The thread: Two routes, one number

What a plant might be doing

Turing's last problem

Turing's final work was on phyllotaxis and it was unpublished when he died. Its core is a ring of cells and two diffusing substances, and the thing it does is select a number of peaks — which can be predicted from the equations before anything is integrated, and then counted from what the integration produces.

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.

Stems and cones

Two numbers out of the points

A seed head's divergence angle can be recovered from its spiral counts only to within an interval, because a range of angles gives the same counts. On a stem the counts come with lengths attached, two measurements pin two unknowns, and the lattice comes back to the last digit it was built with.

Stems and cones

The Fibonacci ladder

Lower the rise on a cylinder and the parastichy pair climbs — 1 and 2, then 2 and 3, then 3 and 5 — each rung the sum of the two before it. Nothing in the arithmetic mentions Fibonacci, the transitions sit at computable rises, and consecutive ones stand in the ratio 1/φ².

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.

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.

Packing and tiling

Two laws that want opposite tissue

Lewis's law and Aboav's relation are quoted side by side as properties of cellular tissue. Measured on the same two tilings they point opposite ways — the ordered head satisfies Aboav's with the textbook value of 1.18 and fails Lewis's completely; the random set does exactly the reverse.

Where the angle comes from

The tree and the attractor

The dynamical model settles on the golden angle over a range of one parameter and on the Lucas angle outside it, and it cannot reach the low-growth end at all. The lattice tree reaches everywhere, has no dynamics in it, and produces the same two angles as limits of two paths. Two routes, one pair of numbers.

Stems and cones

The forks are exact

Where a stem's pattern has to choose between two futures, three spiral families are equally short and the lattice is exactly equilateral. A numerical solver found those points; the numbers it returned turned out to be rational, and chasing that gave a closed form — including the fact that every fork sits at a rational divergence, and the golden angle at none of them.

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.

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.

What a plant might be doing

A comb is evidence of a rule

Build the same lattice kinematically — every node at an exact multiple of the divergence, an independent error on each azimuth, no feedback anywhere — and the spectrum is empty. The photograph is identical and the parastichy pair is identical. The comb is not a property of the arrangement.

Where the angle comes from

The lag that is not there

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

Stems and cones

Transitions a factor of φ² apart

The ladder's rungs are a factor of 1/φ² apart in rise. A disc's rise falls as one over radius squared and a cone's as one over distance, so the same rungs land a factor of φ apart on a seed head and a factor of φ² apart on a cone — measured, on both, by a counter that has never heard of either.

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.

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.

Where the angle comes from

Noise is not a slow rate

A stem seeded on the Lucas branch keeps it below ninety nodes per rung and abandons it above — which invites the objection that a real apex's fluctuations would knock it off regardless. Measured across a hundred and sixty runs of two independent kinds of noise, one escapes, at the amplitude where the pattern is already coming apart.

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.

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.

Shells and growth

One angle decides contact

Seen from the apex of its coiling axis, every whorl of Raup's shell subtends the same half-angle, and two whorls touch exactly when the sine of that angle exceeds (W − 1)/(W + 1) — with no disagreement against the drawn discs at 400,000 random shells. Two of the three numbers enter only through the angle and the third only through the threshold, which decides which picture of a shell can answer the question: a spire's outline carries no W, a plan carries no T, and an axial section carries all three.

The 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.

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.

What a plant might be doing

The forgery needs a history

A disturbance passed between touching organs manufactures the comb, the second comb and the parastichy pair on an arrangement with no rule in it — which is why the comb stopped being evidence. Give the organs the same correlation with no accumulation in it and the forgery collapses: one seed in eight returns a pair, and the comb is the noise floor.

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.

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.

What a plant might be doing

A difference forgets a drift

This collection proposed a second observable and priced it as free: if a plant's errors are inherited between touching organs, the divergence sequence should carry a slow wander as well as a comb. The disturbance with the largest wander of any built here leaves none at all in the sequence, because a divergence is a difference and differencing is what removes a drift.

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.

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.

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.

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.

Branching and transport

A cube law with a lever arm

Murray's exponent of three comes from moving fluid for the least work, and Da Vinci's two has had no derivation here, only the name of the mechanical answer. Size every branch so that the same wind on every tip bends it to the same stress, and a junction conserves r to the power 3/(1 + log₂(1/λ)), where λ is how much shorter each branch is than its parent. A crown that fills a plane gives exactly two; halving lengths gives one and a half; no shortening gives three. Murray's flow rule gives three at every λ, so the lengths of a tree's branches say which mechanism sized it.

