The thread: Counted, not admired

Stems and cones

A stem is a cylinder

The sunflower is the photograph, and it is the hard case. Nearly all real phyllotaxis happens on a stem, where the geometry is a lattice on a cylinder with two parameters — and where the spiral counts, which on a disc change with radius, are the same the whole way up.

The pattern itself

A head is a set of points

The nth primordium at n times an angle, and a radius of root n. Two lines of arithmetic produce a sunflower head, which is either remarkable or suspicious depending on how carefully the claim is stated — and stating it carefully is most of the work.

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.

Stems and cones

Counting up the stem

The same counting machinery, pointed at a stem instead of a seed head, returns one answer three times where the head returned three answers. That contrast is a measurement rather than a preference, and it is the one the whole cylindrical argument rests on.

Packing and tiling

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 pattern itself

The counts change with radius

The same head gives 13 and 21 near the centre, 21 and 34 further out, 34 and 55 beyond that, and 55 and 89 at the rim. The transitions are at computable radii, and a photograph captioned with one pair is a statement about one annulus rather than about a flower.

The claims, measured

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.

Packing and tiling

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.

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.

Shells and growth

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.

Packing and tiling

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.

Stems and cones

A disc is a cylinder

Vogel's seed head makes the rise fall as one over radius squared, so a disc is not one lattice but a family of them. Feed that into the cylinder's ladder and it predicts where a sunflower's spiral counts change — with nothing fitted, and against a counter that never sees either model.

The pattern itself

Two at a time

Every counter in these essays asks how far it is from element i to element i plus m, and that index is a claim that the elements arrived one at a time. For teasel, for Cephalaria and for a real minority of plants the claim is false — and what those patterns turn out to be is an ordinary lattice, wrapped twice.

Where the angle comes from

Half a turn, four at a time

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

Shells and growth

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.

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.

The claims, measured

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.

Branching and transport

The band decides the answer

A fit over a whole tree's junctions returns the exponent the tree was built at, even though most of its junctions are from bands that on their own return 1.6. Least squares is already weighting by leverage. The dangerous sample is not the mixed one — it is the one a person can reach.

Packing and tiling

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.

Shells and growth

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.

Stems and cones

A cone has a rise that falls

A stem holds one parastichy pair for ever and a seed head changes its pair with radius. A cone does both — it is a cylinder whose rise falls as one over the distance from the apex, and the same blind counter that finds one answer up a stem finds four up a cone.

Packing and tiling

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.

Branching and transport

The exponent an error moves

Every real measurement of a branch radius carries error and no synthetic tree does, so the question is what a symmetric error does to a fitted exponent. It does two things — a bias and a spread — and the bias runs downward at every error level and in every band, by an amount derivable from the daughter ratios alone.

Shells and growth

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.

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.

What a plant might be doing

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.

The claims, measured

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. Every rung of the ladder is worth a factor of φ², and the counting radius this collection has asked published counts for since its foundation adds ten per cent.

Branching and transport

The fragile junctions are the informative ones

That is the obvious worry once the radii are uncertain, and it is false. Across the whole range of asymmetry a junction's contribution to the bias moves by a factor of 1.36 while its leverage moves by a factor of 308,352, so the junction that says nothing damages the answer as badly as the one that says everything — and a sample is spoiled by counting rather than by weight.

Shells and growth

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.

Packing and tiling

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°.

Packing and tiling

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°.

What a plant might be doing

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.

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 claims, measured

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 ladder is spaced as a cone's or an ogive's. The sample sizes are small, and that is the uncomfortable part.

Branching and transport

The window that closes

The spread of a fitted branching exponent falls as the reciprocal root of the sample and its displacement does not fall at all, so there is a count past which every further junction buys confidence and no accuracy. Between three and five per cent of radius error the count arrives before the answer does, and no sample size both states a claim and contains the truth.

Shells and growth

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 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.

Packing and tiling

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 claims, measured

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.

Stems and cones

Why a cone can be counted once

A pineapple is described as 8 and 13 and the description holds. A sunflower is described as 34 and 55 and the description is a statement about one annulus. Both organs have the same ladder in element number — what differs is where an organ puts its elements.

Shells and growth

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.

Branching and transport

Where three and two become one

Two trees, one built to obey Murray's law and one to obey Da Vinci's, are measured through the same fifty junctions with the same instrument. At twelve per cent of error on each radius the two answers overlap, and above twenty and a half the tree built at three measures lower than the tree built at two.

The pattern itself

A periodicity is not a lattice

Give a lattice's errors a period of eight and a comb appears at spacing eight, on an arrangement with no rule in it. But the partner it names is 10, then 12, then 11, then nothing — an accident of the disturbance rather than a measurement of the pattern. The forgery is caught by reading a second stem, and by nothing else.

What a plant might be doing

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 claims, measured

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.

Packing and tiling

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.

Branching and transport

The angle the cost chooses

Murray's exponent falls out of minimising a cost over the radius of a tube. The same cost minimised over the position of the branch point instead fixes both fork angles, and 281 networks minimised on a grid told nothing about any formula agree with the closed form to under two ten-thousandths of a degree.

Shells and growth

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.

The pattern itself

A period that is not a count

Eighteen wrecked stems settle into a block whose period is one of their own spiral counts, and one settles into a block of four on a lattice counted 8 and 13. The odd one is not noise. It is the case that shows what the rule is actually conserving, and it is the reason this thread is about lattice steps rather than about spirals.

Packing and tiling

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.

What a plant might be doing

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.

Where the angle comes from

Two degrees of scatter

A lattice tolerates about two degrees of wander in its divergence angle, and two kinds of noise sharing no code agree on the number to within a third of a degree. It is not a constant: carried finer, the same stem survives 0.8°, and the tolerance tracks the band of angles that produce its pair at all.

