Every essay — page 2
The pattern itself
A seed head is a set of points, and nearly every claim about one is really a claim about how many spirals run through it. The counting can be done from the points alone, and the answer is not what the captions say.
The centre the spirals give
Reading a seed head's divergence from its organs' positions needs the head's centre, and a photograph does not give it. A misplaced centre turns every organ's angle by an amount that varies round the head and grows toward the middle, so it could read as a twist. It does not, until it is large: a 900-organ head tolerates a centre a sixth of a spacing off, a 2,400-organ head a quarter. The centroid of the organs finds a whole head's centre to a hundredth of a spacing, and read about it the positions separate every twist shape exactly as well as with the centre given. The radius law, which looks like the natural fit, does not find it at all. And when half the rim is missing the centroid moves three to five spacings and turns an untwisted head into a twisted one, while the centre about which the counted spirals are most coherent still lands within a hundredth of a spacing.
A head photographed from the side
A seed head photographed off its axis is squashed by the cosine of the tilt across one bearing, and the squash turns every organ's angle by an amount that goes round the head twice. Read square on, a 900-organ golden head tolerates twelve to fourteen degrees of tilt before its two annuli disagree by more than any untwisted head's do, and the spiral counts never notice at all. Past that, the reading is not a smooth drift but a scatter of heads misread one at a time. Stretching the head back along the short axis its own second moments find restores the square-on reading exactly to thirty degrees, even though the moments cannot see a tilt under three or four degrees. And the square-law twist stays hidden: it is a changed angle organ for organ, so no camera, at any angle, sees it differently.
A whorl feels the far side of its stem
An ordinary stem keeps a long Lucas seed to an edge-to-seed ratio of 2.185 and a bijugate stem to 2.216, and the question was whether the difference follows the stem's angle or its timing. Folded, it is neither. A bijugate stem grown by the free rule is, node for node, an ordinary stem whose repulsion counts every neighbour twice round — once where it is and once a whole folded turn away — and that second copy is the whole of the difference. It carries an eighth of the repulsion where the seed is won or lost and a fortieth where the stem ends, and turned up from nothing it does nothing for its first quarter and then carries the ratio steadily from one level to the other.
A head seen in perspective needs a camera, not an ellipse
A seed head photographed off its axis from far away is squashed, and stretching it back along the short axis of its own second moments restores its reading exactly. Photographed from close up, the near side of the head is larger than the far side, and the stretch stops working: at twenty degrees off the axis it reads every head right from ten head radii and none from five. The organs' centroid drifts towards the near side as the camera comes in, but that is not why — the same stretch about the head's true centre fails at the same distances. What fails is the taper. A camera fitted from the organs alone, its tilt, bearing, distance and the head's centre chosen so that the head it un-projects is round, centred and evenly packed, reads seven to ten heads in ten right at every distance down to two radii.
What a plant might be doing
Turing's last work was on this, and it was unpublished when he died: a ring of cells, two diffusing substances, and a spacing that selects itself. The modern account uses auxin and a pump that works uphill. Both are made to predict a number and then to produce it.
Turing's last problem
Turing's final work was on phyllotaxis and it was unpublished when he died. Its core is a ring of cells and two diffusing substances, and the thing it does is select a number of peaks — which can be predicted from the equations before anything is integrated, and then counted from what the integration produces.
A ring cannot make a spiral
The peaks on a Turing ring do not all appear at once — there is a first and a second. But which two lead is decided by the starting disorder, so the angle between them comes out at 177°, then 47°, then 109°, then 151°. A divergence angle is a relationship that repeats, and this one does not.
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.
What a mechanism would have to show
Every model of phyllotaxis comes with the caveat that reproducing a pattern is not explaining it. That is easy to repeat and hard to make precise. Here it is made precise — a list of what an account of phyllotaxis would have to establish, with each item marked according to whether the models drawn in these essays establish it.
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.
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.
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.
A disturbance with a memory
That earlier work's control assumed that a plant's errors are independent from organ to organ, and nobody had tested it. Give the errors a memory — each one a fraction of the last, up to a coefficient of 0.97 — and the comb does not appear. The obvious threat to the result turns out to be empty, and the algebra says why before the measurement does.
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.
What a forgery has to know
A lattice with transported errors reproduces the comb and the pair, so one quantity is left: the two combs' relative strength. Weighted by distance the forgery puts more in the second comb than the first; the rule does the opposite. It matches only if the coupling is turned three to one towards the further neighbour, which no falloff supplies.
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.
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.
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.
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.
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.
A difference forgets a drift
This collection proposed a second observable and priced it as free: if a plant's errors are inherited between touching organs, the divergence sequence should carry a slow wander as well as a comb. The disturbance with the largest wander of any built here leaves none at all in the sequence, because a divergence is a difference and differencing is what removes a drift.
What the rule does to a drift
A placement rule was supposed to leave no slow wander in a divergence sequence, because its errors are corrections rather than inheritances. Driven by a disturbance that drifts, it leaves a larger one than a lattice with no rule in it at all — while cutting the per-organ scatter by more than half. The rule removes what is relative between neighbours, and a drift is not.
The window was not the neighbourhood
A placement rule corrects what is relative between neighbours and passes what moves them all together, so how much of a slow disturbance gets through should depend on how deep the neighbourhood is. The obvious knob is how many organs the rule sums over. Swept across a factor of six, it changes nothing at all — and a parameter that is not binding produces exactly the flat sweep a robust result produces.
The drift goes the other way
A rule that corrects what its neighbourhood shares should let through any disturbance slower than its own reach, and should suppress anything faster — a crossover, tracking the depth. Swept over a neighbourhood that changes by a factor of sixty, there is no crossover anywhere, and the deep rule passes nearly four times as much as the shallow one. The prediction is not weakly supported; it is backwards.
What the ratio was hiding
The statistic that says a rule sharpens a drift rises by a factor of nearly four across a sweep of the rule's depth. Undo the normalisation and ask instead how many degrees of drift actually reach the divergences, and the answer changes by a fifth. Nearly all of the effect was in the denominator, and the denominator is the thing the rule is good at.
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.
Four ways to count a neighbourhood
How deep does the placement rule look? Counting the organs that carry nine tenths of its profile gives 182 down to 3 as the falloff steepens. Counting the ones that carry half gives 33 down to 1. The weighted mean lag gives 68 down to 11. The four disagree by an order of magnitude about the size and agree exactly about the order.
The nearest organ is not the nearest neighbour
Rank the terms of the sum the rule minimises and read off which organs the biggest ones belong to. At every falloff exponent from 1.5 to 6 the answer is the same five: lags 13, 8, 5, 21 and 26. The organ placed immediately before is not among them, and counting the neighbourhood in organs was the wrong unit.
The corner that does not move
Read as degrees of drift getting through rather than as a ratio, and compared seed by seed, the deep and shallow rules change hands. The share that goes to the deeper rule climbs from twenty-three per cent under white noise to ninety-four at a correlation length of a hundred organs — and the crossing sits at two or three organs whether the two rules differ by a factor of four or sixty-one.
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.
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°.
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.
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.
The plateau was a prediction
The search for a reference organ found that the largest displacement above a hole is a plateau rather than a peak, and reported it as a failure. A profile constant on each of k residue classes has exactly k levels, so its maximum is attained by a whole class — a ninth to a quarter of every window, forever.
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.
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°.
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.
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.
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.
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.
Four accounts of one angle
The exchanged pair in a wrecked stem is about one divergence step, and about is doing twelve per cent of work. Four candidate units were written down and scored on the same seventeen rows: the cut stem's own step, the surviving family's step, the control's step, and the control's corrected.