What a plant might be doing

The drift goes the other way

A rule that corrects what its neighbourhood shares should let through any disturbance slower than its own reach, and should suppress anything faster — a crossover, tracking the depth. Swept over a neighbourhood that changes by a factor of sixty, there is no crossover anywhere, and the deep rule passes nearly four times as much as the shallow one. The prediction is not weakly supported; it is backwards.

Where the angle comes from

Two shapes, one threshold

Read in the same unit, an exponential falloff and a gaussian one disagree about where the lattice ends by half. The quantity they agree on turns out to be one the earlier work measured for an unrelated reason — and it agrees with a bracket left by a sweep of a completely different parameter.

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.

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.

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.

Branching and transport

Forks on a tree sized by stress

Da Vinci's rule predicts that a tree's forks open wider as they get smaller, because the constant in its angle cost is one number for a tree and enters each fork scaled by its size. A tree sized for equal bending stress conserves an exponent set by how much shorter each branch is than its parent, and on such a tree the same constant enters each fork as its radius to the power 2p − 6. The trend survives at every length ratio short of one and shrinks with it: 105 degrees a decade of radius for a crown filling a plane, 83 at a length ratio of 0.74, 53 for a crown filling a volume, 13 at 0.9. Only a crown shortening about as fast as a volume-filling one fans wider across a tenfold range than one fork's flatness can hide.

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.

Where the angle comes from

The boundary belongs to the pattern

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

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.

Branching and transport

A crown that carries its own wood

Sizing every branch so that equal loads on the tips bend it to one stress gives a crown filling a plane Da Vinci's exponent of two. Move the load onto the wood and the sizing becomes a fixed point, because a branch's load now depends on the radii being solved for. Under the wind on its wood a planar crown still conserves two, but only as a limit its trunk is two tenths short of at fifteen generations. Under its own weight it conserves one — radius rather than area, the stress-similarity law that radius goes as length squared — and a crown carrying leaves and wood reads the leaves' two near its twigs and the wood's one at its trunk, with the handover set by how much of the trunk's load the wood carries.

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.

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.

What a plant might be doing

The corner that does not move

Read as degrees of drift getting through rather than as a ratio, and compared seed by seed, the deep and shallow rules change hands. The share that goes to the deeper rule climbs from twenty-three per cent under white noise to ninety-four at a correlation length of a hundred organs — and the crossing sits at two or three organs whether the two rules differ by a factor of four or sixty-one.

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.

Stems and cones

What the sharing costs a lattice

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

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.

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.

Branching and transport

A crown sized for how far it bends

Stress is one criterion for sizing a branch and stiffness is another. Holding every branch to the same deflection as a share of its own length sizes r to the fourth against the sum of each load's arm squared, where equal stress sized r cubed against the arm, and the junctions of a deep crown then conserve 4/(1 + 2·log2(1/λ)). A single cantilever under its own weight comes out at radius as length to the three halves — McMahon's elastic similarity, fitted here rather than assumed — against the square that equal stress asks for. And the two criteria agree at exactly one length ratio out of the whole family: λ = 2 to the minus a half, the crown that fills a plane, where both give exactly two.

Shells and growth

What the axis distance costs

Raup's contact boundary is a relation between two numbers and only one of them has ever been priced here. The second was expected to be the cheaper — a length against another length. It is not: an assumed centre costs it 79.3 per cent where the same centre costs the expansion 6.88, because a ratio of two distances is first order in the centre and a fitted rate is second. But a tilted camera costs it nothing at all, exactly, and averaging the reading round one whorl is free and worth a factor of 4.91. The two numbers fail at opposite ends, and they cross at 1.12 turns of arc.

Stems and cones

A stem on the other branch

Every stem an organ had been cut from carried Fibonacci counts, which is why two rival explanations of the block a wrecked stem settles into had never disagreed. A stem seeded on the Lucas lattice carries four and seven at the same rise, under the same rule. Cut, it settles on seven — the larger number, and not a Fibonacci one.

Packing and tiling

The empty interval is the rings

No single cut-off on hop ratio turns the contacts a count keeps into the walls a tessellation draws, on any whole head at any size. Read cell by cell against the flip rings the divergence angle puts in closed form, every disputed cell lies within two thirds of a wall spacing of a ring, and with one spacing either side set aside a single cut-off between 1.430 and 1.444 serves every golden head from 900 organs to 9,000.

