The refutation index
This subject arrives with more confident wrong statements attached than any other. Some of them are decorative and some are in textbooks, and almost none of them comes with the number that would settle it — which is what the whole collection is a reply to: state the claim, state the test, report the number, and say plainly when the claim survives.
There are 53 of them below. Each carries the verdict, what was found instead, and — the part that matters — the quantity that settles it, computed while this page is built out of the same libraries the figures are drawn from. None of these numbers is typed in. If one of them stopped agreeing with the essay it points at, the build would stop rather than serve the page.
The verdicts are not all wrong, and two of the most interesting are not. A claim here usually fails by being right about one quantity and quoted about another — the nautilus really is a logarithmic spiral, a sunflower really does show consecutive Fibonacci numbers somewhere on it, Lewis's law really does hold of some tissue. Distinguishing the true half from the travelling half is most of the work, and it is the reason this site computes rather than cites.
Wrong
35 claims
The nautilus shell is a golden spiral.
It is a logarithmic spiral, which is the true half and the reason the claim survives being checked casually. It is not this logarithmic spiral. A golden spiral grows by the golden ratio every quarter turn, which is a factor of φ⁴ ≈ 6.85 per whole turn; a nautilus grows by about 3.2. The gap is a factor of more than two, which is not a measurement dispute — and the one free choice in fitting it, where the centre of the shell is assumed to be, moves the answer by under a percent for a quarter-radius error, so the disagreement cannot be blamed on the fit.
What settles it: 6.854 per turn claimed against about 3.2 measured — a factor of 2.14 — where the same fitting routine recovers a drawn golden spiral's growth as 6.854, to a residual of 4e-15. Argued in The nautilus question.
Lewis's law and the Aboav–Weaire relation are two properties of the same cellular tissue.
They are cited side by side and they want opposite tissue. Aboav–Weaire says a cell with many sides is surrounded by cells with few, and its parameter a comes out near the value the literature quotes — on the ordered head, where Lewis's law has just failed. On the random set, where Lewis's law holds well, Aboav's parameter is roughly half of it. Neither relation is wrong; what is wrong is the sentence that puts them together, and no amount of care about either one on its own would find that.
What settles it: a = 1.176 on the ordered head, the value usually quoted, against 0.697 on the random set — the ordering that breaks Lewis's law is what makes Aboav's hold. Argued in Two laws that want opposite tissue.
Raup's shell morphospace has an impossible region where W·D < 1.
The boundary is real and the label is an interpretation the geometry does not supply. W·D < 1 means successive whorls are in contact rather than free of one another, and whorls in contact is the ordinary condition of most gastropods — a snail, not an impossibility. The region is not empty; it is where a large share of living shells sit. This is the clearest case on the site of a computed diagram acquiring, in transmission, a claim that was never computed.
What settles it: 38.6% of the sampled morphospace has whorls in contact — 261 of 676 cells — and a shell at W = 2, D = 0.4 has W·D = 0.80, which the geometry calls "involute — whorls in contact" and not impossible. Argued in Raup's three numbers.
Counts that share a factor mean the plant is whorled.
The counts cannot decide it. An ordinary single-jugate lattice — one primordium at a time, at a divergence near half a turn — counts a pair sharing a factor, and a genuine bijugate stem counts the same pair. What separates them is the rotational symmetry of the point set, which is a measurement on positions rather than on counts, and it is not recoverable from a published parastichy pair. Every counter this site had before the essays on cylinders asked how far it is from element i to element i+m, which is already a claim that the elements arrived one at a time.
What settles it: an ordinary lattice at 180.5° counts 2 and 4 at rotational symmetry order 1, and a genuine bijugate stem counts 2 and 4 at order 2. Argued in What "whorled" was hiding.
What a stem's divergence angles have to tell you is their average and their spread.
Every summary this collection had taken of a divergence sequence — a mean, a scatter, a classified pair — is invariant to shuffling the angles, so four rounds of work of work had thrown the order away without saying so. The order carries the spiral count. Hold the rise fixed, autocorrelate the sequence, and it is periodic at the smaller parastichy number, with peaks at its multiples — so a list of angles with no coordinate in it names the pattern that a photograph is normally needed for. It reads the Lucas ladder too, which is what shows it is reading the lattice rather than Fibonacci.
What settles it: at a rise of 0.013 the positions count 5 and 8 spirals and the angles alone peak at 5, at a height of 0.68 against a sampling band of 0.11. Argued in The order carries the count.
The autocorrelation of a divergence sequence gives one parastichy number, and the second is out of reach of a list of angles.
The lags whose correlation clears the band are not a list — they are two arithmetic progressions. The first is the multiples of the smaller parastichy number, which is the comb the earlier work read; the second is at the same spacing, displaced by the difference of the pair. So the spacing gives m, the offset gives n − m, and a list of angles with no coordinate in it returns both numbers. It costs about four times the internodes the single number costs, because the second comb is the weaker of the two.
What settles it: at a rise of 0.005 the positions count 8 and 13 and the angles alone return 8 and 13 on 5 of 5 stems. Argued in The second comb.
A comb in a plant's divergence angles is a fact about the lattice it has.
Build the same lattice kinematically — every node at an exact multiple of the same divergence, at the same rise, with an independent error on each azimuth and no rule anywhere — and there is no comb at any lag. The arrangement is identical in every respect a photograph records: same divergence, same rise, same parastichy pair, same scatter. What differs is the order the angles were made in, so the comb is evidence that the organs were placed one at a time in the presence of the ones already there, and it is the first quantity on this site that separates a process from a form.
