Phyllotaxis
the spirals are counted, not admired
Concept

Truncation — where it appears

Cutting a sum short at a fixed number of terms rather than at a stated distance. It manufactures patterns: an inverse-first-power rule with no lattice at all produces a clean one once its loop is cut, and the pattern disappears when the loop is widened.

Named by 10 essays across 3 fields — each of them below, with the objects they name alongside it.

Where this implementation stops converging. Below about G = 0.18 the settled angle wanders over 113° however long the run. That is the model's limit, not a fact about plants.

Where the model stops

Below a growth parameter of about 0.18 this implementation does not converge — the settled angle wanders over a hundred degrees however long the run. That is the range where the literature says the interesting behaviour lives, and it is worth a figure rather than a quietly chosen axis.

emergence · Modellimit A tree built at 3, measured to 2%, reads 2.957 on the informative band. The exponent recovered from 100 junctions of a tree built at exactly 3, against the relative error placed independently on the parent and on both daughters — half a per cent is a machined section under a microscope, one to two per cent is callipers on a clean branch, five is a branch that is not round, ten is a radius read off a photograph. Each line is a band of daughter ratio and the whiskers are the central 90% of 300 replicate samples. Every band is displaced downward at every error and never upward: at 2% the informative band read 2.957 and the informative band read 2.957. Below, the same rows with the displacement and the spread drawn as separate bars, because only one of them falls when more junctions are measured.

The exponent an error moves

Every real measurement of a branch radius carries error and no synthetic tree does, so the question is what a symmetric error does to a fitted exponent. It does two things — a bias and a spread — and the bias runs downward at every error level and in every band, by an amount derivable from the daughter ratios alone.

branching · Exponent error Which neighbours decide where an element goes. Each line is one exponent: how much each shell of neighbours makes the energy profile vary around the circumference, divided by what the nearest shell contributes. At p = 0.5 the nearest shell leads the next by a factor of 1.1 and a node is placed against the whole neighbourhood at once. At p = 3 it leads by 9.7e+3, and a node is placed against its immediate neighbours — which is what a lattice is.

How far a primordium reaches

The placement rule's repulsion falls as an inverse cube because that is what two magnetised droplets do, and nothing about a plant supplies the exponent. Asking what it controls produced one tidy wrong answer and one measured right one — and the difference between them is the difference between a total and a variation.

emergence · The range of the interaction What a swelling does to the exponent read from the informative band, by where the swelling is. 50 junctions of daughter ratio 0.6–1, built at exactly 3 and read with no random error, but with one or more radii measured fat. A parent read fat lowers the reading: 1% gives 2.864, 3% gives 2.631, 10% gives 2.075, and at 11.3% the tree reads Da Vinci's 2. Daughters read fat raise it: 1% gives 3.150, 3% gives 3.499; from 8% some junctions have a daughter measured wider than their parent, which no exponent balances, and the line stops. All three radii read fat by one factor return 3.000 at every swelling — the unswollen reading exactly.

A swelling at the fork

A branch thickens where it forks, so a parent measured just below a junction and daughters measured just above it carry three different amounts of the same swelling. A swelling that fattens all three alike moves a fitted exponent by exactly nothing. A parent read one per cent fat moves it by as much as 3.6 per cent of random error on every radius, in a sign known in advance, and a tenth of a radius turns a tree built at Murray's three into one that reads Da Vinci's two with no noise at all. Added to the noise, it does not bring the two rules together any sooner: the two errors do not add.

branching · Exponent error The same rule at p = 1, cut off at two distances. The top 220 nodes of two stems grown by an identical rule whose energy does not converge. Allowed to see 3/√h neighbours it produces 8/13 at 137.62° with 0.58° of scatter — a lattice no test here would question. Allowed 12/√h it produces 44° of scatter and no pattern. The truncation was doing the work.

