Phyllotaxis
the spirals are counted, not admired
Concept

Equilibrium — where it appears

A state a rule returns to after a disturbance, as against one it merely passes through. This collection distinguishes them by intervening: a settled stem is one that comes back to the same divergence after an organ is taken out of it.

Named by 30 essays across 5 fields — each of them below, with the objects they name alongside it.

What a two-organ cut does at each rise of the 2/3 rung. At every rise the coarse rung is a lattice on, all 36 arrangements of two organs removed, with the ones that never repair counted and split by where they end up. 22 of 324 cuts across the rung reverse the stem's handedness onto the mirror of the divergence they were cut from. 40 fall instead into a cycle whose mean is half a turn, which the lattice they came from has no number for. 4 rises give only the first, 4 give only the second, and at a rise of 0.075 both happen in the same table, which is what says the fate belongs to the cut and not to the rise.

Half a turn, four at a time

At four of the nine coarse rises no wrecked cut reverses. What those stems do instead is stop settling: they repeat 171.09°, 269.53°, 189.14°, 90.23° without end, which adds to two whole turns over four organs. The mean is exactly half a turn and a counter finds four files where the lattice had three.

emergence · Coarse rung 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 The angles against the positions, rise by rise. three rises, five seeded stems each. A filled mark is a run whose angle readout returned the pair the position counter finds in the same stem; an open mark is a refusal. At 0.032 the counter says 3/5 and the angles agree on 0 of 5, refusing 5. At 0.013 the counter says 5/8 and the angles agree on 5 of 5. At 0.005 the counter says 8/13 and the angles agree on 5 of 5. The two instruments share no code path: one is given a list of angles, the other a list of coordinates.

A counter that sees no positions

This site has counted spirals two ways, and both were handed coordinates. A third counter is handed a list of angles and nothing else. It returns one number instead of two, it refuses more often, and where it refuses it would have been wrong every time.

lattices · Counting The lag between two rates, measured against the rate. Each dot is one rate: the mean gap between where the grown pattern changed its count and where the static ladder puts that transition, in rungs. Over rates from 9 to 135 nodes per rung the worst is 0.087 of a rung. A lag of one rung would put a dot on the top line.

The lag that is not there

A pattern built out of its own history should hold its old parastichy pair past the point where a fresh lattice would have changed, and the gap should grow as the shoot is hurried. Over a fifteenfold range of rate it does not — every transition lands within a tenth of a rung of where the static ladder puts it.

emergence · Tracking 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 The measurement is limited by the protractor, not by the plant. The peak falls as the reading error grows, and it falls by an arithmetic factor with nothing fitted: a position error enters two consecutive divergences with opposite signs, adding variance at every lag while the pattern's signal sits at one. At a quarter of a degree the readout is right on all 5 runs; at half a degree on 2; at a degree on 1. Below the dashed floor the peak is the largest of thirty noisy numbers rather than a measurement.

What the protractor has to be

The readout that names the parastichy number costs sixty internodes, which is cheap. It also needs every organ's position measured to better than a quarter of a degree, which is not — and the requirement follows from arithmetic rather than from care, so no amount of averaging relaxes it.

wrong · Sequence The memory of a divergence sequence, at 0.75° of scatter. With no noise at all the lag-one correlation is 0.54: the rule corrects itself, so a lattice arrives with a memory in it. Matched at the same recorded scatter, placement noise leaves -0.10, jostle noise leaves 0.66, field noise leaves 0.47. The band is ±0.13, which is what an uncorrelated sequence of this length gives.

The sequence has a memory

Every measurement this collection has made of a stem's divergence angles throws the order away. A spread is invariant to shuffling. Put the angles back in order and there is a large correlation between one and the next — 0.54 with no noise at all — which is the rule correcting itself, and which nothing had looked at.

cylinder · Sequence The order of the angles carries the count. Three stems, each held at a fixed rise so the pattern sits on one rung of the ladder. At a rise of 0.032 the positions count 3 and 5 spirals and the angles peak at 3; At a rise of 0.013 the positions count 5 and 8 spirals and the angles peak at 5; At a rise of 0.005 the positions count 8 and 13 spirals and the angles peak at 8. Each panel marks the peak and its multiples; the pale strip is what an uncorrelated sequence of this length gives.