What settles it: the rule's own stem has a main comb of 0.64 and a second of 0.41; the same lattice with no rule in it gives 0.02, and is refused at every scatter. Argued in A comb is evidence of a rule.
If rational angles make ordered tissue, the most badly approximable angle makes the most disordered.
The repair this site's own sweep invites, tested where it should hold: the Farey interval between 8/21 and 5/13, whose noble number is the golden angle. μ₂ runs from 0.019 at the ends to 0.508 near the far end, and the golden angle sits in the middle of the range at 0.254, with a sixth of the interval above it. Eighth criterion, eighth negative — and the first one this collection invented in order to test it.
What settles it: on the interval the maximum is 0.508 at 138.4° and the golden angle is 0.253. Argued in The most irrational is not the most disordered.
Which divergence angles look ordered is a property of the angles.
It is a property of the angle and the number of organs, and the dependence is steep. At an exact rational μ₂ halves for every doubling of the head, and the dip around it narrows by a factor of four — the prediction was 1/n and the measurement is 1/n², because the drift a rational has to stay inside is swept around a growing radius while the rings it must not cross are crowding together. A head of three hundred organs cannot tell 138.4615° from 138.48°; a head of twelve hundred tells it from 138.4625°.
What settles it: the dip at 5/13 has half-widths of 0.0162°, 0.0048°, 0.0010° at 300, 600, 1200 points. Argued in A dip belongs to the head.
A branching exponent fitted to a whole tree is dragged towards the answer its uninformative junctions give.
It is not. Two hundred junctions spanning daughter ratios from 0.05 to 1, from a tree built at an exponent of exactly three, fit 2.85–2.99; the twig band alone from the same tree fits 1.6–1.7. Least squares already weights by leverage, and leverage collapses as the sixth power of the daughter ratio, so three quarters of a tree's sits in the junctions above 0.7 and the twigs carry six hundredths of a per cent. The danger is not a mixed sample — it is a sample selected by what a person can reach, which on a standing tree is the twigs.
What settles it: a fit over 200 junctions spanning the whole range returns 2.85 with 167 of them retained, where the twig band alone returns 1.7. Argued in The band decides the answer.
A repeating pattern in a plant's divergence angles shows that the plant computes its arrangement rather than merely having one.
This site made the claim itself and withdrew it later here. The control it rested on built a lattice kinematically with an independent error on each azimuth and found no correlation at any lag — but independence was doing the work. An organ's contact neighbours are the ones m and n places back in the order of production, so a disturbance transmitted between touching organs is correlated at exactly the lags the readout examines, without being told them. Driven into the same rule-free lattice it returns the counted parastichy pair on eight seeds of eight, with the comb strengths of a real stem. What survives is narrower: a comb rules out independent errors, and the ratio of the two combs constrains how the transmission is weighted.
What settles it: a lattice with no rule in it returns 8/13 on 8 of 8 seeds, at a main comb of 0.55 against the rule's 0.64. Argued in Errors that pass between organs.
The parastichy pair can be read from a plant's angles only up to 8/13 — finer patterns are beyond the method.
Another of this site's own claims, written into a round of work plan on the basis of reasoning rather than measurement. The two requirements are real — three teeth of the main comb inside the lag window, and the larger parastichy number outside it — but written as inequalities they give a band of [3m, 3n), which is never empty. What actually stopped the finer rung being read was two constants: a seed of eight nodes, which at that rise is a cluster rather than a piece of lattice, and an azimuth grid of 384 samples, on which the placement rule locks onto its own sample points. Widen the window to 45, lengthen the seed and triple the grid and the rung reads on five stems of five at every disturbance tried.
What settles it: the 13/21 rung returns on 25 of 25 stems at 1,152 azimuths and 1 of 25 at 384. Argued in The rung was not the instrument.
How a plant's divergence sequence divides its correlation between two combs says whether a placement rule made the pattern.
This site's own claim, kept when the comb itself was withdrawn earlier here, and it does not survive being driven. Take the placement rule — unmodified — and disturb it seven ways at the same displacement per organ, all leaving a lattice standing. The ratio runs from 0.45 for a disturbance that repeats at the smaller parastichy number to 1.09 for one inherited from the contact neighbours, against 0.80 for independent errors and 1.24 for a kinematic lattice with no rule in it at all. A plant that both computes its positions and passes displacements between touching organs — which is the likeliest thing a plant is — reads in the band that was to count against a rule. The quantity measures how the errors are related, not what placed the organs.
What settles it: one rule, seven disturbances: 0.45 to 1.09, against 0.80 for independent errors. Argued in The ratio was never about the rule.
The comb ratio a placement rule produces is a constant of the rule.
It is neither constant across a rung nor converged at the grid it was measured on. Swept across the 5/8 and 8/13 rungs at a fixed share of the local spacing, the ratio is a U: a floor near 0.79 about two thirds of the way up a rung, climbing to 2.8 as a transition approaches, and the same shape on both rungs. The explanation offered in advance — that the hop asymmetry changes with the rise and takes the ratio with it — is refuted on its own ground: several rises whose asymmetry agrees within a twentieth give ratios spanning a factor of four. Separately, the azimuth grid every flat run on this site uses has a step of 0.94° against a disturbance of 0.25°, and refining it moves the floor from 0.62 to 0.79.
What settles it: a floor of 0.71 and 0.79 rising to 2.80 inside a rung, and 0.62 against 0.82 between the coarsest grid and the finest. Argued in The ratio was the floor of a curve.
The order a rational divergence angle produces is a matter of how nearly the angle misses a simpler fraction.