The claims, measured

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.

Stems and cones

What one exponent reports

Fit a single shape exponent to an organ that has four of them and it returns a real quantity — the harmonic mean of what its individual steps report. Harmonic means sit below arithmetic ones, so the fit understates, systematically, in a known direction, and invisibly.

Branching and transport

A rule that predicts everything

Leonardo's rule says a fork conserves cross-section, and cross-section is exactly the weight the branch point is minimised against. So the three weights land on the boundary of the triangle inequality, the cosine comes out at one to the last bit, and the rule predicts no angle at all — and the free constant its own derivation leaves behind then walks the prediction across every angle a fork could have.

The pattern itself

A count with a factor in it

Cuts of several organs send a stem to six settled divergences and no more. Three of them are the old lattice with a family left standing. The other three are counted 2/6, 4/6 and 3/6 — and a pair whose numbers share a factor is the classic signature of a pattern that arrives several organs at a time.

Packing and tiling

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.

Stems and cones

How much of a cone to measure

Two rings cannot show a varying exponent — not with difficulty, but in principle, because one gap determines one exponent with nothing left to disagree. Four or five can, if each is found to within a per cent. At three per cent this specimen cannot be told from a power law however many of its rings are recorded.

The claims, measured

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.

Branching and transport

An optimum too flat to reach

One per cent of a branching network's cost buys forty-three degrees of fork angle, covering exponents from 2.44 to 5.34, while the angle the theory predicts moves only fourteen and a half degrees across every daughter ratio there is. The prediction is steep and the cost is flat, and those are the same curve read along its two axes.

Shells and growth

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.

Packing and tiling

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.

Stems and cones

The sequence has a memory

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

The pattern itself

The symmetry that is not there

A wrecked stem counted 2/6 and a stem grown two organs at a time counted 2/6 are indistinguishable to a spiral counter. Rotate them by half a turn and one lands on itself and the other lands nowhere. A cut does not make a whorled pattern; it makes a pattern that counts like one.

The claims, measured

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.

Branching and transport

The trees drawn at no angle

Two branching figures in these essays set every junction's radii from the cube law exactly and every junction's angle from a constant nobody derived. Read as exponents the drawn angles say 2.52 and 2.62, in pictures whose widths say exactly three — and at a lopsided fork the drawing puts a daughter thirty-four degrees from where the same cost puts it.

Shells and growth

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.

The pattern itself

The pair read from the angles

Checking that a jostled stem is still the lattice its panel is about turned up a disagreement. Counted from the point positions every rule's stems return five and eight spirals; read from the divergence sequence, the deepest rule's stems come back as five and seven at every seed. The points are right, and this site's founding rule is why.

Packing and tiling

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.

Stems and cones

What one angle says about the next

A tenth of a degree of placement noise moves a stem's divergence scatter from 0.50° to 0.62°, which nobody would report. It takes the correlation between consecutive angles from 0.54 to below zero. The other two kinds of noise, at scatters where no measurement can separate them, leave it at 0.6.

The claims, measured

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.

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.

Branching and transport

A swelling at the fork

A branch thickens where it forks, so a parent measured just below a junction and daughters measured just above it carry three different amounts of the same swelling. A swelling that fattens all three alike moves a fitted exponent by exactly nothing. A parent read one per cent fat moves it by as much as 3.6 per cent of random error on every radius, in a sign known in advance, and a tenth of a radius turns a tree built at Murray's three into one that reads Da Vinci's two with no noise at all. Added to the noise, it does not bring the two rules together any sooner: the two errors do not add.

Where the angle comes from

A window that makes a pattern

A rule whose energy has no well-defined minimum produces a clean 8/13 lattice at 137.62°, with half a degree of scatter, when its neighbourhood is cut at three node spacings. Let it see twelve and the pattern is gone. Every simulation of this kind truncates something, and truncation manufactures exactly the result it is used to look for.

Packing and tiling

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.

Stems and cones

The order carries the count

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

The claims, measured

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.

Shells and growth

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.

Branching and transport

A correction that keeps the overlap

The duel between a tree built at Murray's exponent and one built at Da Vinci's ended by saying the displacement is the geometry, and that no better estimator removes it. Correcting every replicate by simulation-extrapolation removes 92 per cent of the tree at three's displacement at five per cent of error and 68 per cent at twelve, and the error at which the two means cross leaves the measured range altogether. It pays in spread — the corrected readings are twice as wide at twelve per cent — so the error at which the two trees' intervals overlap does not move. Of the duel's two numbers, the inversion was the estimator's and the overlap is the question's.

The pattern itself

The window nobody varied

Every parastichy pair reported for the settling table was counted over the top two hundred organs of a run, and that number had never been moved. Moved seventeen ways across a factor of forty, on all three hundred and eighty-four settled runs, it changes nothing at all.

The claims, measured

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.

The pattern itself

A plateau the instrument should have had

The counting window decides nothing on the settling table, and it has two bounds that decide everything outside it. A window one organ too narrow returns the previous rung of the same ladder, a family past the offset ceiling is reported as a coarser rung at every width there is, and neither failure produces a refusal, noise or a wide error bar.

Packing and tiling

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.

Shells and growth

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.

Branching and transport

A count carries no error

Fitting r₀ᵏ = Σrᵢᵏ junction by junction puts a measured radius on both sides of every equation, and at twelve per cent of error a tree built at Murray's three and one built at Da Vinci's two stop being told apart, however the fit is corrected. Fit the same measured radii against the number of tips each branch carries instead — a count, which nobody measures with error — and the two trees read 2.996 and 1.997 at twelve per cent and 3.015 and 2.001 at thirty, never overlapping. The twelve per cent belonged to the junction fit, not to the tree. The count fails in its own way, and the way is stated.