A fifth of the hop
The exchanged pair misses one divergence step by up to twelve per cent, and the miss is not scatter: every row keeping a lag of 5 or 7 overshoots and every row keeping a 4 or an 8 falls short. Subtract a fifth of the surviving hop's own angle and the worst row is four per cent.
Six lattices were not enough
The interaction between the two walls of a slot came back at −25.8° to +132.9° on six lattices, three above zero and three below, with no ordering by rise, by counted pair or by branch. A quantity that looks free on six rows is usually a quantity that has been sampled at six rows.
When the second wall is free
On six of thirty lattices, removing both walls of the slot costs exactly what removing the larger one alone costs — 35.9° and 35.9°, 12.0° and 12.0°, agreeing to the last digit of the grid the azimuths sit on. The smaller wall is not a wall on those rows.
The rung decides the sign
Twenty-four lattices where both walls of the slot are really there. Thirteen give a strongly positive interaction, at 85° to 135°; eleven give a negative or null one, at −25° to −0.5°. Nothing lies between. Every rung's lattices fall on the same side as each other.
The exception was already labelled
The larger counted number sorts twenty-two of twenty-four lattices by the sign of their slot interaction. Both misses are on the Lucas 3/4 rung — the one rung a different measurement had already singled out, for reasons with nothing to do with this one.
Twice the run
Five wrecked cuts never reached a pattern inside three hundred organs and one reached it at three hundred and three, which is a number asking to be checked. Run every cut in the census twice as far and three of the thirty change their answer.
An onset at the end of the run
One cut reported that its pattern began 299 organs into a 300-organ run. Given twice the room it reports no pattern at all. The reading was the run stopping, not the disturbance ending, and the definition guarantees one at the last organ of every run.
Three rows change sides
Twenty-five of thirty wrecked cuts have a periodic displacement profile over three hundred organs and twenty-six do over six hundred. The count barely moves and the membership does: two rows join, one leaves, and the gap the threshold sits in narrows from 1.69 to 1.27.
A window nobody aligned
Every reading this thread takes of a wrecked stem is taken inside a window, and there are three of them: a run of three hundred organs, a window of a hundred and twenty at its top, and a search window of fifteen to thirty-nine. None was aligned to anything, and one of them turned out to decide its own answers.
A cycle sums to a whole turn
The test proposed for whether three displaced chains are a three-cycle was that their displacements sum to zero. Three chains rotating into one another's places each move about a third of a turn the same way round, and a third of a turn three times is a whole turn — which the unfolded test calls the worst row in the census.
A spread that grows with its window
A spread over a sample of a steady quantity does not depend on how big the sample is. These spreads triple when the window triples, on nearly every row of the census, which means the classes this thread calls steady are sliding — slowly, and invisibly at any single window.
The lag decides whether it closes
Seven of the thirteen excluded rows have displacements that cancel and six do not. Every row that closes kept a lag of seven or eight and every row that does not kept four or five, thirteen times out of thirteen — and then a lattice nobody had cut broke it.
The rows nobody added up
Seventeen of the census's thirty wrecked cuts come back as one balanced pair of displaced chains, and every claim about the exchange is quantified over those seventeen. The other thirteen were set aside as having three or more exceptions and never looked at again. They are one addition each.
The window nobody moved
Three instrument settings sit between the ablation census and every statement it makes. Two have been varied and both decided answers. The third is a hundred and twenty organs at the top of a run, it has never been moved, and moving it changes the verdict on three rows.
Three rows a window moves
Three of the census's thirty cuts are periodic when a hundred and twenty organs are read and not when a hundred and eighty are. Their spreads do not grow in proportion to the window, they grow by twenty and forty times, and the reason is that a window is the tail of a run.
A median that is an exception
Every number in the ablation thread is measured against the value most of a wrecked stem's chains sit at, taken as a median so that a few exceptions cannot move it. On eight of thirty-six rows the median stands on a chain sitting by itself.
No majority and no pair
Whether a wrecked stem's chains agree on a level and whether its exceptions form a balanced pair are computed from different things. Sixteen rows have neither, sixteen have neither, and fifteen rows are in both — and the two they part on are the two most interesting rows in the census.
The twenty-first row
Recomputing the level moves one row out of the set the exchange sets aside and into the exchange itself. Its hop is four times larger than any the correction was fitted over, and the correction fails on it in the one way it had never failed.
The rule comes back
A rule sorting the awkward rows was right fifteen times in sixteen, with the one failure blamed on a statistic. Recomputing the statistic makes it fifteen of fifteen — and halves the gap the line is drawn in, which is the price.
A destination or a refusal
Sixty-three wrecked runs were regrown to twelve hundred organs and put to the settling table's own criterion, unchanged in every tolerance. The prediction written down before the sweep said twenty-nine would settle; seventeen do, and the prediction is wrong on its own side of the table as well as in its total.
Like with like
Seventeen wrecked runs settle when regrown to four times their length, and every one of them settles at the divergence the short run already reported. The endpoint moves by four thousandths of a degree at worst, which makes the short reading precise about where and silent about whether.
What a run length was hiding
The settling criterion returns an identical verdict on all sixty-three wrecked runs at every length from three hundred organs to twelve hundred, so run length explains nothing. What the sweep does find is that an endpoint is a mean over an orbit, and thirty-four refusers never come within a degree of their own reported endpoint.
A basin with no upper edge
The widest basin in the settling table had a width bracketed between 47.5 degrees and about 57, and closing a bracket means sampling near an edge rather than everywhere. Three basins cut at a quarter of a degree located all six of their boundaries, and the widest turned out to run out of basin at 180 degrees rather than reach an edge on that side at all.
A wall or a fade
A basin's border is either a change of destination or a stretch where the angles stop settling at all, and nothing here could tell the two apart. Two instruments were pointed at the question: the settling clock, which looked obviously right and fails, and the tail spread, which was already being computed on every run and had never been read.
The angles left over
A stem started anywhere on the circle was assumed to end up in one basin or another, so that the settled destinations divided the starting angles between them. Twenty and a quarter degrees swept without a hole at a quarter of a degree find 1.75 degrees between two basins that reaches neither of them and nothing else, and a three-degree void beside it.
What a quarter degree cannot see
Six boundaries were located to an eighth of a degree, three basins were named and one width was quoted, and every one of those readings has the same floor under it. The sweep's grid is one step of the grid the stems are placed on, so nothing here bounds a basin narrower than half a degree — and the widest basin's own middle was never swept at all.
The level was doing the ordering
Four falloff exponents have been put in order by the rise at which half their runs stop settling, and a half is the only level that order has ever been read at. Read at nine levels the same runs give ten different orderings of the same four numbers, and the one comparison in the whole study that resolves runs the other way.
Four walls closer than they looked
Two of the four falloff exponents had a wall with no upper end at all, and the other two were located to factors of two and a half and nearly four. Nine rises at eighty starting angles close every bracket — and the four walls turn out to sit inside a factor of 1.111 of one another, which is narrower than the narrowest bracket.
Two refinements that do not multiply
The design that located the wall did two things at once — doubled the starting angles and halved the rise spacing — and the arithmetic behind it assumed each would buy about a factor of two. The finer rises did ninety-nine per cent of the narrowing and the doubled angles added under one, because a bracket's ends are rises and no error bar can move them.
A maximum in the gap
Four falloff exponents have refused to separate on every quantity this thread has read off them, and the wall that was supposed to tell them apart cannot. Two of the four carry a maximum in the settling share at a rise the published list stepped straight over, and it is there in both halves of the sampling independently.
Shells and growth
A logarithmic spiral is what a thing grows into when it adds material without changing shape. Its one parameter is recoverable from a drawn curve — over enough turns, and from a centre that is known — which is how a century-old argument gets a number attached to it, and how the recovery gets one of its own.
