The thread: The round trip
Shells and growth
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.
The pattern itself
Counting the spirals
Almost every claim about phyllotaxis is a claim about how many spirals run through a pattern, and the count is almost never done. It can be done from the points alone, by a count that is never told what angle built them — and then a count of 34 is evidence rather than a restatement.
Branching and transport
Fitting the exponent
Assuming the exponent is three and reporting the error says how far the data is from that assumption. Fitting the exponent and reporting what it comes out as says what the network is doing — and an estimator has to be shown returning something other than three, or it is not a fit.
Stems and cones
Two numbers out of the points
A seed head's divergence angle can be recovered from its spiral counts only to within an interval, because a range of angles gives the same counts. On a stem the counts come with lengths attached, two measurements pin two unknowns, and the lattice comes back to the last digit it was built with.
The pattern itself
Recovering the angle from the counts
Build a head at a stated divergence angle, forget the angle, and get it back from the spiral counts alone. Four angles, worst error twelve thousandths of a degree — and the only thing that crossed between the two halves was a list of coordinates.
The pattern itself
Counting without an index
A person counting spirals on a cone puts a finger on one scale, follows a family round, and counts how many distinct chains there are. That needs no order of arrival — and building it turns out to be strictly more general than the counter that reads the order of arrival, and to find a bug in the counting of a bijugate stem that nothing had caught.
Stems and cones
The forks are exact
Where a stem's pattern has to choose between two futures, three spiral families are equally short and the lattice is exactly equilateral. A numerical solver found those points; the numbers it returned turned out to be rational, and chasing that gave a closed form — including the fact that every fork sits at a rational divergence, and the golden angle at none of them.
The pattern itself
A counter that sees no positions
This site has counted spirals two ways, and both were handed coordinates. A third counter is handed a list of angles and nothing else. It returns one number instead of two, it refuses more often, and where it refuses it would have been wrong every time.
The pattern itself
Every family but two is a sum
A seed head has six spiral families and everybody reports two. That looks like a convention hiding information and it is the opposite — every family but the two smallest is the sum of two others, so a third count is a prediction rather than a measurement, and a check that catches a wrong pair.
The pattern itself
The angles name the branch
Seed the same rule at the Lucas angle and the readout returns 4 and 7, then 7 and 11 — the pairs the position counter finds, and not Fibonacci numbers. So a list of divergence angles carries not only how many spirals there are but which family of ladders the plant is on.
Shells and growth
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.
Shells and growth
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.
Stems and cones
The comb was never the rule
A control is only as strong as the alternative it builds, and the earlier work built one that varied the rule while holding the disturbance fixed at independence. Five rounds of the angle-sequence thread, with what each claimed and what still stands — and why the next evidence has to come from an intervention rather than from a longer stem.
Stems and cones
The rung was not the instrument
The earlier work said the pair readout has a ceiling one rung above where it works, that this is arithmetic rather than statistics, and that no amount of stem fixes it. The arithmetic is right and gives a band of lag windows that is never empty; what was actually stopping the reading was an eight-node seed and a grid of 384 azimuths.
The pattern itself
A counter that cannot be slid
The counter that needs no order of arrival follows each family into chains and counts them, and on an ideal lattice it agrees with the counter that does. On a stem whose rise falls it works only inside a band of widths, and outside the band it returns a pair rather than refusing: the rung below when the band is too narrow for the larger count, a pair on no rung when the band spans more than about a third of a rung of rise. The upper edge moves with the rate, so a width that is right on one stem is wrong on another, and on the fastest bijugate stem measured no width works at all.
Where the angle comes from
Two windows on one stem
A pair read off a climbing shoot can only be read through a window, and a window can straddle a transition. Read a second window half a length lower and the outcomes fall into four kinds — and agreement between them never happens on a shoot whose rung is shorter than the window, which turns the most awkward of the four refusal causes into something a reading can certify.
Branching and transport
What a scar is worth
Counting the scars a dead shoot leaves does not put a branching count back on the sequence it would have had. A scar records a growing point and a dead growing point takes every branch it would have made, so living points plus scars reach 39.2 per cent of the deathless count after twenty seasons at one death in twenty, and 1.9 per cent at one in five — falling without limit rather than closing. What the scars restore is the other number. Scars per living point settle at q/(x − 1) exactly, so a rate with a scar share beside it recovers the death chance and then the waiting time, where a rate alone is reached by a one-season wait losing a tenth, a two-season wait losing 0.64 per cent and no wait at all losing 27.2 per cent.
The pattern itself
A head displaced before it is counted
The round trip from a head's spiral counts back to its divergence angle was tested on heads whose every organ sat exactly where the rule put it. Displaced by a normal error of up to two and a half spacings, heads of 900 organs keep counting a pair from their own sequence and return intervals holding the true angle to a spacing and a half; heads of 300 organs move to the neighbouring pair by half a spacing and then refuse, nine in ten of them by two spacings. Every moved count brings in the family whose chord was third shortest. Of 898 heads recovered, 19 intervals miss the true angle and 17 of those by about a tenth of a degree — displacement makes the reading coarser and then silent, not confidently wrong.