Where the angle comes from

A hard edge is not a falloff

The prediction was that cutting the neighbourhood at three spacings would reproduce the pattern truncation had manufactured. It does — if the cut is smooth. A hard cut at the same distance produces no pattern at any width, and the reason is that it is the only one of the three whose neighbour set depends on where the candidate is.

The claims, measured

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.

Stems and cones

The second comb

The autocorrelation of a divergence sequence has peaks at the smaller parastichy number and at every multiple of it. It also has a second set of peaks, at the same spacing, offset by the difference of the pair — so a list of angles with no coordinate in it returns both numbers rather than one.

Packing and tiling

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.

The pattern itself

How many organs a pair needs

A count taken over too few organs does not fail. It returns the rung below, which is a perfectly good pair, and nothing anywhere says so. The window that avoids it is not a constant but the counter's own arithmetic, and 384 settled runs sit exactly where that arithmetic puts them.

Stems and cones

A harmonic is a step taken twice

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

The claims, measured

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.

Packing and tiling

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 pattern itself

A sequence that was a reading

One of the settling table's additive sequences rested on a single count that skipped a term, and the account of it was that the count had been taken over the wrong patch. It was not. The counter works out whether its two families wind opposite ways and the reading throws the answer away, on five runs in three hundred and eighty-four.

Branching and transport

One constant for every fork

Da Vinci's rule leaves a free constant in the cost that sets a fork's angle, and running it over its range walks the predicted angle from nothing to 120 degrees, through Murray's 74.93. So no single fork can refute the rule. But the constant is one number for a whole tree, and a fork's share of it falls as the square of the fork's size — the constant is a radius axis. A tree spanning a factor of ten in radius must show forks from 29.4 degrees at its biggest to 111.6 at its smallest, a spread wider than one fork's flatness can hide, while Murray's angle is the same at every size.

Shells and growth

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.

The claims, measured

The experiment this site can specify

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.

Stems and cones

Two readings from one stem

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

What a plant might be doing

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.

Packing and tiling

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.

Branching and transport

A count that has lost tips

Radii read against the tips each branch carries keep Murray's three apart from Da Vinci's two where junction fits cannot, because a count has no measurement error in it. A count of the tips a tree has is not a count of the tips it grew. Losing them lowers both trees' readings by one factor that belongs to the losses and not to the rule, so the count stops being right long before it stops telling the trees apart: on fifty junctions at twelve per cent of error, to seventy per cent of the tips lost one at a time, and only to about a quarter lost in whole limbs. Counting scars repairs single losses exactly. Nothing countable repairs a shed limb.

The pattern itself

A grown stem halves its ladder

A bijugate stem at 68.754° was derived to pass the ordinary transitions at half their rises, because a lattice wrapped twice round is the ordinary lattice at twice the rise. No bijugate stem had been grown through them. Grown by the same rule that grows an ordinary shoot, stems of two, three and four organs a whorl walk the ordinary ladder with every pair multiplied by the jugacy, change pair at the ordinary rise divided by the jugacy between whorls — and by its square per organ — lag behind the static ladder by the same hundredth of a rung, and settle within three hundredths of a degree of 137.5078 over the jugacy.

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.

The claims, measured

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.

Packing and tiling

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.

The pattern itself

A Lucas seed counts whorls

An ordinary stem seeded on the Lucas lattice keeps that ladder when its rise falls fast and gives it up when it falls slowly, with the edge near ninety nodes a rung. A stem that grows two organs a whorl is an ordinary stem folded twice round, so the identity predicts its edge — once it says whether the edge counts placements or whorls. Grown across the edge, stems of one, two and three organs a whorl keep a Lucas seed to 86.6, 87.6 and 87.9 whorls a rung: one number to within a per cent in whorls, and one, two and three times as far out in organs. No grid moves it and imposing exact whorls moves it not at all, even where free trijugate whorls come apart completely as the seed is lost.

Shells and growth

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.

What a plant might be doing

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 claims, measured

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.

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.

Packing and tiling

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.

The pattern itself

A whorl that misses its share

A whorl of k organs is defined by its symmetry, and the rule that grows one places its members one after another, each against the members already there. Nothing tells it to put them a k-th of a turn apart. Grown that way, whorls of two, four and eight members sit exactly on their shares of the turn at every lattice measured, and whorls of three, five, six and seven do not — the pattern a mirror argument predicts, since only a power of two leaves every new member a position that mirrors every member already placed. A trijugate whorl misses by 6.5° at a coarse rise and not at all at a fine one, by an amount the rise sets almost everywhere, and none of it moves a single transition.

Shells and growth

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.

The claims, measured

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.

What a plant might be doing

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.

Stems and cones

The ratio was the floor of a curve

One number was left standing between a placement rule and a transported disturbance, measured at one rise, with the explanation that the geometry there happens to favour the larger parastichy number. Swept across two rungs the number is a U — a floor of about 0.79 two thirds of the way up a rung, climbing past 2.8 as a transition approaches — and the geometry is flat exactly where the curve is steepest.

Packing and tiling

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.

Branching and transport

A count set by a delay

An L-system describes a plant and forbids nothing, because none of its parameters is anything a plant has. One branching grammar is the exception: a mature apex makes a new bud every season, and a bud waits d seasons before it branches. Its counts grow at the root of x^(d+1) = x^d + 1, a delay of one season gives Fibonacci's numbers and nothing else does, and the fourth count already separates a one-season wait from every longer one. So a Fibonacci count in a branching plant is a measurement of how long its buds wait. It is also a fragile one: if one bud in ten waits two seasons instead, eleven counts in a row come out Fibonacci's three times in a thousand.

