Sampling — where it appears
Named by 66 essays across 8 fields — each of them below, with the objects they name alongside it.
Which junctions say anything
Da Vinci's rule and Murray's law differ by 12% at an even fork and by a tenth of a per cent at a twig. So one even fork settles which is right, and two thousand twigs do not — a factor of two thousand across a tree, decided entirely by the shape of the junction and not by how carefully it is measured.
A sample that is confidently wrong
Fifty lopsided junctions from a tree built at an exponent of exactly 3 return 1.7, with an interval that excludes 3 and excludes 2 as well. The sample carrying almost no information does not give a wide answer — it gives a narrow wrong one, and the cause is a selection nobody applies on purpose.
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.
A counter that sees no positions
This site has counted spirals two ways, and both were handed coordinates. A third counter is handed a list of angles and nothing else. It returns one number instead of two, it refuses more often, and where it refuses it would have been wrong every time.
The 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.
The disorder is a staircase
Sweep the second moment of a head's side-count distribution across the divergence angle and it is not a curve. It is flat in stretches with sharp steps between them and narrow deep dips wherever a rational falls — so 0.253 is not the golden angle's number, it is the number of every angle from 137.47° to 137.54°.
A dip belongs to the head
At an exact rational the disorder halves when the head doubles, and the dip around it narrows by a factor of four. So which angles look ordered is set by how many organs were counted, and a head of three hundred cannot tell 138.4615° from 138.48° while a head of twelve hundred tells it from 138.4625°.
The 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.
The background is not one sample
The dip in disorder at a rational angle is a comparison against a background, and the background was one measurement taken two tenths of a degree away. At 21/55 that lands seven thousandths of a degree from 13/34 — inside another rational's dip — and the comparison inverts. Fixed, the dip survives to a denominator of 89.
The width carries the denominator
The earlier work measured three denominators, found the dip's half-width falling as the square of the head size, and could not say whether its coefficient depended on the denominator. Six denominators say it does: the coefficient runs from 1,188 at q = 8 to 14,297 at q = 89, and dividing by q flattens a factor of twelve into a factor of two.
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.
Four fractions with one denominator
The dip in a head's side-count disorder is as wide as 150·q/n², measured over six fractions — every one of them a Fibonacci convergent, which is the emptiest neighbourhood a denominator ever gets. So the law could be about the denominator or about how well the fraction approximates its neighbours. Four fractions of 55 at one head size settle it in one figure.
A width read off a staircase
Two fractions of the fourteen measured return a dip width that moves by a factor of two when the head size changes, where the others hold to three per cent. The cause is not their neighbourhood. It is that the disorder statistic changes only when the tessellation changes, so the curve a half-width is read off is a staircase, and a width narrower than the tread cannot be read at all.
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.
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.
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.
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.
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.
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.
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.
A shoot too fast to remember
Sweep the rate at which a stem climbs the ladder and the correlation between one divergence and the next changes sign — negative below about fifty-five nodes per rung, positive above it, with the flip inside one step of the grid. The instrument the earlier work proposed is unavailable on a fast shoot, and nothing said so.
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 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.
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.
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.
Scars with dates on them
A bad season shared by every growing point wrecks the reading of a branching plant by two totals: over eighty seasons the rate and the scar share name a two-season wait for 22 per cent of plants. Date the scars — by position along a shoot, by growth ring — and each season's death chance is read off its own scars, so the bad years stop being noise and become a known input. Running the branching recurrence through the plant's own seasons names the wait for 94 per cent of plants over twenty seasons whether bad years kill a tenth of the points or seven tenths, and for every plant over eighty. What a plant cannot read from its own scars is the climate: the averaged chance comes only as fast as seasons do.
A frost and a drought in the scars
A branching plant whose buds and apices die at chances of their own, in a climate with bad years, leaves two kinds of scar in every season. Sorted by kind as well as dated, the scars give two shares a season, and the branching recurrence run through both names a two-season wait for 97 to 98 per cent of ten-season plants where one share a season names it for 81 to 83 — even when buds and apices die alike, because a small plant's deaths fall on one kind or the other by chance and the sorted series knows which. The three totals that the two chances were first thought to need fail under either kind of bad year, leaving the bud chance a tenth out however many seasons are read. And the sorted series can say what kind of bad year it had: a frost that strikes buds alone is told from a drought that strikes both at nine plants in ten over forty seasons, and one frost season is seen once the plant carries a few hundred buds.
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.
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.
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.
A window nobody aligned
Every reading this thread takes of a wrecked stem is taken inside a window, and there are three of them: a run of three hundred organs, a window of a hundred and twenty at its top, and a search window of fifteen to thirty-nine. None was aligned to anything, and one of them turned out to decide its own answers.
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 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.
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 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.
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.
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 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.
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.
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.
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.
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.
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 quarter degree cannot see
Six boundaries were located to an eighth of a degree, three basins were named and one width was quoted, and every one of those readings has the same floor under it. The sweep's grid is one step of the grid the stems are placed on, so nothing here bounds a basin narrower than half a degree — and the widest basin's own middle was never swept at all.
The 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.
The level was doing the ordering
Four falloff exponents have been put in order by the rise at which half their runs stop settling, and a half is the only level that order has ever been read at. Read at nine levels the same runs give ten different orderings of the same four numbers, and the one comparison in the whole study that resolves runs the other way.
Four walls closer than they looked
Two of the four falloff exponents had a wall with no upper end at all, and the other two were located to factors of two and a half and nearly four. Nine rises at eighty starting angles close every bracket — and the four walls turn out to sit inside a factor of 1.111 of one another, which is narrower than the narrowest bracket.
Two refinements that do not multiply
The design that located the wall did two things at once — doubled the starting angles and halved the rise spacing — and the arithmetic behind it assumed each would buy about a factor of two. The finer rises did ninety-nine per cent of the narrowing and the doubled angles added under one, because a bracket's ends are rises and no error bar can move them.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Honest limitsClaim testingMeasurementResolutionRiseNegative resultMeasurement errorSummary statisticRungArtefactHandoverAblation