Phyllotaxis
the spirals are counted, not admired
Concept

Mechanism — where it appears

An account of how a pattern is produced, as against a description of what it looks like. A description with enough parameters reproduces any pattern, so the test of a mechanism is what it forbids rather than what it fits.

Named by 25 essays across 3 fields — each of them below, with the objects they name alongside it.

48 cells with a carrier that pumps auxin up the gradient. Each short line is one cell's polarisation — the neighbour it pumps towards, which is always the richer one. 10 peaks come out, at a contrast of 93%, from a start that was uniform to within 6%.

A pump that works uphill

The mechanism that actually has molecular support behind it does not use a diffusing inhibitor at all. Cells move auxin towards whichever neighbour already has more of it, which is the opposite of what transport is supposed to do — and it produces a spacing from a field that started uniform to within six per cent.

mechanism · Auxin Two runs of the same rule from unrelated starting angles. Both settle at 137.0°, within 0.5° of the golden angle, from seeds 166° apart.

Droplets with no biology in them

Douady and Couder dripped magnetised ferrofluid into a dish of silicone oil, and got spiral phyllotaxis with Fibonacci parastichy numbers out of a system containing no cells, no genes and no plant. That is the strongest evidence the pattern is physics — and the clearest warning about what a model can claim.

emergence · Analogue Three disturbances, three places to get in. The rule reads its neighbours, builds a profile of the energy at every azimuth, takes the least of it, and records a position. field noise enters at the profile; jostle noise enters at the neighbours; placement noise enters at the record. Two of the three are upstream of the choice and can change which minimum is taken; the third is downstream and never can.

The noise that arrives through the neighbours

The two kinds of noise this site had were idealisations that bracket the rule's choice. The realistic disturbance is neither: a primordium is placed exactly, and then the organ grows, so by the time the next one forms its neighbours have moved. That is a third kind, and it is invisible in every measurement a plant offers.

mechanism · Noise entry Three disturbances, three places to get in. The rule reads its neighbours, builds a profile of the energy at every azimuth, takes the least of it, and records a position. jostle noise enters at the neighbours; placement noise enters at the record. Two of the three are upstream of the choice and can change which minimum is taken; the third is downstream and never can.

A growing organ is part of the rule

Every model in the earlier essays places primordia on a surface and then treats the surface as furniture. But the surface grows between one placement and the next, and that growth reaches the rule through the only channel it has — where the neighbours are. What looks like a boundary condition turns out to be a term in the model.

mechanism · Mechanism claims The same lattice with no rule in it. A cylindrical lattice at a divergence of 137.826° and a rise of 0.005, built by placing node i at exactly i times the divergence and then displacing each azimuth independently by 0.5°. Its photograph is the photograph of the stem in the figure beside it and its parastichy pair is the same pair. The largest comb mean in it is 0.03 against a sampling band of 0.07, and the readout refuses.

A comb is evidence of a rule

Build the same lattice kinematically — every node at an exact multiple of the divergence, an independent error on each azimuth, no feedback anywhere — and the spectrum is empty. The photograph is identical and the parastichy pair is identical. The comb is not a property of the arrangement.

mechanism · Mechanism claims A lattice with an error inherited from the two contact neighbours. The autocorrelation of 759 divergence angles from a kinematic lattice at a divergence of 137.8261° and a rise of 0.005, with 0.5° of scatter on each azimuth. There is no placement rule anywhere in it: node i is put at exactly i times the divergence and then displaced. The only thing that differs between this figure and the control is the structure of the displacement — here, an error inherited from the two contact neighbours at coupling 0.7. The largest comb mean is 0.514 against a sampling band of 0.073, and the readout returns 8/13.

Errors that pass between organs

An organ's neighbours are the ones eight and thirteen places back — that is what a parastichy pair is. So a disturbance transmitted by contact is correlated at exactly the two lags the readout examines, and it does not have to be told them. Driven into a lattice with no rule in it, it returns the counted pair on eight stems out of eight.

mechanism · Noise transport Take away the organ eight places back, and the next one goes into the hole. The last 34 organs of a stem at a rise of 0.005, unrolled. The open circle is the organ removed — eight 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 16.4° apart, against a local spacing of 25°, and the vacancy itself is 22.7° from the undisturbed answer. Nothing else differs between the two runs: same rise, same history, same rule.

