrangement after a mid-front ablation, that is a striking confirmation; if it heals every time, that is evidence about how a real apex differs from this rule, and both are more interesting than what was available before the intervention was computed.

It is a claim about the model, and a fairly strong one. The rule was introduced on this site as a mechanism whose attractor is the golden angle, and that is right and has been tested from a dozen directions. What is added here is that the golden angle is not its only attractor at a fixed parameter, that the second one is periodic rather than fixed, and that a single deletion is enough to move between them. None of the site’s earlier machinery could have found that, because all of it varies a parameter and reports where a run settles.

It changes what the previous phases’ negative results mean, slightly. The measurement and mechanism-02 phases both concluded that noise cannot move a grown pattern off its branch except by destroying it: placement noise never changes which minimum is chosen, a jostle changes it once or twice in a thousand placements, and field noise changes it only once the lattice is already coming apart. Those findings stand — they are about noise. What they were sometimes taken to mean, here included, is that the pattern’s state is effectively unique at fixed parameter, and that reading is now wrong. The state is not unique; noise is simply not the way to the other one.

And it makes the previous essay’s map more interesting than a table. Which removals a stem can undo is now a question about basins: five of the thirteen offsets inside the front put the pattern over a boundary it cannot come back across, and eight do not. A phase that wanted to draw that boundary would be mapping a basin in a space nobody has yet written down.

There is also a specific thing a plant could be looked at for, which is worth stating because so much of this collection’s output is a specification for an experiment nobody has run. If a mid-front ablation on a real apex sends the shoot two-ranked, the shoot does not need to be measured to a tenth of a degree or counted at all: two ranks of leaves is a thing that can be seen from across a greenhouse. It is the most visible prediction this site has produced, and it comes from the least measurable-looking of its results.

six limit divergences, all of them 137.5078 over a whole numberThe golden angle is the k = 1 member of a family. Real bijugate plants — teasel, *Cephalaria* — are reported near 68.75°, which is exactly half of it, and the pairs they are counted at are the Fibonacci pairs doubled.1-jugate137.5078°counts 3/52-jugate68.7539°counts 6/103-jugate45.8359°counts 9/154-jugate34.3769°counts 12/205-jugate27.5016°counts 15/256-jugate22.9180°counts 18/30limit divergence6 jugacies137.5078 / k
Fig. 15 Where the jugate arrangements sit as limits, from the thread that counted them. The cut’s cycle arriving at a two-jugate reading is a coincidence worth watching rather than a result: three arrangements were checked and two of them read (m, 2m).
Both edges of the front heal; the middle of it does notThe same removals, followed for 300 organs each. A cut one to three places back is undone within fifty organs and a cut ten to thirteen places back within sixty. A cut in between is never undone: the divergence sequence settles into an exactly repeating cycle of 4 or 8 angles and holds it for the rest of the run. The rule corrects a displacement and cannot correct a deletion.organ removed, counted back from the tiporgans placed before the stem is back on its lattice102243484never51256never7never8never9never105311591242137— the front ends here140150160rise 0.005 · pair 8/13generated from a stated rule, not drawn to look right
Fig. 16 The map as it stands: eight offsets that heal and five that do not. Every one of the five ends in a cycle, and this essay’s claim is only about the three that were examined.
The two spiral families a counter finds between 0.43 and 0.67 of the radius21 spirals one way and 34 the other, found from the point positions alone — the counter is never told the divergence angle.21 and 34 spiralscounted, not assumed
Fig. 17 And the counting that all of this is measured against, on an undisturbed head. The counter is shown positions and told nothing else, which is what makes its reading of a wrecked stem a measurement rather than a restatement of what the sequence was built from.

Where this goes next

100200300050100organs placed after the removaldivergence, in degreescycle of 8rise 0.005 · cut 8 backgenerated from a stated rule, not drawn to look right A rule that cannot heal a hole The placement rule corrects itself against a displacement — that is what the lag-one correlation of −0.6 has been saying for three phases. It does not correct itself against a deletion. Which organ is removed decides whether the stem is back on its lattice in twenty-four organs or never, and the boundary between the two is sharp, reproducible and in the middle of the front. 05e+51e+61.5e+60100200300angle around the boundary (°)repulsion from what is theregrowth 0.40 · 14 elements in playthe minimum is where the next one goes The angle is an output 137.5° is not a constant of nature. It is where a rule settles — a rule that places each new element as far as it can from the ones already there, contains no reference to the golden ratio, and reaches the same answer from starting angles a hundred and sixty degrees apart. 137.5°disorder 3177°8 peaksdisorder 747°8 peaksdisorder 11109°7 peaksdisorder 19151°7 peaksdisorder 2347°7 peaksdisorder 31133°8 peaksthe angle a spiral would need to repeatsix runs, identical equationsspread 129° A ring cannot make a spiral The peaks on a Turing ring do not all appear at once — there is a first and a second. But which two lead is decided by the starting disorder, so the angle between them comes out at 177°, then 47°, then 109°, then 151°. A divergence angle is a relationship that repeats, and this one does not.

Shares its objects with

Essays that name at least two of the same things, and that neither author linked.

  • A harmonic is a step taken twice — both name artefact, discretisation, divergence angle, equilibrium, lattice, measurement, parastichy pair, the placement rule
  • What a sample grid decides — both name artefact, discretisation, divergence angle, equilibrium, honest limits, measurement, parastichy pair, the placement rule
  • A disturbance the organs share — both name artefact, divergence angle, equilibrium, honest limits, lattice, measurement, the placement rule
  • A disturbance with a memory — both name artefact, divergence angle, honest limits, lattice, measurement, parastichy pair, the placement rule
  • The rung was not the instrument — both name artefact, counting blind, discretisation, divergence angle, honest limits, measurement, parastichy pair
  • A comb is evidence of a rule — both name divergence angle, equilibrium, lattice, measurement, parastichy pair, the placement rule

Named objects

A flat tag is an object no other essay names yet.

AblationArtefactAttractorBifurcationCounting blindDiscretisationDivergence angleEquilibriumHonest limitsJugacyLatticeMeasurementParastichy pairThe placement ruleWhorl