Phyllotaxis
the spirals are counted, not admired
Concept

Settling — where it appears

A stem's arrangement stopping changing, judged by every later divergence staying within a tolerance of a final value. Where a stem settles at all it does so within about 290 organs, and below a rise of about 0.011 most starting angles never do.

Named by 22 essays across 4 fields — each of them below, with the objects they name alongside it.

All 19 destinations, counted at 17 windows from 20 to 800 organs. One row per destination, one cell per counting window, coarsest on the left. A dark cell is a window returning the same pair as the standing 200; a warm cell is a window returning a different one; a pale cell is a window that reaches below the run's own settling organ and was dropped. 5 of the 19 rows differ anywhere, and every difference sits left of the floor its own pair sets — 20 organs for a pair under 14 parastichies and the larger count plus 6 above it. The census read at 25 organs and at 800 is the same census.

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.

lattices · Read window The five readings of 384 whose two counted families wind the same way. Each row is an additive sequence with the term the counter stepped over drawn open between the two it returned. 4/9 is 1, 4, 5, 9, 14 with 5 skipped, and the pair a crossing requirement would have returned is 4/5 — a sequence the same destination's other runs already sit on. Dropping these readings takes the census at 40 starting angles from 31 arrangements and 15 sequences to 28 and 12, on 5 of 384 runs.

A sequence that was a reading

One of the settling table's additive sequences rested on a single count that skipped a term, and the account of it was that the count had been taken over the wrong patch. It was not. The counter works out whether its two families wind opposite ways and the reading throws the answer away, on five runs in three hundred and eighty-four.

lattices · Read window Predicted 29 of 63 settle; 17 do. The 63 wrecked runs of the slot design regrown to 1200 organs and put to the settling table's own criterion, unchanged in every tolerance. Rows are whether the 300-organ endpoint sat within 1 degree of a destination; columns are whether the regrown run settles. The prediction written down before the sweep was that the 29 agreeing runs would settle: 12 of them do and 17 refuse, and the whole set settles 17 rather than 29. Agreement carries information without being a rule — 41.4 per cent of the agreeing runs settle against 14.7 per cent of the others, an odds ratio of 4.09 and a phi coefficient of 0.300 — so an agreeing run is still likelier to refuse than to settle.

A destination or a refusal

Sixty-three wrecked runs were regrown to twelve hundred organs and put to the settling table's own criterion, unchanged in every tolerance. The prediction written down before the sweep said twenty-nine would settle; seventeen do, and the prediction is wrong on its own side of the table as well as in its total.

mechanism · Settling The endpoint over four times the run: 60 of 63 do not move at all. One cell per wrecked run, dark where the divergence reported as its endpoint at 300 organs is the same divergence at 1200, warm where it moved. It moves on three of 63 runs, by a median of 0.0000 degrees and a worst of 0.0039. That worst is 128 times smaller than the 0.5 degrees at which two settled values are counted as one destination, and the movement has no sign — 1 up, 2 down. The short reading is precise about where a run finishes; it is silent about whether the run stays there.

Like with like

Seventeen wrecked runs settle when regrown to four times their length, and every one of them settles at the divergence the short run already reported. The endpoint moves by four thousandths of a degree at worst, which makes the short reading precise about where and silent about whether.

mechanism · Settling 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 Six boundaries on three basins, five kinds between them, and three that are edges. Each basin's stretch of starting angle, with both its boundaries located to ± 0.125° by sweeping ten degrees at a quarter of a degree. The pale bar behind each is the interval the forty-angle table could bracket it in, three to five degrees at a time. The widest basin, at rise 0.03 and exponent 2, is a fringe at 124.375° and a puncture at 179.625°. The same-rise basin, at rise 0.03 and exponent 3, is a sliver at 124.875° and a fade at 167.625°. The narrow basin, at rise 0.02 and exponent 3, is a wall at 144.875° and a fade at 161.625°. Only 3 of the 6 are edges in the sense of a side: the widest basin's upper boundary is a hole 0.75° wide centred on 180°, with the same destination beyond it, so that basin runs out of basin at the reflection point rather than reaching an edge. The interval the forty-angle table bracketed each basin in is drawn behind it, from the sweep at 1200 organs a run.