The organ that was taken away

Every observable this site has is read off an arrangement that was finished before the reading began, and earlier work here showed what that costs. So remove one primordium from a settled stem and place the next one against what is left. The rule has to answer. The rival account cannot, because in it no organ's position was ever computed from its neighbours.

mechanism · Ablation The ratio follows the disturbance, not the rule. The ratio of the second comb to the main comb on stems grown by the placement rule and jostled by seven different disturbances, all at 0.25° of displacement per organ and all on the same rule. Independent errors and errors with a memory return 0.76–0.81, which is the value this site measured for the rule. A periodicity at the smaller parastichy number takes it down to 0.45; errors inherited from the contact neighbours take it up to 1.09, most of the way to the 1.24 a transported disturbance gives with no rule in it at all. So the quantity separates arrangements by how their errors are related, not by whether anything computed the positions.

The ratio was never about the rule

The comb has already been retracted here as evidence that a plant computes its pattern, and one quantity was exempted from the retraction: the ratio of the two combs, which a placement rule and a transported disturbance divide differently. Drive seven disturbances through the same rule and the ratio spans 0.45 to 1.09. The exemption does not hold, and the angle sequence has nothing left.

mechanism · Noise transport What the model settles on, against how fast the meristem grows. A broad golden branch, a transition, and then the two-whorl regime at exactly half a turn. 16 of 40 converged settings land within 4° of the golden angle; 17 land more than 20° away.

Two-ranked, by two different routes

The rule produces a two-ranked stem at a coarse rise, where 180° is the only thing available, and that has been in the bifurcation diagram since the beginning. It also produces one at a fine rise, at a rise whose own answer is the golden angle, if a single organ is removed. The diagram cannot show the second, and the reason it cannot is how it is drawn.

mechanism · Attractor Two combs, at a rise of 0.005. The autocorrelation of 760 divergence angles from one stem held at a rise of 0.005. The filled teeth are the lags at multiples of 8; the open teeth are the second comb, at the same spacing offset by 5. Reading the spacing off the first and the offset off the second gives the pair 8 and 13, which is what the position counter reports for the same stem — from angles alone, with no coordinate anywhere in the calculation.

A disturbance that is not passed on

The disturbance that forges every observable measured here does two things at once — it correlates an organ's error with its contact neighbours', and it hands that error on to be handed on again. Every result about it has been unable to say which half did the work. This is the control that takes the second half away and keeps the first.

mechanism · Noise colour The damage is the sharing; the forgery is the history. Three disturbances of the same size, measured four ways. The two left columns are stems grown by the placement rule and jostled at 0.25° per organ: a disturbance shared between the contact neighbours scatters the lattice by 0.71° against white noise's 0.57°, and one inherited from them — the same sharing, passed on again at every organ — by 0.97°. The two right columns are kinematic lattices with no rule in them at all, where the whole question is what a disturbance can manufacture. The inherited one returns the pair on 8 seeds of 8 with a main comb of 0.205 against a band of 0.073; the shared one, at the same coupling and the same scatter, returns it on 1 and makes a comb of 0.099, which is the band. So sharing an error with the organs you touch does the damage, and only passing it on and on forges the evidence.

The forgery needs a history

A disturbance passed between touching organs manufactures the comb, the second comb and the parastichy pair on an arrangement with no rule in it — which is why the comb stopped being evidence. Give the organs the same correlation with no accumulation in it and the forgery collapses: one seed in eight returns a pair, and the comb is the noise floor.

mechanism · Noise transport Which arrangements carry a comb, and what each one reports. The largest comb mean in five arrangements at a rise of 0.005, all read by the same instrument at the same length, with the sampling band of 0.073 marked. Only the first is a placement rule; the other four are kinematic lattices with no rule in them, differing from one another only in how their azimuth errors are structured. Independent errors and errors with a memory leave nothing to read. A repeating error puts up a comb and names a partner that is not the lattice's. Errors inherited from the contact neighbours reproduce both the comb and the pair.

The comb was never the rule

A control is only as strong as the alternative it builds, and the earlier work built one that varied the rule while holding the disturbance fixed at independence. Five rounds of the angle-sequence thread, with what each claimed and what still stands — and why the next evidence has to come from an intervention rather than from a longer stem.

cylinder · Noise transport What a removal costs the next organ. One mark per wrecked cut in the census: how far the first organ placed after the removal ended up from where the control put it. The rows split by which organ was taken. Removing a direct chain-neighbour of the growing tip — an organ at a multiple of one of the two counted numbers — moves the next organ by between 8.9 and 30.7 degrees. Removing anything else inside the front moves it by between 62.8 and 167.6. Nothing lands between the two groups and the ratio across the gap is 2.05, so the line is a gap rather than a threshold. Taking away a neighbour is the cheap removal, which is the opposite of what the words suggest.