Growth as a rule
A logarithmic spiral is not a shape somebody admired. It is what a thing grows into when it adds material at its opening without changing shape, its one parameter is how much it grows per turn, and that parameter can be recovered from any drawn curve to the last digit.
Raup's three numbers
Nearly every coiled shell that has ever existed is a point in a three-dimensional space — how fast the whorl expands, how far the opening sits from the axis, how far it travels along it. That is a remarkable compression, and the most useful line in the space is the one where the whorls come apart.
The nautilus question
A golden spiral grows by 6.854 per turn. Measured nautilus sections give about 3.2. That is a factor of 2.14 — not a rounding error, not an artefact of where the centre is assumed to be, and not close.
The line was already exact
The boundary between shells whose whorls run into one another and shells whose whorls run free is quoted everywhere as D = 1/W, and a survey designed to measure how far off it sits found that it is not off at all. Located by bisecting the drawn circles at 481 expansions, the residual is 2.2 × 10⁻¹⁶ — the last bit a double holds, over the whole range.
What a spire buys
Translation along the coiling axis enters the contact boundary as its square, so a shell that has only just begun to walk along its axis has not moved the boundary at all. It is also strictly one-way, and it has a threshold above which no distance from the axis whatever puts the whorls in touch.
A boundary with no edge
Two continuous measures cross the line where a coiled shell's whorls begin to touch, and they disagree about whether it is sharp. One falls to zero as a straight line and makes the line a kink; the other leaves it as a three-halves power and makes it a tangency.
A fraction of nothing
Six boxes differing only in where their edges were drawn give between 4.64 and 52.81 per cent for the same geometry, and sampling one axis geometrically rather than uniformly multiplies the answer by 4.60. The share tends to zero as the box widens, because the region under a hyperbola is a logarithm and a box is a line.
What the centre costs
The fit that recovers a shell's growth factor needs a centre, and no shell has one marked. Displacing it by a quarter of the innermost whorl's radius moves the answer by 4.56 per cent at two turns, which is about five times the figure published earlier.
How far a centre must move
Four hundred and eighty-two thousand assumed centres, at nineteen spans and a hundred and eighty directions each, asked whether a spiral drawn at 3.2 can be made to read as the golden 6.854. It can, at every span up to 1.15 turns and at none from 1.2 upward, and every centre that manages it is refused twice over.
The residual is not the test
The fit that recovers a growth factor also hands back a residual, and that residual has been read as what separates a genuine logarithmic spiral from something that merely looks like one. Pointed at twelve curves it fits a circle exactly, accepts an Archimedean spiral, and refuses a golden one that is right to three decimal places.
A measurement in steps
Walking a pair of dividers along a shell's spiral is the oldest way to measure it and the best one available once there are enough steps, because it puts the points where the curve is. Under a count that follows exactly from the geometry it inflates the answer instead, and it is the only route measured here that pushes a nautilus towards a golden spiral.
One angle decides contact
Seen from the apex of its coiling axis, every whorl of Raup's shell subtends the same half-angle, and two whorls touch exactly when the sine of that angle exceeds (W − 1)/(W + 1) — with no disagreement against the drawn discs at 400,000 random shells. Two of the three numbers enter only through the angle and the third only through the threshold, which decides which picture of a shell can answer the question: a spire's outline carries no W, a plan carries no T, and an axial section carries all three.
The fourth number divides the third
Every boundary on Raup's cube was located for a circular opening, and three essays ended on the same hedge: the numbers would move with a differently shaped aperture by an amount nothing had measured. Measured on the drawn outlines of eleven openings, the boundary with no translation does not move at all for any convex opening symmetric about the plane of coiling; an ellipse's height divides the translation and does nothing else; the square law in the translation belongs to a round tip; and a turned opening frees ground only in the D a plan reads.
One number for a shell that changes
An animal is under no obligation to grow at one rate from hatching to maturity, and the fit that recovers a shell's growth factor returns one number whatever it is given. Handed a shell whose expansion rises steadily from 2.8 to 3.6 a turn, it returns 3.17490 — the geometric mean of the two ends, exactly — with a residual it accepts. A change of sixty-four per cent over three and a half turns passes as one logarithmic spiral, and at the aperture, where contact is decided, the one number and the last whorl give opposite verdicts.
A centre that invents a life history
The collection's advice for a shell that might have changed how it grew was to fit it twice, over different arcs, and compare. On a spiral that does not change at all, a centre displaced by a quarter of the innermost radius splits the two halves by 4.09 per cent — the split a genuine 8.35 per cent change from apex to aperture produces — in either sign, depending only on which way the centre is wrong. Point noise of the same size splits them by less than half as much, and averages away where the centre does not. The floor under the test is the centre, not the noise.
Three points on a diameter
Ammonoid workers measure a shell's expansion without a centre: two diameters half a volution apart, squared. On a logarithmic spiral that is exact, and the centre is needed only to aim the line. A quarter-radius aim error costs the fit 1.341 per cent and the diameters 0.0045, because the aim error enters as its square. Reading noise is another matter: at a thousandth of the outer radius the fit's four hundred points beat the calipers' three readings at every expansion up to the nautilus's, and which instrument is better depends on which error the section actually has.
What the septa count
A nautilus's chambers are each a scaled copy of the last, and an earlier essay gave their ratio as about 1.3 — what a growth factor of 3.2 gives over a third of a turn. It does not: a third of a turn at 3.2 is 1.474 in length. A ratio of 1.3 is 4.43 septa a whorl as a length, 8.87 as an area and 13.30 as a volume, so the dimension decides the count threefold. And the count is an exponent in any reading of the growth factor taken from one chamber to the next: one septum miscounted at thirteen moves it by 9.14 per cent. A chamber and the one a whorl out give W³ with no count at all.
A spiral with no clock
The growth factor a shell's curve gives up is a rate per turn of the shell's own coiling, and a turn is not a unit of time. Four clocks — the aperture advancing at a constant angular rate, adding a constant length, a constant area, a constant volume — trace the identical curve: every mark one of them leaves lies on r = W to the power theta over two pi exactly, so a fit through any of them returns the same factor. What differs is where the marks are, and the difference is enormous: at 3.2 per turn the outermost of three whorls holds 33.4, 71.0, 90.3 and 97.0 per cent of the record. It also breaks the instrument. The routine that recovers a growth factor unwraps the angle by assuming successive points advance less than half a turn, and five of the twenty readings here leave gaps past that — returning 3.33 where the curve was built at 3.20, and 8.19 where it was built at 6.85.
What the growth lines carry
A shell's curve says nothing about how fast the animal grew, and its growth lines say all of it. Under a law that holds the pth power of the radius constant per unit time, the time spent crossing one whorl is proportional to the change in that power across it — so the lines in successive whorls stand in the ratio of the growth factor raised to p, and each whorl holds the count the closed form predicts to within the one line rounding can move. Dividing two counts and taking the logarithm against a growth factor the curve already gives returns p: 17 of 20 readings name their own law, the furthest 0.012 from a whole number. The other three are not wrong, they are uncountable — at 4.5 per turn under a volume clock the inner whorl of the pair holds one line.
A shell that changed its law
An animal that grew as a juvenile under one deposition law and as an adult under another leaves a sequence of whorl ratios rather than one, and the sequence says where the change happened. The ratio across the change is a closed form that is neither law's — 6.72 between a length clock and an area clock at 3.2 per turn, exactly the average of 3.2 and 10.24 — and it is monotone in where inside its whorl the change sits, so it inverts. On a seven-whorl shell of 18,466 lines a change at 3.5 whorls comes back at 3.5001, in a band 0.027 whorls wide that holds the true position. The reading refuses a change in the outer three whorls or the inner three, because a plateau it will trust is two agreeing ratios and two ratios need three untouched whorls.