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.

Shells and growth

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 claims, measured

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.

Where the angle comes from

A window inside a rung

A stem that climbs the ladder has no comb in it at any rate, because the quantity the comb is periodic in changes as it goes. Read a window instead and it comes back, on one condition: the window has to be shorter than a rung — which makes the shoot's rate the thing that decides whether a plant can be asked.

Stems and cones

The grid was in the number

The rule places each organ at the least of a profile sampled at a fixed number of azimuths, and every flat run in these essays samples 384 of them — a step of 0.94°, against a disturbance of a quarter of a degree. The quantisation is the larger of the two, it is white, and it moves the discriminator from 0.79 to 0.62. The convergence study this collection had asked for and never done, in the place it turned out to matter.

Packing and tiling

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.

Branching and transport

A count that loses its growing points

The branching grammar behind the Fibonacci claim has no deaths in it, and a stem that loses shoots is the common case. Giving every growing point a chance q of dying each season leaves the counts a linear recurrence and does exactly one thing to it: the growth rate becomes the deathless root multiplied by 1 − q, at every delay and every death chance, to the last bit a double holds. So each waiting time has a death chance above which its lineage shrinks — a half with no wait, 0.3820 at one season, 0.2451 at four — and a longer wait tolerates less. What does not survive is the count itself: a plant losing one growing point in ten a season shows eight Fibonacci counts in a row one time in ten thousand, against one time in eight for a bud that occasionally waits an extra season.

The pattern itself

A band that follows the rise

The index-free counter reads a growing stem only inside a band of widths, and the upper edge is a third of a rung of rise rather than a number of organs — so a width right on one stem is wrong on another. A counter that fits the decay of the rise through the spacings of its own band's whorls, and takes the band spanning a third of a rung, chooses 49, 97 and 193 organs on stems falling over 150, 300 and 600. Given no width at all it reads within eight points of the best of eighteen fixed widths on six of the seven stems, matching it exactly on one and beating it on two. Refusing any band too narrow to have shown the rung above the pair it counted removes every reading of the rung below, on every stem — and costs between five and fifty-one points of correct reading to do it.

Shells and growth

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 claims, measured

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.

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.

Packing and tiling

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.

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 seed measured in whorls

A Lucas seed's length counts whorls, and the number published earlier for the rate edge counts nothing at all. Grown from seeds of fifteen to eighty whorls at one, two and three organs a whorl, stems of every jugacy lose the seed at the same rate in whorls a rung for the same seed length in whorls — 30.80 at fifteen, 63.52 at thirty, identically across the three — and at the same seed length in organs they differ by a factor of 3.24. So the seed is measured in whorls, as the edge is. The other half is worse for the earlier reading: the edge is not a constant but 2.24 times the seed less three, straight to within 2.7 whorls a rung over a fivefold range, so the eighty-seven whorls a rung reported everywhere is a property of the forty-whorl seed nobody varied.

Shells and growth

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 a plant might be doing

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°.

The claims, measured

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.

Stems and cones

Two accounts of one number

A stem that never recovers from an ablation settles into a repeating block whose length was the smaller of its two spiral counts, at both arrangements it had been measured at. Two different explanations predicted exactly that and could not be told apart. Swept across four rises on the ordinary branch the premise itself fails: at one arrangement the block is the larger number, at another both appear, and the rule that seemed to be there was two measurements.

Where the angle comes from

What a sample grid decides

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

Packing and tiling

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 pattern itself

The window was not carrying it

The ratio of a Lucas seed's rate edge to its length rose from 2.05 at fifteen whorls to 2.20 at eighty, and the suspect was the counting window, which is most of a short seed. Read through windows of 20, 26, 39 and 52 folded nodes, seventeen of the eighteen stems lose the seed at exactly the same organ of rate, so the window carries almost none of it. Part of the rise was the grain of rate the edge was found on, worth up to six hundredths of the ratio. What is left rises by a tenth below thirty whorls and has stopped by sixty, at a level an ordinary stem reaches about one and a half per cent lower than a bijugate or trijugate one — and a trijugate edge is not always a line.

The claims, measured

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.

What a plant might be doing

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.

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.

What a plant might be doing

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.

Where the angle comes from

The pattern the cut leaves behind

A stem that never recovers from a removal is not disordered. Its divergences settle into a cycle of eight angles and repeat it exactly for the rest of the run, and a counter reading the positions calls the result 8/16 — a two-jugate lattice. The rule has a second attractor at the same growth parameter, and an ablation is how you get to it.

Stems and cones

What a cut costs a whorl

A bijugate pattern is an ordinary lattice seen twice over, so the account that says a wrecked stem's repeating block is the repeat unit of the lattice underneath has a specific prediction here: three and five. It gets six and ten. And the thing a single missing organ does destroy on a whorled stem is the one property its counts cannot see.

Packing and tiling

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.

The claims, measured

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.

Where the angle comes from

A disturbance the organs share

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

Stems and cones

Seven rises and two seeds

One organ removed from a stem is felt out to the larger of its two spiral counts. Every test of that has confounded the count with the rise, because on one branch the two move together. Grow a second branch beside the first at the same rise and they come apart — and doing it at seven rises turns a matched pair into a design whose last column changes hands four times.

What a plant might be doing

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.

Where the angle comes from

A front with no middle

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

Stems and cones

The hole on the other branch

Near a transition, the run of offsets a stem notices stops being a run: there is quiet past the front and then one isolated offset, felt as hard as anything inside it. Where that offset sits was pinned down on Fibonacci lattices, where the numbers to check it against are 5, 8 and 13. On the Lucas branch they are 4, 7 and 11 — and the rule holds there too.