The pattern itself
A twist is a divergence
Recovering a head's divergence angle from its spiral counts survived independent displacements of whole spacings, moving to a neighbouring pair and then refusing rather than misleading. Displacements with a direction are harder on it in only one case. A head pressed to an aspect ratio of 2.25, spread at the rim by eighty per cent or sheared with a slope of 1.6 is still counted as its own pair or its neighbour, and recovered inside its interval. A head twisted — each organ turned about the centre in proportion to its radius — is not: past a turn of the rim of about a radian and a half the counts leave their sequence and the recovered angle misses, by up to sixty-one degrees, because a twist changes the angle between one organ and the next. The round trip is not fooled. It is reporting the angle the twisted head has.
The pattern itself
The count sees the twist first
A twisted head recovers a changed divergence, and the check proposed for it was two annuli: the twist's extra angle falls with radius, so an inner and an outer annulus should disagree. Read on golden heads of 900, 2,400 and 9,000 organs, they never do in time. Their intervals separate at eight radians on 900 organs and never on the larger heads, always after the ordinary reading has been misled — from six radians on 900 organs and from two on 2,400 and 9,000. What catches the twist first, at every size and on every seed, is the count: at half a radian to three quarters some band stops returning two consecutive Fibonacci numbers — 34 and 89, 89 and 233 — which no untwisted golden head, clean or displaced, ever does.
The pattern itself
The flag reads the angle, not the twist
A twisted seed head is caught first by its counts: at half a radian some band stops returning two consecutive Fibonacci numbers, which an untwisted golden head never does. Twist the head in proportion to the square of the radius instead and the organs land exactly where a head grown at the golden angle plus a/(N − 1) puts them — to a billionth of a spacing — so the two read the same pairs in every band, and the flag fires on both. The flag detects a divergence that is not golden, and it has a resolution: 0.015° on 900 organs, 0.01° on 2,400, 0.003° on 9,000. Every twist shape from a half to four is flagged exactly when its change to the divergence in the outer annulus passes that resolution. So a flagged head is not golden, and nothing in its counts says whether it was twisted or grown that way.
Branching and transport
Two ways to die, three things to count
Giving a branching plant's waiting buds a death chance of their own leaves its counts a linear recurrence, but breaks the collapse onto the survival: the rate becomes the apex survival times the root of y^(d+1) = y^d + r^d, where r is the bud survival over the apex survival. The one-chance reading then names the wrong waiting time on 171 of 477 plants with waits of two to four seasons, shorter when the buds are the fragile ones and longer when the apices are. The two chances are separable from a rate and a scar share, exactly — but the two counts' loci cross at eight to sixteen degrees, so a one per cent error lets the chances wander by a factor of two. A third count is owed, and it is the scars sorted by kind.
The pattern itself
The positions read the twist's shape
A skipped Fibonacci count says a seed head's divergence is not golden, and cannot say whether it was twisted or grew that way; only the radial profile of the angle between neighbours can, and the counts are too coarse to see it. The organs' positions are not. The birth order is not in them — consecutive organs at the rim of a 900-organ head are a hundred and sixth of a spacing apart in radius, so a hundredth of a spacing of displacement scrambles a quarter of them — but each counted family is, and the sum of its phase angles across the organs peaks at the local divergence with no organ named. Displaced by a tenth of a spacing, a 900-organ head's outer annulus reads its divergence to 0.0001°, a hundred times finer than the flag resolves it, and the two annuli tell every twist shape from a changed angle at a quarter of a radian or less, no later than the flag fires. The square-law twist stays inside the untwisted heads' spread at every size and displacement: it is a changed angle, organ for organ.
Branching and transport
A bad year does not average out
Let every growing point on a plant share one season's death chance, bad one season in ten and good otherwise, with the average held at 0.1. Averaged over every sequence of seasons the counts are exactly the independent model's. But no plant is an average over sequences: with bad years at 0.5 a plant settles on a rate of 1.3001 against the expected 1.3190, the median plant holds 54 per cent of the expected count by season forty, and the scar share never settles, because it is set by how many seasons ago the last bad year was — 1.35 the season after one, 0.17 twelve seasons on. So the reading of a plant's waiting time from a rate and a scar share gets worse the longer it runs: right for 57 per cent of plants over ten seasons, 22 per cent over eighty.
Shells and growth
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 pattern itself
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.
Branching and transport
Scars with dates on them
A bad season shared by every growing point wrecks the reading of a branching plant by two totals: over eighty seasons the rate and the scar share name a two-season wait for 22 per cent of plants. Date the scars — by position along a shoot, by growth ring — and each season's death chance is read off its own scars, so the bad years stop being noise and become a known input. Running the branching recurrence through the plant's own seasons names the wait for 94 per cent of plants over twenty seasons whether bad years kill a tenth of the points or seven tenths, and for every plant over eighty. What a plant cannot read from its own scars is the climate: the averaged chance comes only as fast as seasons do.