A law that never stopped changing
A shell whose deposition law moved evenly from one end to the other gives a sequence of whorl ratios that is a straight ramp rather than a plateau, a crossing and a plateau, and the two are separated by more than the counts' own rounding on every shell holding three countable ratios. Each ratio on the ramp reads the law at the boundary it straddles — 0.3486, 0.6865, 1.0320, 1.3755, 1.7180 against 0.3333, 0.6667, 1.0000, 1.3333, 1.6667 — so the reading is local where a fit to the curve is global, and a fit handed the same shell returns the geometric mean of its ends with no warning. The reading that locates a single change refuses a drifting shell at every size, naming the number of ratios that agree with neither end.
A count that is not exact
Reading a deposition law off two whorls' growth-line counts divides one by the other, so a miscount that is the same in both divides out: four lines in five missed at random moves the answer by five thousandths and costs only scatter. What biases it is a miscount that varies along the shell, and there is one that always does. The arc between successive lines rises or falls with the radius according to whether the law is shallower or steeper than a length clock, so a section's resolution limit eats the inner whorls of a shallow shell and the outer whorls of a steep one, and eats evenly at exactly p = 1. Where the limit binds, a whorl's surviving count is its arc over the limit, and whorl arcs stand in the ratio W — so a shell too worn to read reports a length clock whatever law it had.
A section seen from the wrong angle
A photograph of a shell section taken off the normal is the coiling plane compressed along one direction by the cosine of the angle, and nothing in the picture says so. The fit that recovers a growth factor is moved by it — half a turn seen twenty degrees off gives a band of answers 23.7 per cent wide as the span's starting point moves round the shell, centred almost exactly on the right answer, so it is a spread and not a bias. The caliper measure is exactly immune at every tilt and every aim, because a projection scales all three points on a line through the centre by the same factor. And the fit's residual names the tilt to three decimal places, which makes this the rare error a section reports about itself.
The error budget for a nautilus
Every way a growth factor read off a shell section can be wrong has been priced here, one essay at a time. Added up they come to 332.6 per cent in the worst case and 279.6 in quadrature, against a golden-spiral claim that is 114.2 per cent away — so the budget does not refuse the claim at all. One entry decides it: the dividers, at 278.2 per cent on their own, and the dividers are the historical method and the only route measured that pushes a nautilus towards a golden spiral. Set them aside and the budget falls to 54.4 per cent and the claim is refused twice over. What the same budget cannot settle is anything smaller than half: 3.2 against 3.4 is inside it, and stays inside it until six of the seven sources are controlled.
The band nobody can be placed in
The boundary between shells whose whorls run into one another and shells whose whorls run free was located here to the last bit a double holds. A specimen is not a point on that line, it is a measurement with an error, and carrying the whole measured error budget onto the boundary turns the line into a band running from 1/(W(1+b)) to 1/(W(1−b)). At the budget with the dividers set aside that band covers 48.2 per cent of the box the morphospace figure here is drawn on, and its share runs from 6.7 to 59.5 per cent across the six boxes in use — the same box-dependence the contact region itself showed. The angle criterion carries the same error better above an expansion of 1 + √2 and worse below it, exactly.
What the axis distance costs
Raup's contact boundary is a relation between two numbers and only one of them has ever been priced here. The second was expected to be the cheaper — a length against another length. It is not: an assumed centre costs it 79.3 per cent where the same centre costs the expansion 6.88, because a ratio of two distances is first order in the centre and a fitted rate is second. But a tilted camera costs it nothing at all, exactly, and averaging the reading round one whorl is free and worth a factor of 4.91. The two numbers fail at opposite ends, and they cross at 1.12 turns of arc.
A floor no better fit can lift
The band of shells nobody can place was built from one of the two numbers the contact boundary relates, and the other has now been priced. Carried together they widen the band by a third and take 48.2 per cent of the morphospace box to 59.7. The number that matters is further down: with the whorl expansion measured perfectly, 12.6 per cent of the box is still undecidable, and at an expansion controlled to a hundredth 94 per cent of what remains belongs to the second number. And the two errors are not independent — they come out of one guessed centre, which traces a curve across the boundary rather than a rectangle around it.
Whether a section can see its own limit
A shell section that cannot resolve growth lines closer than some distance reads every deposition law as nearer a length clock, and says nothing about it. Given an animal whose days vary, the section can often catch itself: a limit changes how irregular successive spacings are in the whorls where it binds, and one animal is not steady in one whorl and irregular in the next. On an angular clock a test comparing whorls flags every limited section, before the reading has even moved. On a volume clock it can miss a limit that has pulled the reading from 3 to 1.87 — when the animal's own days vary by a tenth, which is exactly as irregular as the limit leaves the whorl it binds.
Three entries and one span
The error budget for a nautilus added seven ways a growth factor read off a section can be wrong, and asked whether they were independent. Three of them are not: the displaced centre, the span of arc and the oblique view are one error priced three ways, at two turns, over a turn and more, and at half a turn — 32.2 per cent together. A section has one span. Read together at one span, a quarter-radius centre and a ten-degree tilt come to 7.1 per cent at two turns and 188 at half a turn, and in their worst orientation they always add to more than their sum. So the budget refuses the golden spiral from three quarters of a turn of shell upward, and below that it cannot.
The dividers belong to the opening
The nautilus error budget's largest entry, dividers walked along the shell at 278 per cent, was set aside as the historical method. Priced at one span it turns out not to belong to the span at all. A person sets a pair of dividers to an opening and walks until the curve runs out, so the step count grows with the arc exactly as fast as the floor on it does: opened to less than the square root of the growth factor less one — 0.789 of the innermost radius for a nautilus — they read the factor exactly over every span from half a turn to six, and opened wider they read it too high over every span, least over the longest. The 278 per cent was nine steps along five turns, an opening of 37 innermost radii. With the dividers opened to anything up to five radii, the whole budget refuses the golden spiral from three quarters of a turn upward.
The rim sets the opening
A pair of dividers reads a nautilus's growth factor exactly when its opening is under 0.789 of the radius of the whorl it starts on — a quarter of a millimetre at the true centre of a real shell, which no hand can set. A real section starts where its whorls can be read, and a person sets the dividers against the shell in front of them. Measured that way, the rule becomes a span: dividers opened to a share f of the outer radius are exact over the last log((√k − 1)/f)/log k turns of any shell — 3.76 turns at a hundredth, 3.16 at a fiftieth — and the change-over falls exactly there at every opening tried. Held against the rim, their error grows with the span rather than falling, so a section with its centre broken away is read more exactly, not less; and the whole budget still refuses the golden spiral at every span from three quarters of a turn to six for any opening up to a fiftieth.
The outline finds its own centre
A worker with a sawn shell has an outline and nothing else: the centre, the expansion W and the distance from the axis D all have to come off it at once, and three numbers fitted to one outline can trade against each other. Fitted together, they do not trade where it matters. Started from a centre guessed a quarter of the innermost radius off, the drawing locates its own centre to about a thousandth of that radius, and over a turn of section drawn to a thousandth of the rim it returns W to 0.12 per cent and D to 0.045 — where reading the same drawing at the guessed centre gives 16.8 and 0.98. W's error rides the centre's; D's does not, and W and D do not trade against each other at all. The trade the round trip was set to watch for appears only where the model is wrong: a view five degrees off the section's plane moves W by three times its own noise error while the fit's residual stays within the noise, because the free centre moves to absorb the squash.