The objection was this site's own, raised against its own law: the dip's half-width had been measured only at Fibonacci convergents, which sit in the emptiest neighbourhood a denominator has, so a law reading w ≈ 150q/n² could as easily have been about approximation quality. Four fractions of each of three denominators, at one head size each, settle it — the widths inside a family agree within a seventh to a quarter, where the same quantity across denominators spans a factor of four. The convergent is not the odd one out in any family, and the largest partial quotient of the continued fraction orders the families backwards from what the alternative requires.
What settles it: within a denominator the widths span ×1.15, ×1.14, ×1.28, where n²·w across the three spans ×3.9. Argued in Four fractions with one denominator.
Nothing a botanist can do to a plant would distinguish a pattern that was computed from one that merely has a history in its errors.
An observation cannot; an intervention can, and the prediction is over-determined before the experiment is designed. Remove one organ from a settled apex and a placement rule must answer, because the neighbourhood it minimises over has changed — the next organ moves by between 2.6° and 168°, against a plant's own divergence scatter of about half a degree. An account in which organ i sits at i times the divergence with an inherited error predicts no displacement at any offset, because nothing in it is a function of which organs exist. And the number of organs whose removal matters is the larger parastichy number, measured at three rungs, so the experiment returns a spirals count as a by-product.
What settles it: the front is 5, 8, 13 organs wide at pairs 3/5, 5/8, 8/13 — the larger parastichy number every time — against a predicted displacement of 0° from the rival. Argued in The organ that was taken away.
The offsets a stem cannot recover an ablation from are a property of the placement rule, so any stem it grows has them.
This site's own claim, made from one rung and corrected at three. The band that never heals is nought offsets wide at the 3/5 rung, two at 5/8 and five at 8/13, so it is neither a constant nor a fixed share of the front. What is constant is the edges: a cut within three organs of the tip is undone at every rung and so is one within three or four of the far edge of the front, and the band is whatever is left in the middle. A front of five has no middle, so a stem at a low count cannot be permanently disturbed by a single ablation at all — which is a selection criterion for the experiment rather than a curiosity.
What settles it: the unhealing band is 0 of 5, 2 of 8, 5 of 13 organs at pairs 3/5, 5/8, 8/13, with a tip edge of 5, 3, 3 at all three. Argued in A front with no middle.
The periodic orbit a wrecked stem settles into is what this rule does to a deletion, so its block is a property of the rule.
The block is chosen at the rung. A stem cut in the middle of its front at the 8/13 rung ends as a block of eight organs precessing by 22.7°; one cut at the 5/8 rung ends as a block of five precessing by −36.1°, at an effective divergence of 208.8° rather than 182.8°. The block is the smaller parastichy number of the lattice that was cut, at both rungs, so the second attractor carries the first one's count. And the precession is one part in twice the block at both — 360/36.1 = 9.97 and 360/22.7 = 15.9 — which makes each of them a two-jugate arrangement with twice the block's rows.
What settles it: 5/8 → a block of 5 precessing -36.1° (10.0 rows), 8/13 → a block of 8 precessing 22.5° (16.0 rows). Argued in The block is the count it was cut from.
The disorder dip at a rational divergence has a width, and integrating the profile rather than crossing a level would measure it.
The integral fixes what it was meant to fix and reveals that the quantity does not exist. On an axis scaled as δ·n²/q the equivalent width agrees between two head sizes to within a few per cent, and it measures the two fractions the level-crossing method had to refuse. But it never settles: at windows of 50, 100, 200, 400 and 800 scaled units it climbs and then turns over, because past a certain reach the profile it is subtracting from is the next rational's shoulder rather than a background. The dip has a depth and a scale that goes as q/n²; it has no outer edge, so the number reported is the window.
What settles it: 13/34's equivalent width is 100, 152, 325, 445, 242 scaled units at windows of 50, 100, 200, 400, 800 — a spread of ×4.5 with no flat stretch anywhere. Argued in A dip with no outer edge.
After the width laws are taken out, what is left is ordered by how crowded the fraction's neighbourhood is.
This site's own claim, withdrawn on the ground that both instruments' free parameters are set by the crowding, and now settled by a construction rather than by an instrument. Seven fractions whose nearest other rationals sit within 3.1% of one distance, with denominators spanning a factor of 2.47, each measured at a head size chosen so that all of them share one scaled unit — which makes a window the same window in degrees and the same fraction of the way to the neighbour for every member. At a window inside the dip their equivalent widths agree to a per cent. There is no residual: after the depth and the q/n² scale, nothing about the dip belongs to the fraction. The spread that appears at wider windows follows the denominator, not the crowding, and follows the crowding backwards.
What settles it: the neighbourhood at 0.31°: ×1.010, ×1.022, ×1.075, ×1.147 at windows of 25, 50, 100, 200; the neighbourhood at 0.36°: ×1.001, ×1.028, ×1.091, ×1.112 at windows of 25, 50, 100, 200 — rank correlation with q 0.86 and 1.00, with the crowding -0.67 and -0.20. Argued in The residual was the window.
A pattern too coarsely arranged for a single ablation to disturb is robust to that kind of damage.
It is immune to one organ and not to two. At the 3/5 arrangement every single-organ removal repairs itself, at three cut points; a two-organ cut leaves four of sixty-four pairs of offsets unrepaired. What those stems settle into is not damage either — three of the four take the divergence 360° minus the one they were cut from, which is the same lattice with the opposite handedness, and a counter shown their positions returns the pair they were cut from. The fourth takes a 3/4 lattice. The front is unchanged at five whatever the second organ does, so the count the intervention returns is not affected by the extra removal.