A window that makes a pattern

A rule whose energy has no well-defined minimum produces a clean 8/13 lattice at 137.62°, with half a degree of scatter, when its neighbourhood is cut at three node spacings. Let it see twelve and the pattern is gone. Every simulation of this kind truncates something, and truncation manufactures exactly the result it is used to look for.

emergence · Truncation Three cut-offs at the same nominal width of 3 spacings. The weight the interaction is multiplied by, against distance. They halve at 2.08 (exponential), 2.50 (gaussian), 3.00 (hard) spacings — so a rule described as "cut off at 3 spacings" is three different rules until the falloff is named. Every later figure is read in half-weight radii for that reason.

A neighbourhood is a hypothesis

Every simulation of this kind stops summing somewhere. The earlier work found that where it stops decides what pattern comes out — so the stopping place is not a detail of the program but a claim about how far a primordium's influence reaches, and it should be written down as one.

emergence · The range of the interaction The landscape the rule chooses over, at a cut-off of 3 spacings. One height of an ideal lattice, swept around the circle. The exponential cut-off hands the rule a smooth landscape; the hard one hands it a landscape with steps, because a neighbour enters the sum as the candidate slides past it. Halving the sample resolution multiplies the largest jump between neighbouring points by 1.99 on the smooth curve and by 1.19 on the hard one — which is the definition of the difference, since a smooth function's steepest step is bounded by its derivative and a discontinuity's is not. An argmin taken over steps is pinned to the steps.

A hard edge is not a falloff

The prediction was that cutting the neighbourhood at three spacings would reproduce the pattern truncation had manufactured. It does — if the cut is smooth. A hard cut at the same distance produces no pattern at any width, and the reason is that it is the only one of the three whose neighbour set depends on where the candidate is.

emergence · Truncation The loop bound is not the neighbourhood. The wander a placement rule leaves in its divergences, against how many recently placed organs the rule sums over, at four correlation lengths of the disturbance driving it. The loop runs from 15 organs to 85 and nothing moves: the largest change along any line is smaller than the change between random seeds at one setting. That is the shape a parameter has when it is not binding, and it is the same shape a robust result has, which is why the sweep is drawn with the seed spread rather than reported as a number.

The window was not the neighbourhood

A placement rule corrects what is relative between neighbours and passes what moves them all together, so how much of a slow disturbance gets through should depend on how deep the neighbourhood is. The obvious knob is how many organs the rule sums over. Swept across a factor of six, it changes nothing at all — and a parameter that is not binding produces exactly the flat sweep a robust result produces.

mechanism · Noise colour Which lattices survive a fifth of a degree of noise. The share of runs that still have a lattice. The prediction was that a cut-off at this range would be as fragile as the truncation it replaces; it is not. Stating the neighbourhood as a function of distance did not merely make the old result honest — 100% of runs survive against 33%, at a scatter an inverse-cube rule cannot be told from.

The fragility belonged to the window

A pattern that exists only because the rule cannot see far was expected to be held together by that cut, and to fall over when nudged. It does — while the cut is a loop bound. Written down as a falloff at the same range, the same rule keeps every run under the same nudge, at a scatter an inverse-cube rule cannot be told from.

emergence · Noise amplitude The same verdict at every length from 300 organs to 1200. The settling criterion applied to the first 300, 400, 600, 800, 1000, 1200 organs of each of the 63 regrown runs. It accepts 17 at every one of them and refuses 46 at every one, and not one run changes sides. A truncated twelve-hundred-organ run is the shorter run organ for organ, so this is a comparison of lengths rather than of growths. The slowest arrival in the whole set is 138 organs, so 140 is the measured requirement and 300 carries a factor of 2.1. The column drawn dark is 300 organs, where the verdict is 17 settled and 46 refused.

What a run length was hiding

The settling criterion returns an identical verdict on all sixty-three wrecked runs at every length from three hundred organs to twelve hundred, so run length explains nothing. What the sweep does find is that an endpoint is a mean over an orbit, and thirty-four refusers never come within a degree of their own reported endpoint.

mechanism · Settling

Named alongside it

The objects these essays reach for when they reach for this one.

RepulsionArtefactThe placement ruleHonest limitsThe range of the interactionNoiseCut-offDivergence angleNegative resultNeighbourhoodRiseDiscretisation

All concepts