The cut along the axis
A median section cannot see how far a shell travels along its axis, so the third of Raup's numbers seemed to need a second cut. It does not need the first. One cut through the axis shows every whorl twice a turn as a whole circle, and fitted with the axis free it returns W, D and T together — to 0.023, 0.085 and 0.031 per cent on a high spire drawn to a thousandth of its rim, from an axis laid a quarter of a radius off and two degrees out. On the shell the median section was read on, it reads the expansion twice as well over a turn and the distance from the axis as well. Laid by eye and left there, the axis costs T four per cent and a high spire's D thirteen, and the residual says so. What it does not always say is that the saw missed the axis: on a low spire a cut a fifth of the innermost radius off moves W by two and a half times its own noise error with the residual at the noise.
An opening that is not round
One cut through a shell's axis returns Raup's three numbers to a few hundredths of a per cent when every section is a circle. A snail's opening is closer to an ellipse. Read with circles, a cut of a shell whose opening is a quarter taller than it is wide gives W exactly and puts the whole misfit into the other two: D comes back 29 per cent low and T 4.9 per cent low, by a first-order form in which the circle takes the ellipse's mean half-axis. A nearly round opening is the dangerous one — an aspect of 0.99 moves D by fourteen of its own noise errors while the points miss their circles by only a quarter more than the noise. Fitted with ellipses, the cut reads all five shape numbers with W, D and T nearly as precise as before, and the opening's height does not trade against the spire's translation.
A limit is a length, and a short day is a share
A shell section that cannot resolve its finest growth lines names a steep law as a shallower one, and so does a shell whose animal really did change its law. Matched to name the same law from the same whorls, the two agree in their counts, and the test that compares how irregular each whorl is goes blind on both across the same band of limits. What separates them is what a limit is. Every spacing it leaves is at least one length, wherever on the whorl it sits, while an animal's short days are short in proportion to the spacing around them. Asked whether the shortest spacings along the outer whorl follow the local spacing, forty sections of a drifting or stepped law are flagged at most twice, and a limited volume clock in the blind band thirty-three to forty times.
A photograph squashed along the axis is another shell
A cut shell is photographed, not measured with calipers, and a camera a few degrees off the face's normal squashes the whole picture across one bearing. Read as a shell with elliptical openings, a squash straight across the axis or along it is absorbed without a trace: the picture is exactly that of another shell, its openings ellipses and its translation off by the squash — six per cent at a twenty-degree tilt — and no fit can tell. At any bearing between, the squash turns every opening the same way on both sides of the axis, which a real opening, mirrored across the axis, never does: a ten-degree tilt at fifteen to seventy-five degrees from the axis is caught on every section. The common photograph, the shell upright in the frame and the camera off to one side, is the one that cannot be caught.
Packing and tiling
How evenly a pattern fills its disc is a statement about cell areas, and it can be measured four ways that disagree. Cells average six sides because Euler's formula leaves them no choice.
Why the average cell has six sides
Not because hexagons are efficient. Because Euler's formula leaves a tiling no choice — count the edges two ways and the mean comes out at six, whatever the cells would prefer. The efficiency argument is a different claim about a different thing.
Packing, measured four ways
The claim is that the golden angle packs best, and it is measurable. Read on the interior of a head, the two criteria about distance put the golden angle first among the angles near it and the two about cells are won by rational angles — which makes the claim half right, and makes the right half a statement about a class of angles. An earlier reading of the same four criteria, divided by the cells at the head's edge, said the opposite.
The gap that grows
A rational divergence angle develops an empty wedge that grows without bound as the head fills — a factor of 2.9 between two hundred primordia and sixteen hundred, and 3.8 from a hundred and fifty to two thousand. An irrational one does not. That is the division between rational and irrational angles that survives measurement.
Lewis's law wants disorder
Cell area rises linearly with side count — measured on cucumber epidermis in 1928 and quoted ever since as a property of packed tissue. It holds beautifully on a random point set, with a fitted constant of 1.64 against Lewis's 2. On a phyllotactic head it does not hold at all: the slope is 0.009, and area and side count are almost independent.
Two laws that want opposite tissue
Lewis's law and Aboav's relation are quoted side by side as properties of cellular tissue. Measured on the same two tilings they point opposite ways — the ordered head satisfies Aboav's with the textbook value of 1.18 and fails Lewis's completely; the random set does exactly the reverse.
The second moment is the measurement
The mean number of sides in a cellular tissue is six, and Euler's formula leaves it no choice — so it takes the same value on a golden-angle head, a whorled head and a set of random points. On heads of nine hundred organs the mean squared departure from six varies by a factor of eighty across the same three, and almost nobody reports it.
The six are the spirals
Label every contact between two cells in a seed head with the difference between the two nodes' placement indices. The labels are the parastichy numbers — 34, 55, 21, 89 — and the six sides Euler forces turn out to be about two from one family, one and a half from the next, and one each from two more.
The disorder is a staircase
Sweep the second moment of a head's side-count distribution across the divergence angle and it is not a curve. It is flat in stretches with sharp steps between them and narrow deep dips wherever a rational falls — so 0.253 is not the golden angle's number, it is the number of every angle from 137.47° to 137.54°.
A dip belongs to the head
At an exact rational the disorder halves when the head doubles, and the dip around it narrows by a factor of four. So which angles look ordered is set by how many organs were counted, and a head of three hundred cannot tell 138.4615° from 138.48° while a head of twelve hundred tells it from 138.4625°.
The background is not one sample
The dip in disorder at a rational angle is a comparison against a background, and the background was one measurement taken two tenths of a degree away. At 21/55 that lands seven thousandths of a degree from 13/34 — inside another rational's dip — and the comparison inverts. Fixed, the dip survives to a denominator of 89.
The width carries the denominator
The earlier work measured three denominators, found the dip's half-width falling as the square of the head size, and could not say whether its coefficient depended on the denominator. Six denominators say it does: the coefficient runs from 1,188 at q = 8 to 14,297 at q = 89, and dividing by q flattens a factor of twelve into a factor of two.
Four fractions with one denominator
The dip in a head's side-count disorder is as wide as 150·q/n², measured over six fractions — every one of them a Fibonacci convergent, which is the emptiest neighbourhood a denominator ever gets. So the law could be about the denominator or about how well the fraction approximates its neighbours. Four fractions of 55 at one head size settle it in one figure.
A width read off a staircase
Two fractions of the fourteen measured return a dip width that moves by a factor of two when the head size changes, where the others hold to three per cent. The cause is not their neighbourhood. It is that the disorder statistic changes only when the tessellation changes, so the curve a half-width is read off is a staircase, and a width narrower than the tread cannot be read at all.
A dip with no outer edge
The disorder of a head dips at every rational divergence, and how wide that dip is has carried a long argument. Reading the width as a level crossing has a resolution problem, and the obvious repair is to integrate instead. The integral reproduces beautifully across head sizes and never settles on a value, because there is nothing out there for it to settle against.
The window is the neighbour
An integral needs a limit, and this one has two conditions on it that pull opposite ways. It has to scale with the dip, so that two head sizes are comparable, and it has to stay clear of the next rational, which is a fixed distance in degrees. Between them there is no stretch where the answer holds still — and the limit that decides it is the crowding.
Fractions with the same neighbours
Every instrument this collection has for the width of a disorder dip has a free parameter set by how close the next rational sits — which makes a hypothesis about the neighbourhood untestable with any of them. The repair is not a better instrument. It is a set of fractions whose neighbourhoods are identical and whose denominators are not, and the arithmetic supplies twenty-four of them.
An interior that is nearly neutral
Give every cell a charge of six minus its number of sides and the total over a tessellated head is fixed by its own boundary, exactly, with nothing left over for the interior. On a golden head that freedom is spent on 264 exceptions among 1,631 cells which cancel to six.