What settles it: 4 of 64 pairs never repair; 3 settle at 220.31° against a mirror of 220.31° for a stem grown at 139.69°. Argued in The stem that changed hands.
The repeating block a wrecked stem settles into is the smaller parastichy number of the lattice that was cut.
This site's own reading, drawn from two rises and refuted by four more. At a rise of 0.020 the pair is 3/5 and the block is five, the larger; at 0.008 the pair is 5/8 and both five and eight appear at different offsets; on the Lucas branch at 4/7 the block is seven at four unrepaired offsets of five. Pooled over six lattices, eleven offsets give the smaller number and seven the larger. What survives is that the block is A parastichy number of the lattice that was cut, at eighteen of nineteen offsets, and that which of the two is decided by the offset rather than by the pattern.
What settles it: golden, rise 0.020: 3/5 → 5; golden, rise 0.013: 5/8 → 5; golden, rise 0.008: 5/8 → 5, 8; golden, rise 0.005: 8/13 → 4, 8; Lucas, rise 0.020: 4/7 → 4; Lucas, rise 0.013: 4/7 → 4, 7. Argued in Two accounts of one number.
A wrecked stem's orbit is a two-jugate arrangement, so its block is the count of the ordinary lattice underneath it.
Tested where the ordinary lattice underneath is not hypothetical. A rule that places two organs at a time grows a genuine bijugate lattice — rotational symmetry of order two, measured on the positions — carrying 6/10 at a rise of 0.0065, whose underlying ordinary lattice is 3/5. Cut, the unrepaired stems settle into orbits of six and ten. Neither three nor five appears at any offset. The relation the reading also required, that an orbit precesses by one part in twice its block, holds at three of five lattices and misses by fifteen per cent on the Lucas branch.
What settles it: pair 6/10 with an underlying 3/5: blocks 6 and 10, symmetry 2 before the cut and 1/1/1/1/1/1/1/1 after. Argued in What a cut costs a whorl.
If a plant's disturbances are inherited between touching organs, its divergence sequence carries a slow wander as well as a comb.
This site's own proposal, and the arithmetic that kills it is one line. A divergence is the difference of two organs' errors, and differencing removes low-frequency power — so the wander is in the disturbance and not in the sequence a plant hands over. The inherited disturbance's own block means carry forty-nine times an independent stream's variance and the divergences it produces carry 0.83, where independent errors give 0.96. The statistic is in fact the disturbance's autocorrelation evaluated at the block size, so it is not a second reading at all; what an inherited disturbance leaves in a sequence is a hole at the offsets it couples at.
What settles it: independent: ×0.93 in the stream, 0.96 in the divergences; a memory, ρ = 0.9: ×16.24 in the stream, 10.25 in the divergences; inherited, a = 0.7: ×49.09 in the stream, 0.83 in the divergences; shared once, a = 0.7: ×2.55 in the stream, 0.95 in the divergences. Argued in A difference forgets a drift.
A stem knocked off its lattice ends up somewhere unpredictable.
It ends up at the divergence it was cut from plus a whole number of turns divided by the period of the family that survived, which is a list rather than a range. Over the wrecked offsets of six lattices the slip closes on a whole turn to within three degrees at every one, and on exactly one turn at seventeen of nineteen. So a wreck is a dislocation with a stated size: the old lattice with one extra turn threaded through every p organs. Swept over cuts of one organ through five at one arrangement, more than three hundred wrecked stems reach eight settled divergences between them, and the largest cut reaches nowhere the smallest had not.
What settles it: 17 of 19 wrecked offsets slip by exactly one turn over the lag they kept, all of them closing to within 2.97° of a whole turn — and cuts of one to five organs at one rung reach 6 destinations, 3 of which keep a lag of the lattice they were cut from. Argued in One turn per survivor.
How deep a removal is felt is set by how finely the stem is patterned.
It is set by the lattice and not by anything the rise decides. Grow a stem on the Fibonacci branch and one on the Lucas branch at the same rise, with the same rule, the same heights and the same azimuth grid, and their fronts differ at every rise tried. Do it at seven rises and the ordering of the two fronts changes hands four times, so no function of the rise gives the column; and one lattice appearing at four rises across a factor of two gives the same front at all four. Two lattices at one rise give two fronts, and one lattice at four rises gives one front.
What settles it: at 7 rises the two branches never agree — gaps of 1, -1, 1, 1, 1, -2, 1 — with 4 reversals, while the golden branch carries 5/8 at 4 of them across a factor of 2.00 in the rise and gives a front of 8 at every one. Argued in Seven rises and two seeds.
Enough damage will turn any stem into its own mirror image.
The mirror belongs to the coarse lattice rather than to the dose. Two organs out of a front of five reverses a 3/5 stem's handedness exactly, counted pair and hop order unchanged. Five organs out of a front of eight is a larger share of a larger neighbourhood, wrecks sixty-three of sixty-four arrangements, and never gets within eleven degrees of the mirror; three organs at 8/13 wrecks seventy of seventy-two and does not either. What the dose decides is whether a stem falls off its lattice, and not where it lands.
What settles it: 3/5 at 40.0% of its front reaches the mirror exactly (1 of 20), while 5/8 at 62.5% wrecks 63 of 64 and comes no closer than 11.8°, and 8/13 at 23.1% wrecks 70 of 72 and comes no closer than 5.6°. Argued in The share was not the thing.