The order carries the count

Take the divergence angles off a stem, throw away every coordinate, and autocorrelate what is left. The result is periodic at the smaller parastichy number — peaks at it and at every multiple of it. A list of angles, with no picture and no position in it, carries the spiral count.

cylinder · Sequence The memory belongs to the rise, not to the lattice. The lag-one correlation of a noiseless rising stem, against how fast it climbs the ladder. Below about sixty nodes per rung it is negative; above it, 0.54, 0.74, 0.58, 0.58, 0.55 — flat across a fivefold change in rate. The horizontal line is the same rule with the rise held FIXED, where the correlation is -0.68. So the +0.74 the earlier work called the sequence's own memory is the pattern chasing an equilibrium that is moving under it.

The memory was the rise

The earlier work measured a lag-one correlation of 0.54 in a noiseless divergence sequence and called it the sequence's own memory. Hold the rise fixed and there is no sequence at all — every angle identical — and under a disturbance the correlation is negative. The 0.54 belongs to the pattern chasing an equilibrium that is moving under it.

cylinder · Sequence Two combs, at a rise of 0.013. The autocorrelation of 760 divergence angles from one stem held at a rise of 0.013. The filled teeth are the lags at multiples of 5; the open teeth are the second comb, at the same spacing offset by 3. Reading the spacing off the first and the offset off the second gives the pair 5 and 8, which is what the position counter reports for the same stem — from angles alone, with no coordinate anywhere in the calculation.

A harmonic is a step taken twice

The spectrum contains the larger parastichy number, their sum, and echoes of the smaller one, and no ranking of peak heights separates them. What separates them is arithmetic: a harmonic is a multiple of the spacing and a family is not, and the two kinds sit in different residue classes.

cylinder · Sequence Both statistics, on the same stems, at a rise of 0.005. Five seeded stems at each disturbance, held at a fixed rise. Bars are how many returned the pair the position counter finds; open portions are refusals. The pair comes out from 0.1 to 0.25, and across that whole range the lag-one correlation of the same sequences is -0.33, -0.58, -0.59 — decisive, negative and flat. There is no trade between the two: one stem supplies both. Below the window the sequence has locked onto the sampling grid and is a cycle rather than a sample; above it there is no lattice left, at 116° of scatter.

Two readings from one stem

Three note left with the work in a row have recorded that the two statistics of a divergence sequence want opposite plants — one quiet, one disturbed. Measured on the same stems they do not. The conflict was in the interpretation of a sign, and the window in which both are readable is wide.

cylinder · Sequence A sixfold neighbourhood, and nothing to dilute. The prediction was that a rule with fewer neighbours would convert a jostle into divergence scatter more efficiently. Across a sixfold widening the ratio sits between 0.97 and 1.00, and the one point that differs is the narrowest, at 0.82 — smaller, where the prediction wanted larger. The row underneath is why: past four spacings the rule builds the identical lattice, internode for internode, so there is no neighbourhood left to widen.

The neighbourhood was already settled

The earlier work explained a small difference between two kinds of noise by saying a jostle is diluted among some thirty neighbours. Sweep the neighbourhood sixfold and the difference does not move — because past four spacings the rule builds the identical lattice, internode for internode. There was nothing to dilute.

mechanism · Noise entry Placements that went to a different minimum, per thousand. The 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" — 0.2°, 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.

Which minimum was chosen

The rule takes an argmin, so there are two completely different things noise can do to it: move the answer, or move the question. One of them can change what is chosen and the other cannot, ever — and the difference is exactly zero against one or two placements in a thousand, at amplitudes where every other measurement says the two are identical.

emergence · Noise entry Where each kind's lattice gives way. The largest amplitude at which every run still has a lattice, and the scatter it produces there. The amplitudes are incomparable — field 0.015 (fraction of the barrier), jostle 1 (degrees of azimuth), placement 0.8 (degrees of azimuth) — and the scatters agree to 19%. The boundary belongs to the pattern rather than to the disturbance: a lattice fails at about a degree and a half of scatter, and which of three mechanisms produced it does not move where.