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.

Stems and cones

A file has to close

The three destinations counted with a shared factor sit near a half turn, a half turn and two thirds. Measuring how near is the trap: by distance from the fraction, the golden angle is closer to two fifths than two of them are to anything, and would be reported as having five files it does not have.

What a plant might be doing

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.

Where the angle comes from

The response with a hole in it

Removing an organ is felt out to the larger parastichy number and no further — that is the intervention's headline, and it holds in the middle of a rung. Swept towards a transition the run of felt offsets stops early and one lone offset past it comes alive, with three quiet organs in between. The lone offset is one place inside the count the stem is about to have.

Stems and cones

A stem coarse enough to cut

Below the 3/5 rung is a 2/3 rung, and it runs from a rise of 0.050 to 0.120. It is not a lattice across all of it: from 0.090 to 0.115 the divergence stops settling and sticks on exactly three eighths of a turn, wobbling by a degree and a half — while a counter goes on reporting 2/3 as though nothing had happened.

What a plant might be doing

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.

Where the angle comes from

The block is the count it was cut from

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

Stems and cones

The shallower front turns over

If reversing a stem means rearranging its whole front, then a stem with a shallow front should reverse more often. Measured across three rungs and four hundred and seventy-three cuts: 6.8 per cent at a front of three, 4.7 at five, and none at all at eight — where the nearest approach is two tenths of a degree away and stays there.

What a plant might be doing

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.

Where the angle comes from

A cut of two organs

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

Stems and cones

Where a handover sits

Inside every rung there is a rise at which the two contact steps change places, so that the shorter hop belongs to the other family below it. Six of the eight rungs on this ladder have one, each has exactly one, and every one of them sits in the coarse half.

What a plant might be doing

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.

Where the angle comes from

The stem that changed hands

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

Stems and cones

Two lines that cross once

The divergence at which a lattice's two contact steps are exactly equal is a curve across each rung, computable from the geometry with nothing grown. The rule's own settled divergence is a second, shallower curve, and where they cross is where the step ordering changes hands.

Stems and cones

A band that holds the angle still

Around every handover the settled divergence has a shallow floor, so a run of rises either side of it share a divergence to a twentieth of a degree while their two contact steps change places. That is a matched pair with one quantity varying, and it is the design the ablation thread had no way to state.

Where the angle comes from

The hop that survived

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

Stems and cones

How long a stem takes to settle

Every result here is grown on a stem that has settled onto a lattice, and settling has always been tested for and never timed. Timed, it takes between nothing and two hundred and ninety organs — against the four hundred every ablation run grows before it cuts anything, and the nine hundred the noise runs carry.

What a plant might be doing

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.

Stems and cones

A wall and not a budget

Below a rise of about 0.005 this collection's stems stop settling onto a lattice, and the limit has been written up four times without anybody asking which kind of limit it is. Grown three times as long, the table is identical row for row: not one stem that failed to settle succeeds. The floor is a wall.

Where the angle comes from

Three organs and no mirror

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

Stems and cones

The angle the ladder returns to

Down the golden branch the settled divergence climbs across one rung and falls across the next, turning three times in four rungs. That is why the same angle is reached at two different rises — and why the Lucas branch, which turns once, almost never offers the same thing.

What a plant might be doing

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.

Where the angle comes from

A wreck has a short list

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

Stems and cones

Two rungs, one angle

Five pairs of rises settle on the same divergence while a counter returns different pairs at them, and four of the five agree to 0.0000° — the same value of a quantity read on a grid of 1,536 azimuths. The rises differ by factors of 1.48 to 5.09.

Where the angle comes from

A survivor has to be a neighbour

A stem that never repairs after a removal keeps exactly one lattice hop rigid, and nothing predicted which one. Sweep every offset at twelve lattices and the answer narrows sharply: at twenty-nine of thirty the surviving hop is one of the two families a counter returns, and the one exception is a step six times too long to be one.

What a plant might be doing

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.

Where the angle comes from

Not the shorter of the two

If a damaged stem keeps one contact family standing, the obvious guess is that it keeps the nearer one. Across thirty wrecked offsets that is true twelve times and false seventeen, and on one lattice the two steps differ by a quarter of a per cent — where the words shorter and longer are doing no work at all.

Stems and cones

Where the survivors meet

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

Where the angle comes from

One offset, two answers

Which contact family a wrecked stem keeps is decided by where the cut landed, at twenty-five of thirty offsets, by the simplest rule anybody would write down. It is refuted by two runs: the same counted pair, the same offset, two different rises, and two different surviving families.

What a plant might be doing

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.

Stems and cones

Four crossings nobody visited

Six rungs of the ladder carry a handover and two of them had a band built on them. The other four are here: 70, 126, 16 and 124 rises wide, found by sweeping at a ratio rather than at a fixed step in the rise, which is why the fine ones had been stepped over.

Where the angle comes from

One rung, two answers

The offset accounts for twenty-five wrecked stems of thirty and is refuted by a single pair of runs that differ in nothing but the rise. Sweep one rung at a thousandth and the refutation stops being an anomaly: the same offset on the same lattice keeps one family at the coarse end and the other at the fine one.

Where the angle comes from

The family that lost a member

The offset rule restated in the arrangement predicts that the chain whose organ was taken is the chain that breaks. Scored on the nine offsets where the question can be asked, it is right none of the time and its opposite is right all nine.

What a plant might be doing

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.

Where the angle comes from

The front deepens down a rung

The offsets that never repair grow from one to five across a single rung, while a counter returns the same pair at every rise. The extra offsets are not a random extension of the ones already there: they are the ones past the smaller counted number, and they are the ones that keep the larger family.

What a plant might be doing

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.