A placement rule passes a disturbance that outlasts its own neighbourhood, so the crossover tracks how deep the rule looks.
Neither half survives. The obvious knob — how many organs the sum runs over — is not binding at all: swept from fifteen organs to eighty-five it moves the wander by less than changing the random seed does. The parameter that does bind is the falloff exponent, which sets the neighbourhood at between three and a hundred and eighty-two organs, and across that factor of sixty there is no crossover anywhere and the deep rule passes 3.7 times as much as the shallow one. Read without the normalisation, the drift that reaches the divergences changes by a fifth while the scatter it is divided by changes by 1.57 — so nearly the whole effect was in the denominator.
What settles it: the loop bound over 15–85 organs moves the wander by ×1.30 against ×3.28 between seeds, while the neighbourhood over 3–182 organs moves it by ×3.68 and the drift through by ×1.20. Argued in The drift goes the other way.
A damaged stem keeps the family whose step is shortest, because a placement rule holds its nearest neighbours.
It keeps a family that is one of the two shortest — at twenty-nine of thirty wrecked offsets over ten lattices — and not the shorter of the two. At seventeen of the twenty-nine the survivor is the second shortest step, by between a quarter of a per cent and eleven. What does most of the work is where the cut landed: the smaller count while the removal is no further back than it, the larger beyond, which holds at twenty-five of thirty. And that is refuted in turn by a pair of runs — two 4/7 lattices, cut five places back, keeping different families — so no function of the counted pair and the offset is the account either.
What settles it: 29 of 30 survivors are a contact family and 17 of those are the second-shortest step, the offset rule accounts for 25 of 30 — and 4/7 cut 5 places back keeps lag 4 at a rise of 0.02 and lag 7 at 0.013. Argued in Not the shorter of the two.
A pattern whose two spiral counts share a factor arrives several organs at a time.
The counts are the same and the symmetry is not. Cuts of several organs send a 5/8 stem to three destinations a counter reports as 2/6, 4/6 and 3/6 — the signature of a whorled pattern — and not one of them has any rotational symmetry at any order from two to eight, while a stem the whorled rule actually grew has exactly the order it was grown at. What the shared factor reports is where the divergence landed: each of the three has a file of two or three organs closing to within twenty degrees, against thirty-five and more for every destination without one, and thirty-two for the golden angle.
What settles it: 3 destinations counted 2/6, 4/6, 3/6 have rotational symmetry of order 1, where a stem grown two organs at a time is counted 2/6 at order 2 and the undisturbed stem is counted 5/8 at order 1 — and the three close a file to within 19.9° against 35.0° for the rest. Argued in The symmetry that is not there.
A stem that never repairs after a cut settles on some divergence, whichever one it is.
At the coarse rung it often settles on nothing at all. Of seventy-one two-organ cuts that never repair across the 2/3 rung, twenty-two reverse the stem onto the mirror of the divergence it was cut from — and forty fall into an exact four-cycle whose mean is half a turn, with a counter finding four files where the lattice had three. No cell at either finer rung comes within four degrees of it. The share that reverses falls with the width of the front, 6.8 per cent at three, 4.7 at five and none at eight, which is what a reachability account predicts and is the smaller of the two results.
What settles it: 2/3 at a front of 3: 22 of 324 cuts reverse (6.8%), 3/5 at a front of 5: 6 of 128 cuts reverse (4.7%), 5/8 at a front of 8: 0 of 121 cuts reverse (0.0%) — with 40 of the coarse rung's 71 wrecked cells cycling about half a turn instead, over the 13 rises of 19 whose divergence settles at all. Argued in Half a turn, four at a time.
A deep rule and a shallow one never change places, however slow the disturbance.
They change places at a correlation length of two or three organs. Read as degrees of drift getting through rather than as a ratio, and compared seed by seed so that the seed spread is paired out, the share of comparisons the deeper rule wins climbs from under a quarter under white noise to over nine tenths at a hundred organs of memory. The earlier sweep could not see it because the ratio it read has the divergence scatter in its denominator and the scatter moves the other way. The crossing does not track the depth: every pair of exponents crosses inside the first ten organs, including the pair whose neighbourhoods are 182 organs and 3.
What settles it: pooled over 48 paired comparisons the deeper rule wins 22.9% under white noise and 93.8% at 99 organs of memory, crossing between 1.4 and 2.8 organs — and the pair whose neighbourhoods are 182 and 3 organs crosses at 2.8. Argued in The corner that does not move.
A wrecked stem keeps whichever of its two contact families has the shorter step.
The census scored it at twelve of thirty and could not say why, because on twenty-five of those rows the shortest step belongs to the larger family and the two readings make one prediction. A band settles it: around the rise where the two steps change places, the counted pair holds and the settled divergence holds to a twentieth of a degree while the ordering reverses. Cut at every offset across two such bands, on two branches and two counted pairs, and the family left standing never changes — so the reading is right on part of each band and wrong on the rest, for an answer that moved nowhere.
What settles it: the 5/8 band on the golden branch keeps the 5 at all 24 wrecked cuts, with the divergence held to 0.047° and the ordering changing hands at 0.0156, the 4/7 band on the Lucas branch keeps the 4 at all 31 wrecked cuts, with the divergence held to 0.020° and the ordering changing hands at 0.0225 — the shortest-hop reading right at 20 of 55 cuts for an answer that never changed. Argued in The ordering was not the actor.
The middle rung has no corner: a deep rule beats a shallow one there at every correlation length.