The boundary belongs to the pattern

Three kinds of noise, in three incommensurable units, destroy a lattice at the same place — about a degree and a half of divergence scatter. The earlier work measured that of two kinds and called it a scale rather than a constant. With a third it looks less like a coincidence and more like a property of what a lattice is.

emergence · Scatter tolerance The memory belongs to the rise, not to the lattice. The lag-one correlation of a noiseless rising stem, against how fast it climbs the ladder. Below about sixty nodes per rung it is negative; above it, 0.54, 0.74, 0.58, 0.58, 0.55 — flat across a fivefold change in rate. The horizontal line is the same rule with the rise held FIXED, where the correlation is -0.68. So the +0.74 the earlier work called the sequence's own memory is the pattern chasing an equilibrium that is moving under it.

A shoot too fast to remember

Sweep the rate at which a stem climbs the ladder and the correlation between one divergence and the next changes sign — negative below about fifty-five nodes per rung, positive above it, with the flip inside one step of the grid. The instrument the earlier work proposed is unavailable on a fast shoot, and nothing said so.

emergence · Rising 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.

What the sharing costs a lattice

A disturbance inherited from the contact neighbours destroys a stem's lattice at half the displacement independent noise needs, and it moves the comb ratio a fifth of the way to a forgery's. Take the inheritance out and keep the sharing, and the damage stays and most of the ratio shift goes — so the two effects have different causes.

cylinder · Noise transport What the finer grid does to the rises already published. The 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 earlier work 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.

What a sample grid decides

The rule takes its minimum over 384 sampled azimuths, and that number has been a constant since this site's first commit. Tripling it changes nothing at the two rises the collection argues from — five runs of five, identical readings — and changes which answer appears at the one rise published as having no answer. A parameter of the program, measured rather than assumed.

emergence · Instrument ceiling A cut eight back is never undone. The 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.

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 since it was measured. 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.

emergence · Ablation A cut four back is never undone. The divergences of a stem whose organ four places back was removed, against the same stem uncut. It never returns. What it settles into repeats exactly every 8 organs — 139°, 137°, 138°, 138°, 138°, 271°, 231°, 271° — and holds that cycle for the whole 300-organ run, with a mean of 186° and a spread of 60°. A rule that corrects a displacement does not correct a deletion.

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.

emergence · Attractor What a lattice survives depends on the colour of the disturbance, sixfold. five stems for each of seven disturbances at each of six displacements, every stream normalised by its own measured spread so that a displacement of half a degree is half a degree in every row. A filled mark is a stem that still has a lattice — a divergence scatter under 2° — and an open one is a stem that does not. Independent errors survive to 0.5°; errors that remember the last one to 1.5°; errors inherited from the contact neighbours only to 0.25°. The number beside each row is the scatter a protractor would record where the lattice is standing, and it is the quantity that explains the table: what destroys a lattice is not how far an organ moves, but how far it moves relative to the organs it is placed against.

A disturbance the organs share

This collection has put three kinds of noise into the placement rule and found the lattice fails at about the same recorded scatter whichever kind it was. None of them asked what happens when the displacements are correlated between organs. At equal displacement per organ, a lattice survives three times as much of a disturbance the organs share — and what a protractor records is the part they do not.

emergence · Noise amplitude Both edges of the front heal; the middle of it does not. The 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.

A front with no middle

Take one organ out of a stem and the pattern sometimes never comes back — but that was measured on a front thirteen organs wide, where five of the thirteen offsets are beyond repair. Repeat it on a front of five and every single ablation heals. The band that cannot be undone is not a number the rule carries; it is what two fixed edges leave over.

emergence · Ablation Two answers 138° apart, and one organ holding the second one up. The repulsion the rule minimises, around the circumference of a stem at a rise of 0.008, at the height the next organ will sit at. It has two low points 138.3° apart: the slot the next organ takes, and the slot the organ after it will take. The runner-up is 13.6% higher. The organ 13 places back carries 14.6% of the energy at the winning slot and twelve places back carries 16.3% at the runner-up — and that is more than the gap, so taking that organ away makes the runner-up win and the next organ appears a whole divergence away. Neither guard is a contact of the organ being placed; twelve is one place inside the larger number of the pair this stem is climbing towards.

