Parastichy pair — where it appears
Named by 132 essays across 6 fields — each of them below, with the objects they name alongside it.
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.
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.
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.
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.
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.
A comb is evidence of a rule
Build the same lattice kinematically — every node at an exact multiple of the divergence, an independent error on each azimuth, no feedback anywhere — and the spectrum is empty. The photograph is identical and the parastichy pair is identical. The comb is not a property of the arrangement.
A disturbance with a memory
That earlier work's control assumed that a plant's errors are independent from organ to organ, and nobody had tested it. Give the errors a memory — each one a fraction of the last, up to a coefficient of 0.97 — and the comb does not appear. The obvious threat to the result turns out to be empty, and the algebra says why before the measurement does.
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.
What a count is worth
A reported parastichy pair pins the divergence angle to a band 221°/mn wide — so 2 and 3 says almost nothing and 34 and 55 fixes it to a tenth of a degree. Each step up the Fibonacci sequence is worth a factor of φ², and recording the radius a pair was counted at adds only ten per cent.
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 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.
What a forgery has to know
A lattice with transported errors reproduces the comb and the pair, so one quantity is left: the two combs' relative strength. Weighted by distance the forgery puts more in the second comb than the first; the rule does the opposite. It matches only if the coupling is turned three to one towards the further neighbour, which no falloff supplies.
An organ has no single exponent
The earlier work measured that a surface whose circumference grows as a power of arc length puts its transitions a fixed factor apart, and checked it on five surfaces. Every one of them had a single exponent, and no organ does — a fir cone is an ogive, whose exponent runs from 2 at the tip to nearly 0 at the shoulder.
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.
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.
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.
The ratio was never about the rule
The comb has already been retracted here as evidence that a plant computes its pattern, and one quantity was exempted from the retraction: the ratio of the two combs, which a placement rule and a transported disturbance divide differently. Drive seven disturbances through the same rule and the ratio spans 0.45 to 1.09. The exemption does not hold, and the angle sequence has nothing left.
Two 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.
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.
A disturbance that is not passed on
The disturbance that forges every observable measured here does two things at once — it correlates an organ's error with its contact neighbours', and it hands that error on to be handed on again. Every result about it has been unable to say which half did the work. This is the control that takes the second half away and keeps the first.
The 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 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.
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.
What the pair costs
The single parastichy number cost sixty internodes. The pair costs two hundred and fifty, and a protractor error of three quarters of a degree takes it to eleven hundred. The arithmetic that predicts the second of those is right about the shape and wrong about the scale by a consistent factor, which is recorded rather than fitted away.
The 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.
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.
A refusal with a reason
Three note left with the work running have recorded that a refusal has four causes and the sequence separates none of them. With a second window and a protractor, three are separated: silence at 0.38° of scatter is a quiet plant, silence at 56° is a disorderly one, and agreement certifies the rate. The fourth survives, and so does a worse discovery — agreement is not correctness.
The 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.
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.
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 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.
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.
An experiment a needle could run
For eight instalments the outstanding item has been a survey — photographs, a protractor, hundreds of specimens — and it has not been done. The intervention is a different kind of ask, and a cheaper one: a needle, one apex, and a yes-or-no per ablation. Here is what it would cost, what it would settle, and the four ways it could come out.
The 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.
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.
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 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.
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 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.
The nearest organ is not the nearest neighbour
Rank the terms of the sum the rule minimises and read off which organs the biggest ones belong to. At every falloff exponent from 1.5 to 6 the answer is the same five: lags 13, 8, 5, 21 and 26. The organ placed immediately before is not among them, and counting the neighbourhood in organs was the wrong unit.
The 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.
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.
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.
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.
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.
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.
Removing a neighbour costs least
Take away an organ that is a direct chain-neighbour of the growing tip and the next organ moves by under thirty-one degrees. Take away anything else inside the front and it moves by at least sixty-three. Thirty cuts, two groups, a factor of two between them and nothing in the gap.
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.
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.
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.
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.
A rule that cannot heal a hole
The placement rule corrects itself against a displacement — that is what the lag-one correlation of −0.6 has been saying since it was measured. It does not correct itself against a deletion. Which organ is removed decides whether the stem is back on its lattice in twenty-four organs or never, and the boundary between the two is sharp, reproducible and in the middle of the front.