Removing a neighbour costs least

Take away an organ that is a direct chain-neighbour of the growing tip and the next organ moves by under thirty-one degrees. Take away anything else inside the front and it moves by at least sixty-three. Thirty cuts, two groups, a factor of two between them and nothing in the gap.

mechanism · Ablation A period of 5, and the two classes that are not with the rest. The same wrecked stem, folded on the lag it kept: one row per residue class, each drawn at the mean displacement of its own organs against the level the rest of them share. The bar through each row is the spread inside that class, and the widest of them is 10.28° — so within a class the displacement is a constant. three classes sit at the common level. The two that do not sit at 143.0° and -147.6°, equal and opposite to within 3.2 per cent, and they are neighbouring residues. The stem's own divergence is 137.97°, so an exception is one organ's step.

The damage has a period

Every wrecked stem in the census has had two numbers read out of its displacement profile and the profile itself read out of none of them. Folded on the lag the stem kept, twenty-five of the thirty are constant inside each residue class to between 0.12° and 6.09°.

mechanism · Damage shape A period of 8, and the two classes that are not with the rest. The same wrecked stem, folded on the lag it kept: one row per residue class, each drawn at the mean displacement of its own organs against the level the rest of them share. The bar through each row is the spread inside that class, and the widest of them is 0.87° — so within a class the displacement is a constant. six classes sit at the common level. The two that do not sit at 134.3° and -134.2°, equal and opposite to within 0.0 per cent, and they are neighbouring residues. The stem's own divergence is 137.44°, so an exception is one organ's step.

One level and two exceptions

Inside a wrecked stem's period most residue classes sit at one level and a couple do not. On seventeen of the thirty cuts the exceptions are exactly two, equal and opposite to within five per cent — and on all seventeen they are neighbouring residues, which was not looked for.

mechanism · Damage shape The exceptional pair, measured in divergences. One row per wrecked cut whose profile has exactly one pair of exceptional classes, drawn at the size of that pair divided by the stem's own settled divergence. Every row sits between 0.882 and 1.076, so the two chains that came apart moved by one organ's step rather than by two or by half of one. The residual is not scatter: rows are grouped by the lag the stem kept, and every lag sits wholly above the line or wholly below it. Why a surviving 5 or 7 overshoots and a surviving 4 or 8 falls short is not answered here.

A step of one organ

The balanced pair inside a wrecked stem's period measures 88.0° to 147.2° against divergences of 99.1° to 138.0° — one organ's step, to within twelve per cent on every row. The residual is not scatter: every stem keeping a 5 or a 7 overshoots and every stem keeping a 4 or an 8 falls short.

mechanism · Damage shape How far above the hole the damage becomes a pattern. One row per wrecked cut, drawn at the first organ from which every residue class stays at its own level for the rest of the run. On the 25 rows that reach it at all, it runs from 7 to 303 organs above the removed one; five rows never reach it inside the 300 organs each run is continued for. Below that point the stem is still moving, and the displacement of the first organ after the cut — the quantity that tells a cheap removal from an expensive one — is measured there. Above it, nothing changes again.

Two regimes above a hole

Below the repeating pattern there is a transient, and the boundary between them is measurable: the first organ from which every class stays at its own level runs from 7 to 303 organs above the hole on twenty-five of thirty cuts, and five never reach it inside the run.

mechanism · Damage shape Removing one wall of the slot, then the other, then both. The growing tip sits in a slot between its two chain-neighbours — the organ 5 places back and the organ 8 places back. Each bar is how far the next organ placed moves when those are removed, against a control sharing the same history. Either alone is a cheap removal: 26.3° and 4.9°, both inside the 45° that separates taking a neighbour from taking anything else. Together they move it 164.1°, against 31.2° for the two effects added, so the interaction is +132.9°. The slot is not two independent walls.

Both walls of the slot

The growing tip sits between its two chain-neighbours. Removing either alone is a cheap removal on all six lattices — 2.3° to 41.7°. Removing both together throws the next organ past the expensive line on three of them, and the interaction runs from −25.8° to +132.9°.

mechanism · Both walls Removing one wall of the slot, then the other, then both. The growing tip sits in a slot between its two chain-neighbours — the organ 5 places back and the organ 8 places back. Each bar is how far the next organ placed moves when those are removed, against a control sharing the same history. Either alone is a cheap removal: 25.8° and 12.0°, both inside the 45° that separates taking a neighbour from taking anything else. Together they move it 12.0°, against 37.7° for the two effects added, so the interaction is -25.8°. The slot is not two independent walls.