The organ that guards the second slot

An organ twelve places back is the furthest of any from where the next one goes, and removing it moves the next one by a whole divergence. The reason is that the rule's profile has two low points rather than one, the second is the slot after next, and that organ is holding it up. The comparison between what it holds up and how far behind it is decides the whole thing.

emergence · Ablation The block a wrecked stem settles into is the count it was cut from. A stem is cut in the middle of its front and followed for 300 organs. It does not come back to its lattice; what it does instead is repeat a fixed sequence of divergences exactly, to the resolution of the azimuth grid. Each row shows that sequence twice over, one bar per organ, drawn to the same scale. At the 5/8 rung the sequence is five angles long and advances -36.1° per block; At the 8/13 rung the sequence is eight angles long and advances 22.5° per block. Every block is the smaller number of the pair that was cut, so the second attractor carries the first one's count — and every precession is one part in twice the block, which makes each of these a two-jugate arrangement with 10 and 16 rows.

The block is the count it was cut from

A stem that never recovers from an ablation settles into a repeating block of eight angles precessing by 22.7°. Eight was also the smaller parastichy number of the lattice that was cut, which left two possibilities and no way to choose between them. Cut a stem one rung coarser and the block is five.

emergence · Attractor A second cut moves the next organ, and does not move the boundary. Every pair of organs that can be taken out of a settled stem at a rise of 0.032, where the pattern is 3/5. A row is the offset of the nearer organ removed, counted back from the tip; a column is how many further places back the second one sits. The shade is how far the next organ ends up from where it would have been, from under a degree in the palest cells to a half-turn in the darkest. The run of shaded rows ends at 5, which is the larger parastichy number, and every row below it is blank across the whole width — the largest displacement anywhere past the front is 2.34°. So the nearer of the two organs decides whether the removal is felt at all, and the second one, wherever it is put, cannot make the pattern notice an organ it was not going to notice.

A cut of two organs

One organ removed from a stem is felt out to the larger parastichy number and no further, and at the coarsest arrangement the stem always repairs itself — so the one rung where the interesting prediction could be checked had no experiment that could reach it. Two organs can. The second cut brings a parameter with it, and that parameter turns out to be a control.

emergence · Ablation

The stem that changed hands

A stem that never recovers from an ablation is supposed to end up somewhere worse than it started — a repeating block of angles, a pattern with the wrong counts in it. At the coarsest arrangement it ends up somewhere that is not worse at all: at 220.3125°, which is 360° minus the divergence it was cut from. The lattice is intact and its handedness is reversed.

emergence · Attractor

The hop that survived

A stem that never repairs after an organ is removed settles into an exactly repeating block of angles, and the period of that block is a spiral count of the lattice it was cut from. Nobody could say why. Read the wrecked stem by lags rather than by neighbours and the answer is one line: one family of the original lattice is still standing, organ by organ, and the block is its period.

emergence · Ablation

One turn per survivor

If a wrecked stem keeps one family of its old lattice exactly, then the angle it settles at is not free. Over the period of the family that survived, the pattern has to come back to where that family left it — which means the whole change in the divergence is a whole number of turns spread over a small whole number of organs. Measured, it is one turn, at seventeen of nineteen.

emergence · Ablation

Three organs and no mirror

A coarse stem cut of two organs can end up as its own mirror image — the same lattice wound the other way, counts unchanged, handedness reversed. Finer stems never do it, and two accounts of why were on the table: coarseness, or the share of the neighbourhood removed. A three-organ cut at the finer arrangements settles it, and the answer is the first.

emergence · Dose

The share was not the thing

Two organs out of a front of five reverses a stem's handedness; three out of eight does not, and neither does five out of eight, which is a larger share of a larger neighbourhood. The hypothesis under test was that the dose decides the destination. It decides whether a stem falls off its lattice and nothing about where it lands.

emergence · Dose

A wreck has a short list

Cuts of one organ through five, over two hundred and forty-six stems that never came back, land on six settled divergences between them. Removing five organs instead of one wrecks nearly everything and reaches nowhere the single cut had not already found — and half the list turns out to be the old lattice slipped by a turn, while the other half is not the old lattice at all.

emergence · Dose

Named alongside it

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

MeasurementThe placement ruleDivergence angleRiseEnsembleHonest limitsParastichy pairLatticeAblationDiscretisationNoiseArtefact

All concepts