Where the angle comes from

The shortest hop was a coin flip

The reading that a wrecked stem keeps its shortest hop was refuted at twelve of twenty-nine across the census. Re-scored along a single rung, where the counted pair is held and the step ordering reverses, it is right at sixteen of thirty-one — which is not a refutation but an absence of information.

What a plant might be doing

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.

Where the angle comes from

The organ that was nobody's neighbour

Twenty-one of the thirty wrecked offsets remove an organ that lies on neither contact chain through the tip. The reading that explains the other nine has nothing to say about them, and the honest thing is to say so rather than to widen the definition until it does.

Stems and cones

Every rise of a band

A band is cut at nine rises because the quantity it was built to test is a constant, and a constant is checked at the ends and at the crossing. On the widest band that quantity turned out not to be constant, which makes nine the wrong number. This is all hundred and twenty-six.

What a plant might be doing

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.

Where the angle comes from

The corner moves with the rise

The corner was either the contact scale or simply any memory at all, and nothing in the thread had ever varied the rise — the one knob that moves the contact numbers while leaving the rule, the amplitude and the run length alone. Swept over it, the comparison does not keep its shape.

Stems and cones

The alternation is not a period

Nine sampled rises gave 8, 4, 8, 4 at one offset of one band, and a period was the obvious thing to look for. At full resolution it is thirteen islands one to three rises wide, with gaps of 1, 2, 3, 6, 7, 8, 9, 16, 31, 44 and 48 — and a fitted period buys exactly nothing.

What a plant might be doing

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.

Where the angle comes from

A stem too fine to settle

Below a rise of about four thousandths the counter stops returning contact families and starts returning pairs like 2/13 and 13/24. Lengthening the stem does not fix it. That is a ceiling on every sweep this collection runs up the ladder, and it has never been written down.

What a plant might be doing

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.

Where the angle comes from

The band was not the sampling

Five rises in the middle of the coarse rung stick on three eighths of a turn, and the ladder that found them is swept at five thousandths — coarse enough that a band of the same kind could sit inside any finer rung unsampled. Swept at a tenth of that across a whole finer rung, nothing locks.

What a plant might be doing

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.

Stems and cones

The offsets that never change

Three of the six offsets that wreck anywhere on the band keep the same family at every rise they wreck at — 98, 22 and 81 rises of the 126. And which offsets wreck at all is a function of the rise, which no reading of a band had drawn.

Where the angle comes from

One rise per rung is a sample

Every census on this site takes one rise from each rung, because the question was always which pair. Any rule later scored on those rows inherits a variable that was never varied — and two of this collection's results turn out to be about the sampling as much as about the rule.

Stems and cones

A band with nothing inside it

Five offsets wreck on the Lucas 7/11 band and every one of them keeps the same family at every rise it wrecks at. There are no islands, no transition region and no period to look for, which is what makes the picture from the other band a picture of that band.

What a plant might be doing

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.

Where the angle comes from

What a count cannot decide

A spiral count is the measurement this whole subject is built on, and it is deliberately blind to everything that varies inside a rung. Four results this collection now holds are results about that blindness rather than about the arrangements.

Stems and cones

A fifth cluster

A correction to the exchange's size was fitted over four hop clusters and its own file said so. A search turned up a fifth, at a hop smaller than any of the four, and the rule is right on it — which is what a prediction being confirmed looks like when the confirmation is worth having.

What a plant might be doing

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.

Where the angle comes from

The panel with no corner

Sweeping the rise gave three shapes where one was expected, and the middle one is the informative panel: at the 5/8 contact scale the deeper rule wins at every correlation and there is no crossing to locate. That is either a fact about the lattice or a fact about the pair of exponents, and one measurement separates them.

Stems and cones

A transition and not a slope

The question was whether a fourth cell's cost declines smoothly to nothing or falls in one step. It falls in one step, and the answer decides whether a word in the collection names something or is a threshold on a continuum.

What a plant might be doing

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.

Where the angle comes from

The slide a counter holds constant

Inside one rung the settled divergence moves by more than a degree, monotonically, with no flat stretch anywhere — measured at a thousandth on one rung and at half a ten-thousandth on another. A rung is a plateau in one reported number laid over a geometry that never stops moving.

Stems and cones

One rise below the census

Ten lattices were cut at every offset and their surviving lags came back as four numbers. One rise further down a rung the census already sweeps, three cuts keep a lag of eleven — which is a fifth number, on a lattice nothing about was unusual except that nobody had cut it.

What a plant might be doing

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.

Where the angle comes from

The column that cost no stems

Every table in this collection records the rise a stem was grown at. None records where inside its own rung that rise sat, and the fraction turns out to be computable from numbers already written down — which makes it the cheapest column anybody here has ever added and the one that changes the most about how the tables read.

Stems and cones

The end of a wrecked run

The obvious follow-up to a transition in where one organ goes is whether the stem also finishes somewhere different. On this rung the question has no answer: a wrecked run's final divergence takes four values and changes between rises a thousandth apart, three times inside a nine-rise sweep.

What a plant might be doing

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.

Where the angle comes from

The side the census sat on

Eight of the ten lattices the ablation census wrecks at were grown past their rung's handover, one before it, and one so close that the ordering it quotes differs by parts in a thousand. A reading scored over the step ordering was therefore scored against a quantity the census was nearly holding fixed.

Stems and cones

The lag that never survives

A correction to the exchange's size rests on four hop clusters, and a fifth would be the first real test of it. The prediction was written for a golden lattice at a lag of eleven. No golden lattice on this ladder reaches one, and the reason is a fact about the rule rather than about the search.

What a plant might be doing

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.