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.
A head displaced before it is counted
The round trip from a head's spiral counts back to its divergence angle was tested on heads whose every organ sat exactly where the rule put it. Displaced by a normal error of up to two and a half spacings, heads of 900 organs keep counting a pair from their own sequence and return intervals holding the true angle to a spacing and a half; heads of 300 organs move to the neighbouring pair by half a spacing and then refuse, nine in ten of them by two spacings. Every moved count brings in the family whose chord was third shortest. Of 898 heads recovered, 19 intervals miss the true angle and 17 of those by about a tenth of a degree — displacement makes the reading coarser and then silent, not confidently wrong.
The pattern 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.
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.
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.
A twist is a divergence
Recovering a head's divergence angle from its spiral counts survived independent displacements of whole spacings, moving to a neighbouring pair and then refusing rather than misleading. Displacements with a direction are harder on it in only one case. A head pressed to an aspect ratio of 2.25, spread at the rim by eighty per cent or sheared with a slope of 1.6 is still counted as its own pair or its neighbour, and recovered inside its interval. A head twisted — each organ turned about the centre in proportion to its radius — is not: past a turn of the rim of about a radian and a half the counts leave their sequence and the recovered angle misses, by up to sixty-one degrees, because a twist changes the angle between one organ and the next. The round trip is not fooled. It is reporting the angle the twisted head has.
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.
The count sees the twist first
A twisted head recovers a changed divergence, and the check proposed for it was two annuli: the twist's extra angle falls with radius, so an inner and an outer annulus should disagree. Read on golden heads of 900, 2,400 and 9,000 organs, they never do in time. Their intervals separate at eight radians on 900 organs and never on the larger heads, always after the ordinary reading has been misled — from six radians on 900 organs and from two on 2,400 and 9,000. What catches the twist first, at every size and on every seed, is the count: at half a radian to three quarters some band stops returning two consecutive Fibonacci numbers — 34 and 89, 89 and 233 — which no untwisted golden head, clean or displaced, ever does.
The disturbance that travels
If a lattice survives three times the displacement when the organs share it, then a disturbance passed between the organs that actually touch should be the gentlest of all — it is correlated at exactly the offsets the rule places against. It is the harshest. Half the displacement destroys what independent noise leaves standing, and the reason separates two things that had been one.
The front that reads one short
Eleven cells of a fourteen-cell design put the boundary exactly at the larger spiral count. Three put it one offset earlier, and the tempting move is to lower the threshold until all fourteen agree. Measured instead of tuned, the three turn out to be the three cells nearest below their own rung's boundary — and the last offset of a front is weak because it has only just arrived.
A count that drifts by two
A reported pair of 34 and 55 that may be wrong by one allows three sharp bands; allowed to drift by two it allows thirteen, and by three, twenty-nine — and the information lost is exactly the logarithm of that count, because every band is as narrow as the true one. The nearest wrong band stays twenty-four degrees away until a drift of three brings one to eleven. What does not survive is the protection: 21/34 and 34/55 could not be miscounted silently by one, but every Fibonacci pair can be by two, so a counter who drifts by two as readily as by one reports 34/55 silently wrong 9.5 per cent of the time rather than 0.13.
The flag reads the angle, not the twist
A twisted seed head is caught first by its counts: at half a radian some band stops returning two consecutive Fibonacci numbers, which an untwisted golden head never does. Twist the head in proportion to the square of the radius instead and the organs land exactly where a head grown at the golden angle plus a/(N − 1) puts them — to a billionth of a spacing — so the two read the same pairs in every band, and the flag fires on both. The flag detects a divergence that is not golden, and it has a resolution: 0.015° on 900 organs, 0.01° on 2,400, 0.003° on 9,000. Every twist shape from a half to four is flagged exactly when its change to the divergence in the outer annulus passes that resolution. So a flagged head is not golden, and nothing in its counts says whether it was twisted or grown that way.
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.
Both walls of the slot
The growing tip sits between its two chain-neighbours. Removing either alone is a cheap removal on all six lattices — 2.3° to 41.7°. Removing both together throws the next organ past the expensive line on three of them, and the interaction runs from −25.8° to +132.9°.
