Phyllotaxis
the spirals are counted, not admired
Concept

Cut-off — where it appears

A stated distance beyond which the inhibition between organs is negligible, written as a function rather than as a loop bound. A smooth one is a hypothesis about how far a real inhibitor reaches; a hard one pins the pattern to its own edge.

Named by 11 essays across 3 fields — each of them below, with the objects they name alongside it.

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 Two shapes, two ranges, one contrast. The exponential's lattice ends at 3.63 spacings and the gaussian's at 2.25 — ranges 47% apart — and at those two ranges the contrast is 5.74 and 6.09, 6% apart. The band is what the exponent route leaves: 3.98 at p = 1, where the uncut rule makes nothing, and 7.63 at p = 1.25, where it makes a lattice.

Two shapes, one threshold

Read in the same unit, an exponential falloff and a gaussian one disagree about where the lattice ends by half. The quantity they agree on turns out to be one the earlier work measured for an unrelated reason — and it agrees with a bracket left by a sweep of a completely different parameter.

emergence · Falloff exponent 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 A cut-off would have to exceed 2.236 and not exceed 1.441, and nothing does both. Each of the 608 interior cells contributes two marks: its furthest wall, and its nearest partner that is not a wall, both in units of that cell's own shortest lag. A single cut-off would have to sit to the right of every mark of the first kind and to the left of every mark of the second, and the two clouds overlap — the extreme cases are 2.236 at 0.0 per cent of the radius and 1.441 at 60.0 per cent. So the interval is empty by a factor of 1.55, while 606 of the 608 cells have a cut-off that works for themselves.

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.

tissue · Neighbour definition A sixfold neighbourhood, and nothing to dilute. The prediction was that a rule with fewer neighbours would convert a jostle into divergence scatter more efficiently. Across a sixfold widening the ratio sits between 0.97 and 1.00, and the one point that differs is the narrowest, at 0.82 — smaller, where the prediction wanted larger. The row underneath is why: past four spacings the rule builds the identical lattice, internode for internode, so there is no neighbourhood left to widen.

The neighbourhood was already settled

The earlier work explained a small difference between two kinds of noise by saying a jostle is diluted among some thirty neighbours. Sweep the neighbourhood sixfold and the difference does not move — because past four spacings the rule builds the identical lattice, internode for internode. There was nothing to dilute.

mechanism · Noise entry How often a cell on a 2,400-organ golden head has a wall the contact cut gets wrong, by its distance from the nearest flip ring. The 1,680 interior cells of a 2,400-organ golden head, grouped by distance from the nearest flip ring in twentieths of the head's wall spacing, 1.92. The bars are the share of each group with at least one wall a three-family contact cut gets wrong; the dots are the share with five or seven sides. 353 cells have such a wall and the farthest is 0.638 spacings from a ring. Of the 1,292 cells 0.66 spacings or more from every ring, none has.

The empty interval is the rings

No single cut-off on hop ratio turns the contacts a count keeps into the walls a tessellation draws, on any whole head at any size. Read cell by cell against the flip rings the divergence angle puts in closed form, every disputed cell lies within two thirds of a wall spacing of a ring, and with one spacing either side set aside a single cut-off between 1.430 and 1.444 serves every golden head from 900 organs to 9,000.

tissue · Neighbour definition The centre of a 900-organ Lucas head, with the band of two thirds of a spacing drawn round every flip ring. The organs of a 900-organ Lucas head out to a radius of 12.5, with a pale annulus 0.66 of a wall spacing either side of each flip ring, the rings keeping 1, 3, 4, 7, 11, 18. Warm: cells with five or seven sides; dark: six-sided cells a three-family contact cut gets wrong. Ringed: the one cell well inside a band that the cut reads exactly, organ 17 at radius 4.12, 0.23 of a spacing from its ring. Where two annuli overlap, a cell can sit well away from its nearest ring and still inside the next one's band.

The blur was at the centre

On a Lucas head the band of disputed cells round each flip ring looked blurred at its inner edge — exact cells as close as 0.23 of a wall spacing, disputed hexagons out to 0.59 where a golden head's stop at 0.43. Read a ring at a time, the two heads carry the same band on every resolved ring, to a hundredth: disputed hexagons within 0.16, exact cells from 0.64, a ring's own number of fives and of sevens and the number before it of hexagons. Every difference is inside a radius of six, where the Lucas rings of 4, 7 and 11 sit closer together than the band is wide, and the one exact cell is organ 17, which has no organ eighteen behind it.

tissue · Neighbour definition A 900-organ golden head displaced by 0.02 of a wall spacing, with its flip rings and the cells in dispute. Every organ of a 900-organ golden head inside the rim cut, each moved by a seeded gaussian displacement of 0.02 of a wall spacing, with the flip rings the divergence angle puts at radii 11.3, 18.2. Warm: the 150 cells with five or seven sides, 13 of them more than 0.66 of a spacing from every ring. Dark: the 59 six-sided cells with a wall a three-family contact cut gets wrong. Of the cells a spacing or more from every ring, 0 are disputed.

A hundredth of a spacing

Off the flip rings one hop-ratio cut-off turns a seed head's counted contacts into its cell walls, on every head from 900 organs to 9,000. Displace the organs and it is the first thing to go: shut by a fiftieth of a wall spacing on 900 organs and a two-hundredth on 9,000, because it is decided by the worst of thousands of cells. The three-family count survives two to four times further, because each cell only has to beat its own margin, and the rings keep their fives and sevens in between. All three fail from the rim inward, since the margin one spacing from a ring is 9.7 divided by the ring's family number.

tissue · Neighbour definition The disputed cells round the ring of 55 on a 2,400-organ head twisted by 0.2 radians at the rim. Every organ turned about the centre by 0.2 times its radius over the head's. Bars: the six-sided cells a three-family contact cut disputes (grey-blue) and the five- and seven-sided cells (warm) within two spacings of the ring the untwisted divergence puts at 55, by signed distance from it. There are 34 disputed hexagons, with a median at 0.54 of a spacing, and 110 fives and sevens. The dashed line is where the twisted divergence puts the ring in closed form, 0.54 of a spacing out; the band keeps its cells and its width and sits on it.

The band moves, it does not blur

Displaced organ by organ, a seed head loses its single contact cut-off first, its rings' hold on their fives and sevens next and its three-family count last. Displaced by a smooth field that moves neighbours together, the same head keeps its census — the same 353 disputed cells and 264 fives and sevens at every step up to a third of a spacing — and moves the band instead. A twist moves each flip ring exactly to where the twisted divergence puts its tie, the ring of 55 by 0.53 of a spacing, the ring of 34 the other way. Read against strain, correlation helps the cut-off and not the count, and on a 900-organ head the two fail at the same step: the order was an order of blurring.

tissue · Neighbour definition

Named alongside it

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

Honest limitsThe range of the interactionThe placement ruleContact familyNegative resultNeighbourhoodRepulsionVoronoi cellsArtefactDefect ringDivergence angleHop length

All concepts