Where the angle comes from

The ordering was not the actor

Cut an organ out of every rise of a band where the counted pair and the settled divergence are held and the two contact steps change places, and the family left standing does not change. Fifty-five wrecked cuts on two branches, and the ordering reverses underneath every one of them.

Stems and cones

The second band, cut whole

One band was cut at every one of its rises and came back with a transition region — a stretch where three offsets change their answer, in short islands with uneven gaps. The obvious question is whether that is a picture of bands or a picture of that band. The other wide band answers it.

What a plant might be doing

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.

Where the angle comes from

The organ that moved furthest

A reading that works on nine of thirty rows needs a reference organ, and the obvious repair is to measure one rather than to choose it. Measured, the disturbance turns out to have no far edge at all, so there is no organ that moved furthest in any sense the reading can use — and the generalisation that does work needs no reference organ.

Stems and cones

The wrecking set moves again

Which offsets wreck a stem was assumed to be a property of the lattice. On one band it turned out to be a property of the lattice and the rise, changing on nearly a fifth of that band's steps. On the second band it changes more, and one offset's wrecking is broken into five separate stretches.

What a plant might be doing

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.

Where the angle comes from

Six of six is not a measurement

A panel comparing two rules seed by seed reported the deeper one winning every one of six seeds at four correlation lengths out of six, and was read as a lattice with no corner in it. Six of six is the largest number the panel can print, so the flat middle was a reading of the ceiling — and raising the disturbance brings a corner out of it.

Stems and cones

When nine rises are enough

A coarse design was shown to be misleading on one band and it has been criticised on that ground ever since. On the second band it is exactly right, and the difference between the two cases is a property of the band rather than of the design — which is the awkward part.

What a plant might be doing

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.

Stems and cones

Where a slot loses a wall

Two rungs were reported to go free at 84 and 85 per cent of themselves — the second removal stops costing anything over the larger one alone. Three samples a rung cannot say whether that is a transition or a slope, and twenty-nine more say it is a transition one grid step wide.

What a plant might be doing

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.

Where the angle comes from

The clock a share cannot see

Settling takes nothing to 290 organs, and four rounds of this collection have carried that as a property of the rule. At a steeper falloff the slowest is 808 — and not one of the 72 pairs of runs settles at 3,200 organs after failing at 1,200, so the fine end is still a wall.

Stems and cones

The third band, cut whole

Two bands cut at every rise disagreed about whether the family a cut keeps ever changes, and three explanations were available for a difference between two things. The cheapest third band settles which of them survives, and it settles it against the account nobody was betting on.

What a plant might be doing

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.

Where the angle comes from

Destinations only a steep rule reaches

A settled stem's divergence is the sharpest instrument in the sweep and the cheapest: it is already in every run, it does not average, and it is read on one grid at every exponent. Exponents 4 and 5 reach 42.3°, 47.9° and 148.1°, and the two shallower ones reach none of them.

Stems and cones

A family that is a multiple

When a wrecked cut keeps a family that is not one of the lattice's counted pair, the first case looked like a rule: it was half of one of them. The second case is four times the other, which makes the rule a coincidence and leaves a weaker statement that is probably the true one.

What a plant might be doing

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.

Where the angle comes from

A counter on the settling table

The settling table has reported an angle for every run that reaches a lattice, and nobody has ever counted one. A hundred and seventeen settled runs, regrown and counted: every one of them has a parastichy pair, and sixty-five of them land on sequences the ladder does not carry.

Stems and cones

The branch is what is left

Four accounts of why one band's cuts change what they keep and another's do not were written down before a third band was cut. Three of them are now wrong on a band each, and the survivor is the one with no mechanism behind it.

What a plant might be doing

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.

Where the angle comes from

What a steep rule counts as

Exponents four and five settle stems on 42.3°, 47.9° and 148.1°, and the two shallower ones reach none of them. All three count: 8/9, 8/15 and 2/5. None is a rung of either ladder, and one of them is the coarsest rung of a sequence the table already had.

Stems and cones

Nine rises were not enough

A coarse sample of a band had never been wrong about whether anything changes inside it, and that record was the argument for trusting a negative from it. The third band cut whole makes it two of three, and the missed feature is one rise wide.

What a plant might be doing

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.

Where the angle comes from

Ten sequences, two of them the ladder's

Every parastichy pair the settling table produces is two consecutive terms of a sequence in which each term is the sum of the two before it. Ten such sequences account for all fifteen destinations — nine, once one of the ten turns out to be a reading rather than a ladder — and the two the collection is built on are neither the largest nor the smallest.

Stems and cones

A change with nowhere to be

The claim this whole thread rests on is that a survivor does not change where the two contact steps change places. Nineteen located changes never put one there. The twentieth is flagged at a handover, and it is flagged because the offset stops wrecking for thirty-four rises.

Where the angle comes from

A list that was a rounding

Three destinations only a steep falloff reaches, read at two grid steps. Thirteen, read at a tenth of a degree. And four arrangements, read by what the counter returns rather than by the angle — a different four, with one the angle reading hides.

Stems and cones

One step of the grid, again

A search of the fine end stepped from one rise to another a hundred grid steps away, and the family a cut keeps changed somewhere between. Cutting the rises in between puts the change inside one step, with the lattice identical on both sides.

Stems and cones

The hops cross once

Walking a whole rung at the grid its rises are named on costs a few hundred stems and no cuts at all, and it answers a question nobody had asked: whether a rung has one handover or several. It has one, and the ladder had recorded it in the wrong place.

Where the angle comes from

A basin has a width

A destination reached from one starting angle is a presence. A destination reached from seven consecutive starting angles spanning forty-five degrees is a basin with an extent, and nine angles could not have measured one — they were too far apart to have two of them land in the same place.