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.
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.
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.
Two counts that slip together
A counter who closes the circle a few degrees late counts a sliver of the head twice, in both families at once, so the two counts of a reported pair drift together rather than apart. Coupled that way the count is safer than it was: fourteen Fibonacci pairs in twenty-three admit no silent equal shift of one, against seven that admit no silent single miscount, and 34/55 announces every closing error short of 9.82°. The check is what breaks. Two annuli closed at the same wrong mark pass 17.6 per cent of wrong readings of 34/55 and 76.8 per cent of 13/21's, because a linear relation survives multiplication — and what catches them instead is a protractor good to twelve degrees.
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.
The organ that guards the second slot
An organ twelve places back is the furthest of any from where the next one goes, and removing it moves the next one by a whole divergence. The reason is that the rule's profile has two low points rather than one, the second is the slot after next, and that organ is holding it up. The comparison between what it holds up and how far behind it is decides the whole thing.
The census wants a low count
Four specimens separate the geometry's Fibonacci share of 14.7 per cent from the ninety per cent a grown history gives — if every count is right. Counted with a closing error spread over 7.2°, the same census needs six specimens counted at 13/21, fifty-four at 34/55 and 449 at 55/89, because the geometry's own pairs are all small enough that no closing error under 11° moves them, while a grown plant counted high loses its Fibonacci reading first. Counted at 55/89 with a spread of 9.83° the census reads plants as less Fibonacci than random angles. The count that pins the divergence best is the one a census should avoid.
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.
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.
Two marks chosen by one eye
A counter traces each family of spirals from a starting organ of its own, so a reported pair carries two closing errors, correlated because one eye chose both. Letting them differ costs 34/55 its ten-degree margin — 33/56 and 35/54 share no factor, and marks that err 5.3° in opposite directions reach them — while 21/34 keeps its margin whatever the marks do. And it decides the second annulus. At a spread of 7.2° the relation passes right readings 2.8 times as readily as silent ones when the marks are independent, 1.25 times at a correlation of 0.9, and stops telling them apart at 0.98; where it does work it keeps one reading in forty-six.
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.
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.
Counting it again
A reading whose two counts share a factor says the count went wrong, and the specimen is still there to be counted again. Counted afresh, the reading kept is exactly one reading conditioned on not announcing itself — the second chance a silent error gets is matched by the second chance a right reading gets — so a recount changes which specimens a census keeps, not what a kept reading says. At 34/55 with closing errors spread over 7.2° it takes the census from fifteen kept specimens to ten and from about thirty counts to twenty-one, and against scoring every reading it turns 449 counts at 55/89 into 52. It never makes a high count as cheap as counting 13/21 once.
The 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.
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.
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.
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.
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.
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.
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.
When the second wall is free
On six of thirty lattices, removing both walls of the slot costs exactly what removing the larger one alone costs — 35.9° and 35.9°, 12.0° and 12.0°, agreeing to the last digit of the grid the azimuths sit on. The smaller wall is not a wall on those rows.
The 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.
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.
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.
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.
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.
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.
The angle is not the actor
Cut an organ out of two stems that settled on the same divergence and return different counted pairs, and the family left standing is different at every one of the four pairs where both stems wreck. The angle is held to a hundredth of a degree underneath.
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.
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.
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.
The last of three quantities
The pair, the divergence and the step ordering move together when the rise is swept, and for a long time no result could be attributed to any of them. Two designs later, two are ruled out as sufficient and the third has never been held still — because holding it is what a rung already does.
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.
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.
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.
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.
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.
The ordering on six bands
A hundred and fourteen wrecked cuts across four bands, and at every offset of every one of them the family left standing is the same immediately above the handover and immediately below it. Where the answer does change — on the widest band, at three offsets — it changes somewhere else.
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.
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.
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.
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.
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.
Three offsets, three crossings
The claim the band design rests on is that the survivor does not change where the two contact steps change places. It holds at full resolution: nineteen changes and not one at the handover. Where they are is three different rises, eight, nineteen and twenty-nine below it.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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 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.
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.
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.
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.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Honest limitsMeasurementRiseRungDivergence angleAblationNegative resultClaim testingLatticeThe placement ruleCounting blindLattice offset