The panel that said so was pinned. A paired count over six seeds cannot print more than six, and four of that panel's six cells read six of six — so no feature could have appeared there whatever the lattice did, and the flat middle was a reading of the ceiling. Raising the jostle brings the cells down: at half a degree not one is pinned, and at a degree the comparison shows a clean crossing, with the deeper rule taking two seeds of six under white noise and every one of them once the disturbance remembers itself. Every stem at every amplitude is still counted at the pair the rise carries, from its points.
What settles it: at a jostle of 0.25° the cells read 5/6 6/6 6/6 6/6 6/6 4/6 (4 middle cells pinned, no corner), at a jostle of 0.5° the cells read 4/6 5/6 5/6 5/6 5/6 3/6 (0 middle cells pinned, mixed), at a jostle of 1° the cells read 2/6 4/6 4/6 6/6 6/6 6/6 (2 middle cells pinned, crossing) — a crossing appearing at 1°, with every stem still counted 5/8 from its positions. Argued in Six of six is not a measurement.
The rises below the ladder's finest rung would settle onto a lattice given a long enough run.
A wall, not a budget. Every rise and every starting angle grown to 1,200 organs and again to 3,200 gives an identical table: the same runs settle, at the same organ, on the same divergence, and not one that failed at the shorter length succeeds at the longer. Where settling happens at all it happens fast — the slowest anywhere is well inside the stem an ablation run grows before it cuts. What falls with the rise is not the speed but the share of starting angles that reach a lattice at all, which collapses from seven of nine to one.
What settles it: 72 of 72 pairs of runs are identical and 0 settle only at the longer length, with the slowest settling anywhere at 290 organs — while the share of starting angles that settle falls from 7 of 9 at a rise of 0.03 to 1 at 0.003. Argued in A wall and not a budget.
Right about one quantity, wrong about another
14 claims
Sunflower spirals are always consecutive Fibonacci numbers.
Fibonacci is not what the geometry hands over. Sweep the divergence angle at a fine rise and consecutive Fibonacci pairs are a minority of outcomes — and the share falls as the pattern gets finer, which is the opposite of the way the claim is usually told. Reading the census generously, up to jugacy, roughly triples the Fibonacci share, and every point of that increase is one of the coarsest pairs repeated k times, which describes no plant. What actually makes plants Fibonacci is continuity from a coarse start: stems grown from divergences spread over two thirds of a circle end on the same pair, because the pattern is on the ladder before the ladder has many rungs.
What settles it: 14.7% of divergences give a consecutive Fibonacci pair at a rise of 0.008 — 50.1% read up to jugacy — while 1 of 16 stems grown down to that rise from scattered starts end on one. Argued in How often is it Fibonacci.
137.5° is the angle that packs seeds most efficiently.
This collection first reported that no packing criterion singles the golden angle out and that three criteria give three winners. That reading was divided by the mean area of every bounded cell, and the cells just inside the edge of a head are enormous, so the scale belonged to the rim. Divided by the interior's own cell area the answer splits in two. The criteria about distance — how close the nearest pair come and how large the largest empty patch is — put the golden angle at the top of a half-degree grid of angles, and away from the centre of the head they tie it with every other noble angle at values that are Hurwitz's theorem read off a set of points. The criteria about cells — how even their areas are and how many are hexagons — are won by rational angles, whose points sit on rays with gaps between them that grow without bound. So the claim is right about spacing and wrong about cells, and where it is right it is true of a class of angles rather than of 137.5° alone.
What settles it: at 800 points, read on the interior's scale, the golden angle's closest pair is 0.903 of the mean spacing against 0.133 for a divergence of 5/13 of a turn and 0.887 for the Lucas angle, and the rational angle's largest gap is 4.44 times the golden angle's — while on area evenness the two score 0.011 and 0.011, which no reading distinguishes. Argued in Packing, measured against the interior.
A sunflower has 34 and 55 spirals.
One head has several answers and the count depends on where it was taken. Run the same blind counter — one that is never shown the divergence angle — band by band up the radius of a single head, and it returns a different consecutive pair in each band. Every published count is therefore a statement about an annulus, and a count reported without the radius it was taken at is a sample from an unstated mixture. The transitions are not noise: they are computable in advance from the rise, and they are a factor of φ² apart.
What settles it: one head of 800 points gives 21/34 at 0.3 of the radius, 34/55 at 0.5 of the radius, 55/89 at 0.92 of the radius — the same rule and the same counter throughout. Argued in The counts change with radius.
Lewis's law describes the cells of a growing tissue.
It holds, and it holds on the wrong tissue. Lewis's law says a cell's area grows linearly with its number of sides, with a slope of a quarter and an intercept at two sides. On a random set of points it is close to exactly that. On an ordered golden-angle head — the pattern a phyllotaxis paper is likely to be about — the slope is a small fraction of Lewis's, because ordering the points is precisely what removes the area variation the law is a statement about. A law quoted as a property of cellular tissue is a property of disordered cellular tissue.
What settles it: slope 0.249 on a Poisson set against Lewis's 0.25, and 0.009 on a golden-angle head — a factor of 27.0 — with side count explaining 39.1% of the variance in cell area even where the law holds. Argued in Lewis's law wants disorder.
The average cell in a plant tissue has six sides, and that says something about the tissue.
The six is real and it is a theorem rather than an observation. Euler's relation for a planar graph, counted two ways, forces the mean number of sides to six as any tessellation grows — so a golden-angle head, a whorled head, a head at a rational angle and a scatter of random points all average six, and the agreement is a property of the plane rather than of the arrangement. What varies, on exactly the same cells with the same rim cut, is the second moment: the mean squared departure from six spans a factor of eighty across the same set. Reporting the mean and stopping reports the one number the arrangement cannot change.
