Underdetermination — where it appears
Named by 31 essays across 6 fields — each of them below, with the objects they name alongside it.
![An L-system after 4 rewrites of two rules. X → F[+X]F[-X]+X and F → FF, walked by a turtle turning 22.5°. 130 segments, and not one of them knows anything about light, water or auxin.](/f/an-l-system-after-4-rewrites-of-two-rules-cf31ba72.svg)
L-systems describe, they do not explain
Two rewriting rules and a turtle produce something indistinguishable from a plant, and there is no plant in it — no light, no water, no auxin, no mechanics. That is worth demonstrating precisely because the output is so convincing.
Droplets with no biology in them
Douady and Couder dripped magnetised ferrofluid into a dish of silicone oil, and got spiral phyllotaxis with Fibonacci parastichy numbers out of a system containing no cells, no genes and no plant. That is the strongest evidence the pattern is physics — and the clearest warning about what a model can claim.
What a mechanism would have to show
Every model of phyllotaxis comes with the caveat that reproducing a pattern is not explaining it. That is easy to repeat and hard to make precise. Here it is made precise — a list of what an account of phyllotaxis would have to establish, with each item marked according to whether the models drawn in these essays establish it.
A rule that predicts everything
Leonardo's rule says a fork conserves cross-section, and cross-section is exactly the weight the branch point is minimised against. So the three weights land on the boundary of the triangle inequality, the cosine comes out at one to the last bit, and the rule predicts no angle at all — and the free constant its own derivation leaves behind then walks the prediction across every angle a fork could have.
What a refusal does not say
The readout can decline for four different reasons — too quiet, too disturbed, too fast, or a window in the wrong place — and a stem that returns nothing does not say which. That is the third time this thread has failed to close the mixture problem, and the first time the failure has a shape.
A spiral with no clock
The growth factor a shell's curve gives up is a rate per turn of the shell's own coiling, and a turn is not a unit of time. Four clocks — the aperture advancing at a constant angular rate, adding a constant length, a constant area, a constant volume — trace the identical curve: every mark one of them leaves lies on r = W to the power theta over two pi exactly, so a fit through any of them returns the same factor. What differs is where the marks are, and the difference is enormous: at 3.2 per turn the outermost of three whorls holds 33.4, 71.0, 90.3 and 97.0 per cent of the record. It also breaks the instrument. The routine that recovers a growth factor unwraps the angle by assuming successive points advance less than half a turn, and five of the twenty readings here leave gaps past that — returning 3.33 where the curve was built at 3.20, and 8.19 where it was built at 6.85.
Four ways to count a neighbourhood
How deep does the placement rule look? Counting the organs that carry nine tenths of its profile gives 182 down to 3 as the falloff steepens. Counting the ones that carry half gives 33 down to 1. The weighted mean lag gives 68 down to 11. The four disagree by an order of magnitude about the size and agree exactly about the order.
A count set by a delay
An L-system describes a plant and forbids nothing, because none of its parameters is anything a plant has. One branching grammar is the exception: a mature apex makes a new bud every season, and a bud waits d seasons before it branches. Its counts grow at the root of x^(d+1) = x^d + 1, a delay of one season gives Fibonacci's numbers and nothing else does, and the fourth count already separates a one-season wait from every longer one. So a Fibonacci count in a branching plant is a measurement of how long its buds wait. It is also a fragile one: if one bud in ten waits two seasons instead, eleven counts in a row come out Fibonacci's three times in a thousand.
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.
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.
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.
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.
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.
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.
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 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.
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.
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 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 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 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 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.
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.
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.
Negative resultClaim testingHonest limitsParastichy pairControlMeasurementRungRiseAblationFalsifiabilityLattice offsetLattice