Every essay — page 3
What a plant might be doing
Turing's last work was on this, and it was unpublished when he died: a ring of cells, two diffusing substances, and a spacing that selects itself. The modern account uses auxin and a pump that works uphill. Both are made to predict a number and then to produce it.
A disturbance with a memory
The previous phase'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.
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.
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.
Shells and growth
A logarithmic spiral is what a thing grows into when it adds material without changing shape. Its one parameter is recoverable from any drawn curve, which is how a century-old argument gets a number attached to it.
Growth as a rule
A logarithmic spiral is not a shape somebody admired. It is what a thing grows into when it adds material at its opening without changing shape, its one parameter is how much it grows per turn, and that parameter can be recovered from any drawn curve to the last digit.
Raup's three numbers
Nearly every coiled shell that has ever existed is a point in a three-dimensional space — how fast the whorl expands, how far the opening sits from the axis, how far it travels along it. That is a remarkable compression, and the most useful line in the space is the one where the whorls come apart.
The nautilus question
A golden spiral grows by 6.854 per turn. Measured nautilus sections give about 3.2. That is a factor of 2.14 — not a rounding error, not an artefact of where the centre is assumed to be, and not close.
Packing and tiling
How evenly a pattern fills its disc is a statement about cell areas, and it can be measured four ways that disagree. Cells average six sides because Euler's formula leaves them no choice.
Why the average cell has six sides
Not because hexagons are efficient. Because Euler's formula leaves a tiling no choice — count the edges two ways and the mean comes out at six, whatever the cells would prefer. The efficiency argument is a different claim about a different thing.
Packing, measured four ways
The claim is that the golden angle packs best. It is measurable, and the measurement gives three different winners on three criteria — all near 137.5° and none of them it. That does not make the claim wrong; it makes the usual statement of it wrong.
The gap that grows
A rational divergence angle develops an empty wedge that grows without bound as the head fills — a factor of nearly four between two hundred primordia and sixteen hundred. An irrational one does not. That is the statement about the golden angle that survives measurement.
Lewis's law wants disorder
Cell area rises linearly with side count — measured on cucumber epidermis in 1928 and quoted ever since as a property of packed tissue. It holds beautifully on a random point set, with a fitted constant of 1.64 against Lewis's 2. On a phyllotactic head it does not hold at all: the slope is 0.009, and area and side count are almost independent.
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.
The second moment is the measurement
The mean number of sides in a cellular tissue is six, and Euler's formula leaves it no choice — so it takes the same value on a golden-angle head, a whorled head and a set of random points. The mean squared departure from six varies by a factor of eighty across the same three, and almost nobody reports it.
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.
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 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 previous phase 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.
The claims, measured
The nautilus, the sunflower and the golden angle arrive with more confident wrong statements attached than any other subject on this fleet. Each one gets a test and a number — including the one that turns out to be right.
Fibonacci is a branch, not a law
Fibonacci counts come from one branch of the model. The Lucas branch — which the same rule reaches at a different growth rate — gives 47 and 76, and neither is a Fibonacci number. The sequence is a consequence of an angle rather than a property of plants.
The claim that survives
Of the three famous assertions about this subject, one is out by a factor of two, one is true of a branch rather than of plants, and one is right — in a sharper form than the version usually told, and about arithmetic rather than about packing.
What "whorled" was hiding
The expansion phase's census put 35% of divergences in a bucket labelled whorled and moved on. Opened, every pair in it is k and 2k — the coarsest rung of the ladder, repeated k times — and reading the census up to jugacy takes the Fibonacci share from 14.7% to 50.1% without describing a single extra plant.
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.
The angle is not the object
Every popular account of phyllotaxis is organised around a number. After a phase spent on stems, forks and frequencies, the number looks like the wrong thing to organise an account around — it is the limit of one path through a branching structure, it is at no fork, and a plant that has it got there by not jumping.
What a count is worth
A reported parastichy pair pins the divergence angle to a band 221°/mn wide — so 2 and 3 says almost nothing and 34 and 55 fixes it to a tenth of a degree. Every rung of the ladder is worth a factor of φ², and the counting radius this collection has asked published counts for since its foundation adds ten per cent.
How many plants would it take
Fourteen specimens separate the geometry's Fibonacci share from a coin weighted to a half. Four separate it from what a grown history gives. One fir cone measured at three rings settles whether its ladder is spaced as a cone's or an ogive's. The sample sizes are small, and that is the uncomfortable part.
The survey this site cannot do
Four phases of asking for a dataset, and it is still not here. What this phase can do instead is specify it — the fields, the sampling, the sizes, and which of this collection's claims each one would settle. Two of the four fields asked for turn out to be worth less than the asking implied, and one was never asked for at all.
The test a plant could settle
Every other open question in this collection is priced in tens of specimens, and one of them in a hundred and sixty. This one is priced in internodes on a single stem, and the number is fifty-six — because it is a statistic of one sequence rather than a share of a population.
What a quiet plant is worth
Almost every measurement gets easier as the effect gets larger. This one gets harder — a stem's divergence sequence stops carrying information about its noise at precisely the scatter where the noise becomes obvious. The specimens worth measuring are the ones that look least interesting.
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.
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.
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 most irrational is not the most disordered
If rational angles make ordered tissue, the most badly approximable angle should make the most disordered — which would at last give the golden angle a criterion it wins. Swept across the interval it is the most irrational point of, μ₂ peaks at 138.42° and the golden angle sits unremarkably in the middle.
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 refusal with a reason
Three phase plans 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 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 five phases 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.