The defects lie on rings
The cells in a seed head that are not hexagons are not scattered through it. Every one of 264 sits within a cell of a radius computed from the divergence angle alone, the radii are a factor of φ apart, and between two of them lie 422 consecutive cells without a single exception.
Every five is bound to a seven
A five-sided cell beside a seven-sided one is one object in a crystal and two exceptions in a tiling, and the phyllotaxis literature borrows the crystallographic word without measuring the binding. Measured against a seeded permutation null on a 2,400-organ head, every five in the interior shares a wall with a seven, and the loneliest one in the head is 0.927 cell spacings from the nearest.
The rings are not the transitions
A seed head has two ladders on it — the radii where the counted parastichy pair changes, and the radii where the exceptional cells sit — and the obvious guess is that they are the same ladder. They are not: the second sits a factor of the square root of phi outside the first at every rung of two different divergence ladders, which is exactly halfway between two consecutive transitions.
No cut-off makes them one
Two different relations on a head have both been called neighbour: the shortest index lags a count keeps, and a shared Voronoi wall. A cut-off that turns the first into the second exists for almost every cell taken alone, and for no whole head at any size.
Two thirds of a cell
The founding claim of this field is that the six sides Euler forces are the spiral families. Measured against the tessellation it names two thirds of a cell's walls exactly, in every band of a head and at every rise of a stem, and the missing third is the same third everywhere.
Two rankings, one list
An essay in this collection claimed that the four shortest index hops on a seed head and the four largest shares of its cell walls are the same four numbers in the same order, and called the correspondence exact. Measured again from the same points, the two lists hold the same four families and order them differently, and they order them differently in five of the six bands the head can be read in.
The third family
On a seed head no threshold makes the counted contacts and the shared cell walls the same relation. On a stem they are the same relation exactly, at every one of a hundred and eleven rises and to three decimal places of nothing, provided the contact cut keeps three families where a count keeps two.
Packing, measured against the interior
An earlier reading of these heads reported that no packing criterion singles out the golden angle and that three criteria give three winners. Every one of those readings was divided by a mean cell area that, on a head of 150 organs, was 38.8 where the interior's is π. Divided by the interior's own, the criteria about distance put the golden angle first of 72 angles and the criteria about cells go to rational ones.
One over root two
On the interior's scale a golden head's largest empty circle is 0.8435 of a spacing at every size from 150 organs to 2,000, because one triangle at its centre decides it. Everywhere else it is 1/√2 — a square cell at every ring where the lattice flips — and a closed form in the angle's continued fraction says that only noble angles hold it there.
The empty interval is the rings
No single cut-off on hop ratio turns the contacts a count keeps into the walls a tessellation draws, on any whole head at any size. Read cell by cell against the flip rings the divergence angle puts in closed form, every disputed cell lies within two thirds of a wall spacing of a ring, and with one spacing either side set aside a single cut-off between 1.430 and 1.444 serves every golden head from 900 organs to 9,000.
A second moment that goes to zero
The mean squared departure of a cell's side count from six separates a random tissue from a whorled head by a factor of eighty, on heads of 900 organs. Read at thirty-three head sizes it is exactly the share of cells on the defect rings of a spiral head and falls as one over the square root of the organ count, it falls as one over the count on a whorled head, and the factor is 25 at 300 organs and 677 at 8,000.
Nothing in the staircase moves
Disorder swept across the divergence angle is a staircase, and every step of it had been read at one head size — which leaves open whether a step is the lattice changing or a ring of defects crossing the rim as the angle moves it. Read again at 539, 900, 1409 and 3690 organs, 52 of the 53 features present at a smaller head are still there at the same angle at the next size up. Not one slides. A bigger head adds steps between the ones already there — 4, 18, 31, 40 — so the staircase belongs to the angle and the head size decides only how much of it is resolved. The one size every other disorder figure here uses turns out to sit three per cent past a ring entry.
The blur was at the centre
On a Lucas head the band of disputed cells round each flip ring looked blurred at its inner edge — exact cells as close as 0.23 of a wall spacing, disputed hexagons out to 0.59 where a golden head's stop at 0.43. Read a ring at a time, the two heads carry the same band on every resolved ring, to a hundredth: disputed hexagons within 0.16, exact cells from 0.64, a ring's own number of fives and of sevens and the number before it of hexagons. Every difference is inside a radius of six, where the Lucas rings of 4, 7 and 11 sit closer together than the band is wide, and the one exact cell is organ 17, which has no organ eighteen behind it.
A hundredth of a spacing
Off the flip rings one hop-ratio cut-off turns a seed head's counted contacts into its cell walls, on every head from 900 organs to 9,000. Displace the organs and it is the first thing to go: shut by a fiftieth of a wall spacing on 900 organs and a two-hundredth on 9,000, because it is decided by the worst of thousands of cells. The three-family count survives two to four times further, because each cell only has to beat its own margin, and the rings keep their fives and sevens in between. All three fail from the rim inward, since the margin one spacing from a ring is 9.7 divided by the ring's family number.
The band moves, it does not blur
Displaced organ by organ, a seed head loses its single contact cut-off first, its rings' hold on their fives and sevens next and its three-family count last. Displaced by a smooth field that moves neighbours together, the same head keeps its census — the same 353 disputed cells and 264 fives and sevens at every step up to a third of a spacing — and moves the band instead. A twist moves each flip ring exactly to where the twisted divergence puts its tie, the ring of 55 by 0.53 of a spacing, the ring of 34 the other way. Read against strain, correlation helps the cut-off and not the count, and on a 900-organ head the two fail at the same step: the order was an order of blurring.
Lewis's law needs the sides to vary
Lewis's law holds on a random set of points and fails on a golden-angle head. Walked from one to the other by displacing every organ independently, the head's Lewis slope reaches half a random set's at a fifth of a wall spacing and nine tenths by seven tenths, and in between it explains up to 41 per cent of the variation in cell area — more than the 31 per cent it explains in the random set. Moved instead by a smooth field correlated over eight spacings, the head's cell areas become nearly as varied as a random set's and its slope stays at nought, because its side counts stay the lattice's. The law is not about how varied the cells are. It is about how varied their sides are.
One law counts sides, the other pairs
Lewis's law and Aboav's relation point opposite ways at the two ends of disorder, and the obvious guess is that they are one reading of disorder taken from two sides. Measured on the same moved heads, they are not. Displaced organ by organ, Aboav's a first rises — to 1.45 at 0.15 of a wall spacing, as the first new defects arrive as bound five–seven pairs — and falls half-way to a random set's only at 0.45 of a spacing, where Lewis's law had switched on at 0.2. Between the two a tissue satisfies both. A smooth field, which never switches Lewis's law on, lowers a by pulling the pairs apart without making any new defects. Lewis's law reads how varied the sides are; Aboav's reads whether the defects are paired.
Two numbers for a tissue, and which two
Lewis's law and Aboav's relation read different things in a tiling — how varied the sides are, and whether the defects are paired — so a tissue has a place on a plane of both. Move a golden head by a smooth field and then displace it organ by organ, over a grid of both, and the tissues fill that plane rather than lying along a line. No single one of the four numbers a tissue is usually reported by places it on both laws: the variance of the side counts reads Lewis's slope to three times the seeds' noise and misreads Aboav's a, the pairing share reads a to two and a half times and misreads Lewis's slope. The variance with either law's own statistic places both to within one and a half times the noise; the variance with the pairing share, which is what a counter of cells records, to about twice. And the only tissues that fail both laws are heads moved by a smooth field of two spacings or more and nothing else.