Where the angle comes from

A wall that stopped moving

Four falloff exponents were reported not to move the rise below which stems stop reaching a lattice. Their measured walls spanned a factor of two and ordered themselves 2, 5, 3, 4. At twenty starting angles they span a fifth of one and order themselves 5, 4, 2, 3 — so the conclusion was right and its arithmetic was noise.

Stems and cones

Two rises far apart

The whole handover thread rests on the claim that where the two contact steps change places is not where the survivor does. On one rung both rises are now located to a single step of the grid, and they sit forty-five per cent of the rung apart.

Stems and cones

An offset that arrives

The rise where a lattice first keeps a new family is located to one step of the grid, and what happens there is not what the question assumed. No cut changes its mind: a cut that was not wrecking starts, and what it keeps is the new family.

Where the angle comes from

Round numbers are not a sample

The nine starting angles the settling table was grown from reach a lattice four times in ten. Eight angles placed exactly halfway between them reach one a quarter of the time. The difference is not noise and it is not the range — several of the nine sit next door to somewhere a stem could settle.

Where the angle comes from

Twenty angles instead of nine

Every claim in this collection about where a stem ends up rests on nine starting angles a cell, and the file that uses them says so — it computes a binomial error of 0.17 and declines to read a spread against it. Eleven more angles halve that error and change what several of the numbers were.

Stems and cones

The column nobody read

Every cell of the slot design carries where its run finished as well as how far its first organ moved. One rung's worth had been plotted and called unusable. Reading all eight says the endpoint is exact on two rungs, wanders on six, and is worst on the one it was read on.

Stems and cones

A wrecked run goes somewhere

Where a wrecked stem finishes was called unstable. Half of them finish within a degree of a destination measured from intact stems started at arbitrary angles — two tables that share no run, no design and no question.

Where the angle comes from

Forty angles, and a limit

Nine starting angles turned out to be a biased sample of the circle, and doubling to twenty said by how much. Doubling again says the estimate is converging — to a smaller correction than one doubling extrapolated to.

Where the angle comes from

A wall that was never measured

Three samplings of the starting angle give three orderings of the four falloff exponents' walls and a spread that does not shrink, while every error bar behind them halves. The reason is that a wall is a crossing of a nearly flat curve, and nobody had asked how well it is located.

Stems and cones

Two files, and a way back

Twelve wrecked runs finish at exactly a half turn, which is a pattern with no spiral in it, and every one is at the coarse end of the ladder. Three finish at the divergence they would have had anyway, after being thrown a hundred degrees off it.

Where the angle comes from

A list that can only shrink

The destinations only a steep falloff reaches grew from three to five when the sampling doubled, and everyone read it as a list filling in. Doubling again takes two off it, which is the only direction a list defined by an absence can ever move.

Stems and cones

The fourth band, cut whole

Three bands cut at every rise left one account of which bands change their answer standing, and the account was the one nobody had a reason to prefer. The band that would have killed it has now been cut, and it did not kill it.

Where the angle comes from

A basin that doubled

A run of consecutive starting angles reaching one destination is a basin, and its width is a lower bound. Halving the spacing doubled the angles in the widest one and left its width alone, which is what a real basin does and a sampling artefact does not.

Stems and cones

Two bands that wreck nothing

The census that reads a band refuses two of the six, and the refusal is correct: a band with no wrecked cut has no surviving family, so it has no answer to change. Cutting them anyway turns a refusal into a measurement, and the measurement has a third tone in it that the census cannot see.

Stems and cones

Six bands, one table

Every rung of this ladder that carries a handover now has a band grown on it and cut at every rise it holds, and four accounts of which bands change their answer are scored on all six at once. The survivor is right on every band that can test it, and the same table read one cell differently kills it.

Stems and cones

The coarse design scored

Every claim this thread has made about an uncut band rests on a sample of nine rises, and its record was the argument for trusting it. Six whole bands close that record, and two of its six correct verdicts are correct only because there was nothing on those bands to find.

Stems and cones

A wrecking set with a range

Which offsets wreck a stem was taken to be a property of the lattice, and every band cut whole has found it to be a property of the rise instead. Six bands turn that replication into a measured range, and the range is a factor of fifty-one.

Stems and cones

Five rungs walked

Six rungs of the ladder carry a handover and only one of them had ever been walked at the resolution its rises are named on. Walking the other five costs 1,224 grown stems and no cuts at all, and it returns a crossing count per rung — five ones and a five.

Stems and cones

One crossing or two

One rung of the ladder changes hands five times where the other five change hands once, and the difference is not in the geometry. It is a divergence read in steps against an ordering that slides, and the account is a threshold at one that is right on all six rungs.

Stems and cones

The handovers corrected

Six recorded handovers, relocated to the grid against where a one-per-cent sweep put them: all six sit on the fine side of a crossing and all six inside a single sweep step. Nothing about the rung explains the size of the discrepancy, which is what a sampling artefact is supposed to look like.

Stems and cones

A count or a floor

Nineteen changes of surviving family on the widest band have been quoted as a number since the band was cut, with nothing to say whether a finer grid would find more of them. Halving the step finds twenty-one there and nothing at all on the next band along.

Stems and cones

New islands or old edges

Halving a band sweep's step found two more changes of surviving family, and there are two quite different things they could have been. Every coarse change and every coarse island turns out to be carried by exactly one fine one, so the extra pair is a rise the coarse grid stepped over rather than a boundary it misplaced.

Stems and cones

The last unmoved setting

Halving a band sweep's step is only a halving if the sweep steps where it says it does, and this one does not: its rises are rounded to five decimal places, so at one handover the grain is 1.65 parts per thousand against a nominal step of two. Moving the last setting nobody had moved found the setting was never what it was called.