What settles it: across 6 arrangements the mean side count runs 5.97–6.04 — a spread of 0.068 sides — while the mean squared departure from six runs 0.023 to 1.83, a factor of 79.3. Argued in The second moment is the measurement.
A lattice's spiral families follow the Fibonacci recursion: each is the sum of the two before it.
The families of a head are closed under addition — every one but the two smallest is the sum of two others, on every angle measured, which is why two counts have always been enough. What is not general is the recursion. At the golden angle the addends are the two largest so far, which is Fibonacci's rule; half a degree away, at 137.0°, the same measurement gives 8, 13, 21, 29, 50, 71, 92, 113, which adds 21 over and over. Both sequences are the denominators of the angle's continued-fraction convergents and intermediate fractions, and Fibonacci is the case where every partial quotient happens to be one.
What settles it: a golden-angle head's contact families are 8, 13, 21, 34, 55, 89 and a head half a degree away gives 8, 13, 21, 29, 50, 71, 92, 113 — both closed under addition, and only the first is Fibonacci. Argued in Every family but two is a sum.
The second moment of a head's side-count distribution varies with the arithmetic of its divergence angle, so two measured values locate two angles.
The arithmetic is real: swept across the angle, μ₂ has narrow deep dips at every rational, ordered by denominator. What is wrong is reading a difference between two values as a gradient. Between the dips the curve is a staircase — flat over stretches of a few hundredths of a degree with sharp steps between them — so the 0.253 recorded at the golden angle is the value of every angle from 137.47° to 137.54°, and the 0.255 at a rational a hundredth of a degree away is the same step of the same staircase.
What settles it: μ₂ is 0.255 at 137.49° and 0.249 at 137.53° — the same flat stretch, on either side of the golden angle. Argued in The disorder is a staircase.
Correlated disturbances would explain the periodicity in a divergence sequence without any placement rule.
It depends entirely on what "correlated" means, and the two usual meanings behave oppositely. A disturbance with a memory — each error a fraction of the last — manufactures nothing at any correlation coefficient up to 0.97, and the algebra says why before the measurement does: differencing a memory gives a decaying tail with no teeth, and the signature gets weaker as the memory lengthens. A disturbance that returns at a fixed separation does manufacture a comb. So the threat is not correlation but periodicity, which is a much more specific claim about a plant.
What settles it: a memory gives a largest comb of 0.028 against a band of 0.073 at every coefficient tried, while a periodicity at the smaller parastichy number gives 0.43. Argued in A disturbance with a memory.
The width of the order a rational divergence angle produces is set by the head size alone.
The exponent is right and the constant is not. Within a denominator the dip's half-width falls as the square of the head size, to four per cent, at every denominator from 8 to 89 — confirming the previous measurement on three denominators it could not reach. Across denominators the coefficient runs from about 1,200 at q = 8 to about 14,000 at q = 89, and dividing by q collapses a spread of twelve to a spread of two. So the law is w ≈ 150q/n² to within a factor of two, and three denominators were three measurements of one end of it.
What settles it: n²·w spans 12.0 across the six denominators and n²·w/q spans 2.1. Argued in The width carries the denominator.
A disturbance whose displacements are correlated between organs is gentler on a pattern than an independent one.
True of a disturbance the whole neighbourhood shares and false of one passed between touching organs, and the difference is a factor of six. A memory with a coefficient of 0.97 correlates at every nearby lag, so it moves the neighbourhood as a body without changing its shape, and a lattice survives three times the displacement it takes from independent errors. A transport correlated at the two contact offsets and nowhere else is a structure inside the neighbourhood, and it destroys a lattice at half the displacement independent noise does. The control that locates the mechanism is a transport at offsets the rule is not dominated by — seven and eleven — which is as harmless as white noise.
What settles it: a lattice survives 1.5° of shared displacement, 0.5° of independent, and 0.25° of displacement inherited from the contact neighbours. Argued in A disturbance the organs share.
The removal of an organ is felt out to the larger parastichy number and no further.
True in the middle of a rung and false within about a fifth of a rung of a transition, where the set of felt offsets acquires a hole. At a rise of 0.008 the run ends at eight, the organs nine, ten and eleven places back move the next organ by under a degree, and the organ twelve places back moves it by a whole divergence. The isolated offset is one place inside the larger number of the pair the stem is climbing towards, predicted from the hop lengths of the undisturbed stem and found at both transitions the sweep crosses — because the rule's profile has a runner-up slot one divergence away, and that organ is the one holding it up.
What settles it: at a rise of 0.013 the felt offsets are 1–8 against a pair of 5/8; at 0.008 they are 1–8 and 12, with the incoming pair 8/13. Argued in The response with a hole in it.
A placement rule produces no slow wander, because its errors are corrections rather than inheritances.
True of the rule's own errors and false of a rule under a drifting disturbance. What a placement rule corrects is the part of a displacement that is relative between neighbours; a drift moves the whole neighbourhood and passes through untouched. So the rule is a high-pass filter, and a filter that removes the fast part raises the ratio the slow part is measured by: a stem grown under a memory with coefficient 0.97 gives a block-mean statistic of 46 against 30 on a kinematic lattice with no rule in it, while its per-organ scatter is 0.246° against independent noise's 0.553°. Under independent noise the rule does leave a flat curve, which is the half of the claim that stands.
What settles it: W(64) = 0.91 under independent noise and 46.47 under a drift, with scatters of 0.553° and 0.246°. Argued in What the rule does to a drift.