A tissue that was never shaken
Every tissue whose laws have been read here was disordered by moving its points. A growing tissue also disorders itself by dividing, and a division is a wall no set of points generates. Held as a map and divided cell by cell by three rules, a golden head's tiling switches Lewis's law on once a tenth of its cells have divided, at a variance of side counts lower than any moved tissue reaches the law at, because the commonest single division makes two half-sized fives and two full-sized sevens at once. Dividing the largest cell first reaches the corner of the plane no moved tissue reached — Lewis's law on and Aboav's a above its band, at 1.67 — because the largest cells of a golden head are its sevens. And the two numbers that placed every moved tissue on Lewis's law to one and a half times the noise misplace a divided one by fifteen times it: they were a calibration of how the tissue was disordered, not of tissue.
What lies between the steps
The disorder staircase — the spread of a head's side counts against its divergence angle — gained steps with every larger head, forty at 3,690 organs, and nothing said whether it had steps at every scale. Read again at a hundredth of its grid inside its two widest gaps, it has none: no change there reaches the size it counts as a step, and no dip hides between two of its samples. The steps stop. What the gaps hold instead is a sawtooth — μ₂ climbing a cell or two at a time and falling in teeth of ten to thirty-two cells, five of them exactly twenty-one — and a step, read at the same resolution, is not one event but two runs of flips of fifty-five cells each. How many steps a head has is a statement about where the line is drawn; the steps themselves are finite.
A defect ring is where a family of spirals begins
Label every wall between two cells of a seed head with the difference of their placement indices and a hexagon is two contacts from each of three spiral families. The cells that are not hexagons sit on rings, and read by their labels each ring is a handover: one family of spirals ends there and the next begins. On a golden head the ring where 89 enters holds 34 sevens, 21 hexagons that have traded one contact and 34 fives — 34 + 21 + 34 = 89 — because every chain of the entering family begins at one of its cells and every chain of the leaving family ends at a traded hexagon. Joined by their walls, the fives and sevens make 21 clusters, and the clusters run round the ring in the order of the Fibonacci word.
The claims, measured
The nautilus, the sunflower and the golden angle arrive with more confident wrong statements attached than any other subject on this fleet. Each one gets a test and a number — including the one that turns out to be right.
Fibonacci is a branch, not a law
Fibonacci counts come from one branch of the model. The Lucas branch — which the same rule reaches at a different growth rate — gives 47 and 76, and neither is a Fibonacci number. The sequence is a consequence of an angle rather than a property of plants.
The claim that survives
Of the three famous assertions about this subject, one is out by a factor of two, one is true of a branch rather than of plants, and one is right — in a sharper form than the version usually told, and about arithmetic rather than about packing.
What "whorled" was hiding
The earlier work's census put 35% of divergences in a bucket labelled whorled and moved on. Opened, every pair in it is k and 2k — the coarsest rung of the ladder, repeated k times — and reading the census up to jugacy takes the Fibonacci share from 14.7% to 50.1% without describing a single extra plant.
How often is it Fibonacci
The claim that plant spirals come in consecutive Fibonacci numbers is stated as a near-universal. Asked of the geometry, the answer collapses with scale — at a coarse rise 67% of divergences give Fibonacci pairs, and at a fine one 15%, with whorled and unnamed pairs taking the rest.
The angle is not the object
Every popular account of phyllotaxis is organised around a number. After a round of work spent on stems, forks and frequencies, the number looks like the wrong thing to organise an account around — it is the limit of one path through a branching structure, it is at no fork, and a plant that has it got there by not jumping.
What a count is worth
A reported parastichy pair pins the divergence angle to a band 221°/mn wide — so 2 and 3 says almost nothing and 34 and 55 fixes it to a tenth of a degree. Each step up the Fibonacci sequence is worth a factor of φ², and recording the radius a pair was counted at adds only ten per cent.
How many plants would it take
Fourteen specimens separate the geometry's Fibonacci share from a coin weighted to a half. Four separate it from what a grown history gives. One fir cone measured at three rings settles whether its transitions are spaced as a cone's or an ogive's. The sample sizes are small, and that is the uncomfortable part.
The survey this site cannot do
Four rounds of asking for a dataset, and it is still not here. What the work here can do instead is specify it — the fields, the sampling, the sizes, and which of this collection's claims each one would settle. Two of the four fields asked for turn out to be worth less than the asking implied, and one was never asked for at all.
The test a plant could settle
Every other open question in this collection is priced in tens of specimens, and one of them in a hundred and sixty. This one is priced in internodes on a single stem, and the number is fifty-six — because it is a statistic of one sequence rather than a share of a population.
What a quiet plant is worth
Almost every measurement gets easier as the effect gets larger. This one gets harder — a stem's divergence sequence stops carrying information about its noise at precisely the scatter where the noise becomes obvious. The specimens worth measuring are the ones that look least interesting.
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.
What a summary throws away
Four statistics this collection has relied on turn out to be incapable of varying with the thing they describe — one is invariant to shuffling, one is fixed by a theorem, one is a parameter that stopped mattering, one is a fitted number selected into being wrong. In each case the second statistic was free and nobody had taken it.
What the pair costs
The single parastichy number cost sixty internodes. The pair costs two hundred and fifty, and a protractor error of three quarters of a degree takes it to eleven hundred. The arithmetic that predicts the second of those is right about the shape and wrong about the scale by a consistent factor, which is recorded rather than fitted away.
The most irrational is not the most disordered
If rational angles make ordered tissue, the most badly approximable angle should make the most disordered — which would at last give the golden angle a criterion it wins. Swept across the interval it is the most irrational point of, μ₂ peaks at 138.42° and the golden angle sits unremarkably in the middle.
What a refusal does not say
The readout can decline for four different reasons — too quiet, too disturbed, too fast, or a window in the wrong place — and a stem that returns nothing does not say which. That is the third time this thread has failed to close the mixture problem, and the first time the failure has a shape.
A refusal with a reason
Three note left with the work running have recorded that a refusal has four causes and the sequence separates none of them. With a second window and a protractor, three are separated: silence at 0.38° of scatter is a quiet plant, silence at 56° is a disorderly one, and agreement certifies the rate. The fourth survives, and so does a worse discovery — agreement is not correctness.
The control a survey would need
A comb no longer shows that a plant computes its pattern, so the survey this site has been specifying for a long time has to change. What it loses is its headline; what it gains is a measurement a botanist can actually make — six requirements, four of them already in the specification, and a quantity nobody has ever reported.
An experiment a needle could run
For eight instalments the outstanding item has been a survey — photographs, a protractor, hundreds of specimens — and it has not been done. The intervention is a different kind of ask, and a cheaper one: a needle, one apex, and a yes-or-no per ablation. Here is what it would cost, what it would settle, and the four ways it could come out.
The survey loses its second outcome
The survey specification written earlier here names three results the survey could return, and the second — a ratio near or above 1.30, read as evidence against the placement rule — is the one that would have been worth publishing. It does not survive the measurements here. The ratio moves with where the plant sits between two transitions, and it moves again with the colour of the plant's own disturbance.
The order belonged to the method
A residual was left over after the two width laws, and it looked ordered: the most crowded fraction gave the widest dip, in all three families, in the direction a measurement artefact would take. Measured again with an instrument that has no level in it, the order changes with the window, disagrees between families, and in one of them comes out backwards.
The ablation a plant would survive
The intervention proposed earlier returns a spiral count from a yes-or-no answer, needs no protractor, and was specified at one rise. Measured across the ladder it acquires three conditions a real experiment would have to meet — and one of them is that the plant must not be too coarsely patterned, or nothing will go wrong at all.
A period the grid invented
A wrecked stem was reported as settling into a repeating block of three angles — 219.84°, 220.31°, 220.78° — which is the smaller of its two spiral counts and would have confirmed a standing prediction. Those three numbers are three consecutive samples of the azimuth grid. There is no block; there is a constant the grid cannot write down, and the routine that found the block was working perfectly.