Shells and growth
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.
The pattern itself
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.
Shells and growth
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.
The pattern itself
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.
Shells and growth
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.
The pattern itself
A dome is read as a nearer camera
A camera fitted from a photographed seed head's organs alone restored the reading of a flat head seen in perspective from as close as two radii. Lay the same heads on domes and the camera is fooled in a particular way: it finds the tilt, its median within about a degree, and puts itself nearer than it is, by an amount set by the dome's height, so that a dome a fifth of the head's radius high photographed from infinitely far is read as a flat head photographed from seven radii. The reading it hands on is wrong. Off the axis, a domed head is misread once its height times the sine of the tilt passes about four hundredths of the radius — the same number at which a flat head's perspective broke the old correction — and only a photograph taken square on is safe.
Shells and growth
A cut seen in perspective is not another shell
A photograph of a cut shell squashed straight along the axis is exactly the picture of another shell, with nothing to tell them apart. A camera close enough to see the cut in perspective is different. The near whorls come out larger than the far ones, and a whorl's size then changes across the picture in a way no shell's own expansion does — so the reading that fitted a distant photograph misses a near one by more than the noise, even straight along the axis. Before it is seen the error is small, W read too large by about the sine of the tilt over the distance. A camera fitted beside the shell, three numbers more, takes it back entirely: W, D and T within a third of a per cent at every bearing and every distance down to three rim radii.
Shells and growth
A free opening lets the camera trade again
A camera fitted beside a shell recovered every photographed cut whose openings were round. Give the opening its own shape — an aspect and a turn, as a snail's has — and the twelve numbers recover everything only when the camera's tilt runs at a bearing between the axis and the line across it. Straight across or along the axis, a pinhole tilted τ at L rim radii is exactly a pinhole tilted τ′ at L′ followed by a squash, whenever tan τ′/L′ = tan τ/L, and that squash is another shell. Every camera on the line fits a noiseless picture to rounding. The fit keeps W and the perspective's strength, and nothing else about the camera: T and the aspect come back scaled by a ratio of cosines, a turned opening's turn and its D move with them, and on noisy pictures T ran from six per cent low to a quarter high with every picture fitted inside the noise.
Shells and growth
Two photographs pin the shell only from two sides
A photograph of a cut shell whose opening has its own shape, taken straight across or along the axis, fits every camera on a line, and each camera on the line reads a different shell. A second photograph of the same cut shares the shell and brings a camera of its own. Taken at the same bearing — farther away, or steeper — it does not help: the two cameras slide together along a family set by one closed form, cos τ₁′/cos τ₁ = cos τ₂′/cos τ₂, each picture fitted to rounding at every point of it, and the shell slides with them. Turned so that its tilt runs across a different direction, it pins the shell, and on noisy pictures a turn of ten degrees is already enough.
Shells and growth
A tilt undone beats the calipers
A shell section photographed off its plane reports its own tilt in the fit's residual, but not the axis the tilt was taken across, and a correction needs both. Read together from the whole outline — the tilt and axis whose undoing leaves the straightest line — they come back on half a turn of shell to a fifth of a degree and under a degree. Undone, the fit no longer merely matches the calipers that a tilt cannot touch: on a traced outline it beats them, by three times on half a turn and by twelve to forty from one turn up, because the calipers read three points and the fit reads all of them. Correcting a section that was square on costs a little, and a shell whose growth slows is mistaken for a tilted one without harm.
Packing and tiling
A head's holes spell its angle
A head's largest empty circles follow a closed form in its angle's continued fraction, and a closed form runs both ways. Given only the organs of a 2,400-organ head, the largest hole in each ring and no angle, a reading outward through the rings returns 28 of forty undisclosed angles to a thousandth of a degree, quotient by quotient, and a larger head tightens them a hundredfold. The other nine it finds are other angles, and they are not random: their expansions end as the truth's ends and start differently, because the start of a continued fraction lives at the centre of a head. And most of it is gone once each organ has moved by a hundredth of a spacing.
Shells and growth
Two free frames cost one right angle
Two photographs of one cut shell pin it only when their cameras stand at different bearings, and the pair at right angles read best of all. Each of those fits was handed both pictures in one frame, and no real pair comes that way. Give the second picture a frame of its own, rotated and scaled and shifted, and leave the reading to find it: nineteen numbers, not fifteen. It costs almost nothing. The rotation comes back to a twentieth of a degree, the scale to under a tenth of a per cent, and the shell's numbers as well as before, except at one bearing. At exactly ninety degrees a free scale swallows one camera's squash whole, the fit slides, and the pair that had been best reads the translation twenty-three times worse than a pair at forty-five.
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