A comb in a divergence sequence is evidence that something is transmitted between the organs that touch.
It is evidence of something stronger, and the difference is testable. A disturbance shared once with the two contact neighbours — the same correlation at the same lags, built from the neighbours' fresh deviates rather than from their displacements — manufactures nothing: a main comb of 0.099 against a band of 0.073, and the pair returned on one seed of eight, where an inherited disturbance at the same coupling and the same scatter returns it on eight of eight with a comb of 0.205. Raising the shared disturbance's coupling to 0.9, which the inherited one cannot even be run at, does not rescue it. A comb is a residue class of lags, and only a disturbance that is passed on and on populates one.
What settles it: inherited: comb 0.205, 8 of 8 seeds agreeing on 8/13; shared once: comb 0.099 against a band of 0.073, 1 of 8 seeds. Argued in The forgery needs a history.
The repeating block a wrecked stem settles into is one of its spiral counts, and nothing explains which.
It is a spiral count at eighteen of nineteen wrecked offsets, and it is a count for a reason that is not about counting. Read a wrecked stem by lags rather than by neighbours and exactly one lattice hop is rigid — the angle from an organ to the one p places above it is unchanged from the undisturbed control, organ by organ, to hundredths of a degree, while the divergence itself swings by tens. The block's period is that lag. Parastichy numbers are the short hops, and a placement rule holds short hops, so the survivor is usually a count; at one offset it is a lag whose step is 6.8 times a contact hop, which no census would report and which the rule held rigid all the same.
What settles it: 19 of 19 wrecked offsets have the block equal to the rigid lag, whose hop holds to 0.117° where the divergence spreads by 47.9–91.1° — and 1 of them keeps a lag 6.8 times the length of a contact hop. Argued in The hop that survived.
Outside the mechanism
1 claim
A reaction–diffusion mechanism explains the divergence angle.
A ring of cells running Gierer–Meinhardt does select a spacing, and the peak count is predictable from the Jacobian before any integration is run — that part works and is checked here both ways. What it does not do is define a divergence angle. Start the same ring from different small disorders and the angle between the first two peaks to appear is anything at all, while the peak count barely moves. A mechanism of this kind supplies a spacing; what supplies an order is motion, and a stationary ring has none.
What settles it: six starting disorders give first-gap angles of 176.6°, 47.2°, 109.1°, 150.7°, 47.2°, 132.7° — a spread of 129.4° — while the peak count over the same six runs is 7 or 8. Argued in A ring cannot make a spiral.
Underdetermined
1 claim
Teasel's divergence angle is 68.75°.
It is exactly half the golden angle, which is the interesting half of the claim and is not a coincidence: a k-jugate pattern is an ordinary lattice seen k times over, so every k has a limit divergence at 137.5078/k and every one of them is a noble number. But a k-jugate divergence is only defined modulo 360/k. Adding 180° to a bijugate plant's divergence relabels which member of each pair is called first and leaves the point set identical — the same coordinates, checked. So the reported number is a statement modulo 180, and quoting it without that is quoting a coordinate as though it were a measurement.
What settles it: 68.7539° is exactly half of 137.508°, and 68.75° and 248.75° are the same point set. Argued in Half the golden angle.
Survives
2 claims
The golden angle is special because it resists rational approximation.
This is the claim that holds, and it is arithmetic rather than packing. Measured as the smallest q·|qδ − p| over a range of denominators — how badly the angle can be approximated by any simple fraction of a turn — the golden angle scores higher than any other angle tried, approaching the bound Hurwitz's theorem puts on every irrational. The sharper form is the one worth carrying: a sweep over a grid of angles can never land on it, because every grid point is rational and every rational collapses the pattern onto a few rays. The famous number is the one point a search of this kind cannot reach.
What settles it: 0.438 for the golden angle against 0.422 for the Lucas angle and exactly 0 for every rational, with Hurwitz's ceiling at 0.447 — the golden angle reaching 97.9% of the best any number can do. Argued in The claim that survives.
Counting the spirals further out gives a better measurement.
It does, by a great deal, and the size of the gain is worth stating because it is the one piece of advice this collection would give anybody counting a real plant. A reported parastichy pair pins the divergence angle to a band whose width goes as one over the product of the two counts, so a count at the rim is worth a thousandfold in precision over a count near the middle. The caveat attached is the uncomfortable half: the value of a high count is entirely conditional on it being right, and a wrong 34/55 excludes the truth confidently where a wrong 2/3 excludes almost nothing.
What settles it: a report of 34 and 55 pins the divergence to 0.118° of arc against the 38.8° a report of 2 and 3 leaves — a factor of 329 — while knowing the radius it was counted at narrows the high report by a further factor of only 1.10. Argued in What a count is worth.
What this index is not
two things it is easily taken for
It is not a list of things people get wrong about plants. Every claim here is one a careful person says, and most are shorthand for something true — the trouble is that the shorthand travels and the conditions do not. A sunflower has 34 and 55 spirals is a perfectly good sentence with an annulus missing from it, and a rule of thumb with its conditions stripped off is indistinguishable, in a sentence, from a measurement.
Nor is any of this a claim about what a plant is doing. Every computation behind these verdicts is a model of form: a lattice that matches a sunflower does not show that the plant computes one, and the two mechanism models on this site are made to predict a number and then shown what they cannot do. Where a verdict rests on a model rather than on geometry the essay it points at says so, and the one verdict that turns on which model you take — whether a divergence sequence's noise arrived before or after the pattern chose — is recorded as an open question rather than as a finding.