The second statistic was the first
The experiment this collection has been specifying was priced as two readings off one sequence, the second of them free. The two readings turn out to be one function looked at twice, so the specification loses a statistic — and gains a cheaper one, a warning about how observables get priced, and a question it could not previously ask.
The residual was the window
After the depth and the q over n squared scale are taken out of a disorder dip, something looked left over and looked ordered by how crowded the fraction's neighbourhood is. Measured on fractions whose neighbourhoods are identical by construction, seven widths across a factor of two and a half in denominator agree to one per cent. There is no residual; there was a comparison made at different effective windows.
Matching instead of correcting
Two rounds of work failed on one question because every instrument's free parameter was set by the thing under test. The repair was not a better instrument or a model of the bias: it was choosing what to compare so that the confound could not vary. That move is available in four other places here, and three of them have already used it without anybody naming it.
A count that can be wrong by one
A reported parastichy pair pins the divergence angle to a band 221°/mn wide only if both counts are right. Allowing either to be off by one adds the bands of every neighbouring pair whose counts share no factor, and those bands sit where their own lattices live: for 2/3 they swallow the whole range, and for 34/55 they are two bands as narrow as the true one at 109° and 113°, twenty-five degrees away. So a high count that may be wrong is not a blurred reading but a short list of sharp candidates, costing log₂ 3 bits. And on the Fibonacci pairs, two in every six — 21/34 and 34/55 among them — cannot be miscounted silently by one count at all, because every such miscount shares a factor.
What a head can mean by most irrational
Hurwitz's bound, the one famous claim about this subject that survives, is a limit over every denominator, and a head shows only the counts between its innermost spirals and its rim. Over those counts an angle resists approximation like the golden angle exactly when the counts it shows add up, each the sum of the two before, from a pair near the golden ratio — and every such pair has an angle of its own. The golden angle still scores highest over every window measured, by a ten-thousandth: over counts from 34 to 144 the Lucas angle is 99.989 per cent of it and forty-six angles are within one per cent. What separates the golden angle from them is below the counts they share, at the centre of the head.
A count that drifts by two
A reported pair of 34 and 55 that may be wrong by one allows three sharp bands; allowed to drift by two it allows thirteen, and by three, twenty-nine — and the information lost is exactly the logarithm of that count, because every band is as narrow as the true one. The nearest wrong band stays twenty-four degrees away until a drift of three brings one to eleven. What does not survive is the protection: 21/34 and 34/55 could not be miscounted silently by one, but every Fibonacci pair can be by two, so a counter who drifts by two as readily as by one reports 34/55 silently wrong 9.5 per cent of the time rather than 0.13.
The first three hundred organs
Over the counts a head shows, forty-five angles resist approximation within a per cent as well as the golden angle, and what separates them is at the centre. Grown as heads and measured there, the golden angle has the widest closest pair of all forty-six — by organs 1 and 4, the count its arithmetic names — and keeps first place only while the centre is in the reading. Its rivals stay a per cent apart from it out to a radius that tracks where their spiral counts start to add up, and every one of them is within a per cent by the 289th organ. By the largest hole it is never the best.
Two counts that slip together
A counter who closes the circle a few degrees late counts a sliver of the head twice, in both families at once, so the two counts of a reported pair drift together rather than apart. Coupled that way the count is safer than it was: fourteen Fibonacci pairs in twenty-three admit no silent equal shift of one, against seven that admit no silent single miscount, and 34/55 announces every closing error short of 9.82°. The check is what breaks. Two annuli closed at the same wrong mark pass 17.6 per cent of wrong readings of 34/55 and 76.8 per cent of 13/21's, because a linear relation survives multiplication — and what catches them instead is a protractor good to twelve degrees.
The census wants a low count
Four specimens separate the geometry's Fibonacci share of 14.7 per cent from the ninety per cent a grown history gives — if every count is right. Counted with a closing error spread over 7.2°, the same census needs six specimens counted at 13/21, fifty-four at 34/55 and 449 at 55/89, because the geometry's own pairs are all small enough that no closing error under 11° moves them, while a grown plant counted high loses its Fibonacci reading first. Counted at 55/89 with a spread of 9.83° the census reads plants as less Fibonacci than random angles. The count that pins the divergence best is the one a census should avoid.
Two marks chosen by one eye
A counter traces each family of spirals from a starting organ of its own, so a reported pair carries two closing errors, correlated because one eye chose both. Letting them differ costs 34/55 its ten-degree margin — 33/56 and 35/54 share no factor, and marks that err 5.3° in opposite directions reach them — while 21/34 keeps its margin whatever the marks do. And it decides the second annulus. At a spread of 7.2° the relation passes right readings 2.8 times as readily as silent ones when the marks are independent, 1.25 times at a correlation of 0.9, and stops telling them apart at 0.98; where it does work it keeps one reading in forty-six.
Counting it again
A reading whose two counts share a factor says the count went wrong, and the specimen is still there to be counted again. Counted afresh, the reading kept is exactly one reading conditioned on not announcing itself — the second chance a silent error gets is matched by the second chance a right reading gets — so a recount changes which specimens a census keeps, not what a kept reading says. At 34/55 with closing errors spread over 7.2° it takes the census from fifteen kept specimens to ten and from about thirty counts to twenty-one, and against scoring every reading it turns 449 counts at 55/89 into 52. It never makes a high count as cheap as counting 13/21 once.
The recount aims where the counter expects
A counter who recounts an announced reading knows it went wrong, and if the same habit spoils both counts of a head, the first error says where to aim the second. But the reading alone does not say which way the first erred: a reading of 34/54 is as well explained by a whorled 34/54 read right, or by 34/53 read long, as by 34/55 read short. The direction comes from what the counter expects. Expecting Fibonacci, an aimed recount at 7.2° and a habit correlated at 0.9 reads 34/55 69.6 per cent of the time where an unaimed one reads it 41.8, and the census needs seven specimens rather than fifteen. Expecting only a spiral, it aims the wrong way and reads 34/55 5.5 per cent of the time. The belief that helps is the hypothesis the census is testing: uncapped, it reads the geometry's whorled 3/6 heads as 3/5 and a census of a hundred rejects a true null 40 per cent of the time; capped, it still reads a silent 33/53 as Fibonacci twice as often. And no aimed recount spends fewer counts than 13/21 counted once.
The high heads the geometry rarely makes
A Fibonacci census draws its null from the geometry's heads, and at the rise it was specified at none of them counts past sixteen — which is why a counter who aims a recount at the Fibonacci pair they expect, capped at twice their own spread, left the null alone. Drawn at finer rises, where the geometry's heads count into the forties, the null gets easier to beat rather than harder: its Fibonacci share falls from 22.5 to 7.5 per cent and the census needs four kept specimens instead of seven. The capped aim still adds almost nothing to it, because the high heads the geometry makes are rarely the ones it moves: the geometry makes a 33/53 only at a rise of 0.0003, near a divergence of 54°, and it is a twentieth of a per cent of the null. What the census does lose at a fine rise is its five per cent — to the counter's habit, not to the aim, and every recount policy loses it alike.
A disc reads almost no Fibonacci heads by chance
A Fibonacci census draws its null from the geometry, and every null so far was a cylinder's: one rise, one dominant pair per divergence angle. Read the same angles as discs, counted in the band a surveyor uses, and the null's Fibonacci share falls by a factor of ten to forty — because nearly all of the cylinder's Fibonacci heads were the pair 1/2 at the smallest angles, a single tight arm that no band reads as 1 and 2. The census then needs three kept specimens. Half of a large head's bands do straddle a transition, but a band that straddles reads one edge's pair or its own middle pair, and the handful of true mixtures are nowhere near a Fibonacci pair.