Branching and transport

Three rules, one exponent

A measured branching exponent is quoted as evidence for a sizing rule, and it cannot be. Murray's flow rule conserves three at every length ratio and reads nothing of the lengths at all; equal stress conserves 3/(1 + l) and equal deflection 4/(1 + 2l), where l is log2(1/lambda). So an exponent names a rule only with a length ratio beside it, and even then not everywhere: of ninety-six length ratios between 0.3 and 0.99, thirteen have two rules within five hundredths of each other at a precision of 0.05, in three bands with three different reasons — stress against stiffness where they cross at the planar crown, stiffness against flow where the stiffness curve passes three at 0.8909, and stress against flow only as the branches stop shortening.

Shells and growth

A floor no better fit can lift

The band of shells nobody can place was built from one of the two numbers the contact boundary relates, and the other has now been priced. Carried together they widen the band by a third and take 48.2 per cent of the morphospace box to 59.7. The number that matters is further down: with the whorl expansion measured perfectly, 12.6 per cent of the box is still undecidable, and at an expansion controlled to a hundredth 94 per cent of what remains belongs to the second number. And the two errors are not independent — they come out of one guessed centre, which traces a curve across the boundary rather than a rectangle around it.

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.

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.

Branching and transport

The lengths that name the rule

A real crown has no single length ratio, and giving every fork a spread of daughter lengths does not blur what a sizing rule conserves: Murray's flow rule still conserves three at every junction, and the two mechanical rules keep their mean exponent, moved only as the square of the spread. What the spread adds is a second number. Each junction's exponent follows its daughters' summed log length with a slope of 0.30 under equal stress and 0.86 under equal bending at the planar crown, where the exponents are both two — and a two per cent error in every radius moves neither slope, while it swamps the scatter that looked like the obvious instrument.

Branching and transport

A crown that would rather not buckle

A column held below the load at which it buckles and a cantilever held to a fixed deflection need the same radius at every length, because both hold the bending stiffness against a load times a length squared — so the three halves of elastic similarity is also the buckling law, and a crown whose loads all run along its branches conserves the same exponent under either. Gravity does not run along branches. It divides by the cosine of each branch's tilt, so a buckling junction's exponent is set by the direction its parent points, nothing past level is sized at all, and a crown sized by the larger of the two criteria splits by direction into an upright core and a spreading shell whose boundary junctions conserve more than either rule gives.

Branching and transport

A trend that stops at Murray's angle

A crown sized by whichever of flow and bending stress asks for the thicker branch is sized by stress at its trunk end and by flow at its twigs, and the twigs fix the constant that was free in the fork-angle prediction: a twig at the radius the transport cost prefers spends exactly half its upkeep on pumping. On such a crown the fork angle does not change sign at the handover. It rises through every stress-sized generation, meets Murray's 74.93° at the handover and stays there, and no fork anywhere opens wider. The trend turns back only when the twigs are thinner than the cost wants — past a pumping share of (λ^(−2/3) − 1)/(1 − λ^(4/3)), 0.70 at the planar crown and closing on one half as branches stop shortening.

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.

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.

What a plant might be doing

Four accounts of one angle

The exchanged pair in a wrecked stem is about one divergence step, and about is doing twelve per cent of work. Four candidate units were written down and scored on the same seventeen rows: the cut stem's own step, the surviving family's step, the control's step, and the control's corrected.

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.

What a plant might be doing

A fifth of the hop

The exchanged pair misses one divergence step by up to twelve per cent, and the miss is not scatter: every row keeping a lag of 5 or 7 overshoots and every row keeping a 4 or an 8 falls short. Subtract a fifth of the surviving hop's own angle and the worst row is four per cent.

Where the angle comes from

One turn per survivor

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

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.

What a plant might be doing

The exception was already labelled

The larger counted number sorts twenty-two of twenty-four lattices by the sign of their slot interaction. Both misses are on the Lucas 3/4 rung — the one rung a different measurement had already singled out, for reasons with nothing to do with this one.

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

A band that moves nothing

One of the six bands holds the counted pair, holds the divergence, and does not move the ordering: its two contact steps stay within four parts in a thousand of each other across the whole of it, so the ordering changes hands three times and neither end has one worth the name.

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.

Stems and cones

How wide a band should be

A band ends where the divergence has slid a twentieth of a degree, so its width should follow from how fast the divergence slides. Predicted from the rung's slope that is wrong by factors of 0.20 to 5.92; predicted from a stationary point it is 0.41 to 1.08, and the outlier is the rung that has no stationary point.

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.

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

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.

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

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.

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 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

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

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

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.