A removal that changes nothing

On one of the six lattices, taking away both walls of the slot moves the next organ 11.953125° — which is exactly, to the last digit, what taking away the larger wall alone moves it. The smaller wall's removal contributes nothing at all when the larger one is already gone.

mechanism · Both walls Which hops survive one wall, the other, and both. One row per lattice. The last three columns are the lags whose hop the cut stem still holds, unchanged from a control that shares its history — the measurement that identifies what a wrecked stem has become. Removing a single wall always leaves something standing, which is what every single-organ cut in this collection does. Removing both leaves nothing at all on two of six lattices, including the coarse rung that no single removal can wreck. A stem that keeps no rigid hop is not a wrecked lattice with a slip in it; it is a stem that is no longer a lattice.

The rung that two organs wreck

On the coarse 3/5 stem both walls of the slot heal when removed alone and wreck when removed together — and the wreck keeps no rigid hop at all. Two of the six pairs in the design end at a destination single removals almost never reach.

mechanism · Both walls Which chain the backward exception sits on, over the census. Chains are numbered from the removed organ, so chain 0 is the chain the hole was on and chain 2 is two organs along it. The exchange is at the hole's own chain on 10 of the 17 rows that carry one, against 2.8 rows for a chain drawn at random from each row's own period. That is far more often than anywhere else and it is not every row, so the position is a tendency rather than a rule — and the file says so rather than rounding it up.

Which chains changed places

A wrecked stem's displacement profile is a set of levels, one per chain, with two of them out of line — equal and opposite, on neighbouring chains. Nothing said which two. They are the hole's own chain and the one below it, on ten of the seventeen cuts that carry a pair.

mechanism · Damage shape A period of 8, with the hole's own chain at the top. Each mark is one residue class of the displacement profile, placed round a ring at its own residue, with the chain the removed organ sat on at the top. The radius is how far that class sits from the level the rest of them share. six of the eight classes sit together at the middle ring; two do not, and on this row they are one pair, equal and opposite to within a twentieth. The forward one is chain 7 and the backward one is chain 0, one residue above it, which is the order every row of the census puts them in.

One way round, seventeen times

The two chains that change places in a wrecked stem are adjacent, which is symmetric and says nothing about direction. Label them by lag from the hole and the one displaced forwards is always the lower of the two — on every row of the census, without an exception.

mechanism · Damage shape A divergence that does not move across the 5/8 band. Measured at every rise of a band on the golden branch, where a counter returns 5 and 8 spirals throughout. The settled divergence moves by 0.0469 degrees across the whole band, which is a fraction of the azimuth grid step and a fortieth of the slide across the rung it sits in. The ratio of the two contact steps does move: it falls to 1.0013 and the ordering changes hands at a rise of 0.0156, so above that rise the shorter step belongs to the 5 family and below it to the 8 family. Two of the three quantities that vary along a rung are therefore held here and the third is not, which is what makes the ends of this band a matched pair.

The angle is not the actor

Cut an organ out of two stems that settled on the same divergence and return different counted pairs, and the family left standing is different at every one of the four pairs where both stems wreck. The angle is held to a hundredth of a degree underneath.

cylinder · Same angle Shared counted numbers against shared survivors. One row per matched pair, over both branches. The third column is the counted numbers the two rungs have in common and the fourth is the families both stems leave standing; on every row the two are the same set. The row whose rungs share no counted number is the one whose stems share no survivor, which is what makes this a claim about an intersection rather than a restatement that a survivor is usually a contact family. four rows, and the empty case is one of them.

Where the survivors meet

At a matched pair the two stems keep exactly the counted numbers their two pairs have in common — the 5 where 3/5 meets 5/8, the 8 where 5/8 meets 8/13, the 7 where 4/7 meets 7/11, and nothing at all where 3/5 meets 8/13. Four rows, including the empty one.

cylinder · Same angle

The last of three quantities

The pair, the divergence and the step ordering move together when the rise is swept, and for a long time no result could be attributed to any of them. Two designs later, two are ruled out as sufficient and the third has never been held still — because holding it is what a rung already does.

cylinder · Same angle

Named alongside it

The objects these essays reach for when they reach for this one.

MeasurementAblationClaim testingHonest limitsLattice offsetControlNegative resultParastichy pairThe placement ruleDivergence angleRigid hopDescription versus mechanism

All concepts