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.

mechanism · Attractor Two instruments at six boundaries: the clock cannot tell a wall from a fade and the tail spread separates them by a factor of 327. The clock is the ratio of the slowest settling in the last degree inside a basin to the middle of the rest of its window. It rises towards every boundary and diverges at none: the one clean wall rises by 1.313× and the two clean fades by 1.00× and 1.57×, so the wall sits between them and no threshold on a clock separates the two kinds. The tail spread is how far a run's own last two hundred organs wander, and every run has one whether it settles or not. Past the wall it is 0.101°, as steady as the basin just left; past the two fades it is 33.042° and 39.712°. Pooled over all 574 runs everything that settles spreads by 0–0.442° and everything that does not by 30.828–39.712°. The two instruments are drawn side by side on one row per boundary, from the sweep at 1200 organs a run.

A wall or a fade

A basin's border is either a change of destination or a stretch where the angles stop settling at all, and nothing here could tell the two apart. Two instruments were pointed at the question: the settling clock, which looked obviously right and fails, and the tail spread, which was already being computed on every run and had never been read.

mechanism · Attractor 1.75° of starting angle between two basins that reaches neither, and a 3° void beside it. 20.25 degrees of starting angle swept unbroken at 0.25°, from inside the basin at 101.5° to inside the widest basin at 139.3°, one cell per sampled angle. The last angle reaching 101.5° is 114.5° and the first reaching 139.3° is 116.25°; the 1.75° between them holds 6 sampled angles and 0 of them settle anywhere at all. The widest run of angles reaching nothing is 3° wide, at 118–120.75°, and a detached island of 139.3° sits between the two. Starting angle is left over. The stretch that reaches neither basin is bracketed above the strip, from the sweep at 1200 organs a run.

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.

mechanism · Attractor 41.75° in the middle of the widest basin, sampled only at the table's own 3.12–4.38°. The widest basin from its located lower boundary at 124.375° to the reflection point at 180°, with the two ten-degree windows swept at 0.25° shaded and the stretch between them left open. Nothing has looked inside that stretch more finely than the forty-angle table's own 3.12–4.38° spacing. The narrowest feature this sweep found anywhere is the 0.75° wedge inside same-rise, and a feature that size falls between the table's angles 79% of the time — 75% under an even 3° sampling. So an unmeasured sliver or puncture could sit anywhere in the middle of this basin and nothing here would have seen it. The two swept windows are shaded and the stretch between them is left open, from the sweep at 1200 organs a run.

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.

mechanism · Attractor Ten of the 24 orderings four exponents admit, from 15 readings of the wall. Every arrangement of the four falloff exponents is a cell, and a cell is filled when some level of some sampling puts the four walls in that order. 15 readings of runs that are shared cell for cell give ten of the 24, with the most common occurring three times. The marked cells are the orderings read at a half, the only level any round of this collection has published, and each of them occurs once. Exponent 5 is ranked first in 8 of the 15 readings and each of the others in two or three.

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.

mechanism · Falloff exponent Every bracket on the wall at a half, before the refinement and after. One pair of bars per falloff exponent: above, the rises consistent with that exponent's crossing on forty starting angles and the 5 published rises; below, the same on eighty angles and 9 rises. Two of the four have an open end before — at a half that is the coarse end, where no published rise has a share confidently above the level — and none is open after, the widest closing at 1.57 times. The four walls sit inside a factor of 1.111 of one another, against 1.624 on the reading that could not locate them, so they came closer together rather than further apart.

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.

mechanism · Falloff exponent The bracket at each of the four corners of the refinement, at a half. The geometric mean bracket at each corner, and under it how much of the joint narrowing each half of the refinement accounts for. It is taken over the two exponents whose bracket is closed at every corner, with the two halves of the refinement crossed against each other. The rule across the bars is 1.54, the narrowest bracket the published rise list can express at all. On forty starting angles and the published rises it is 3.06 times with two of the four brackets closed; doubling the angles alone gives 1.96, halving the rise spacing alone gives 1.46, and both together give 1.44. In the logarithm, where the two effects add, the rises account for 98.8 per cent of the joint narrowing and the angles for 59.3 per cent — summing to 158.1 per cent, which is what overlapping causes look like and not two independent factors of two.

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.

mechanism · Falloff exponent The settling share against the rise at exponent 5, on the refined rises. One line, over 80 starting angles at exponent 5. The rise runs coarse on the left and the level the wall is read at is the dashed rule. Between 0.02 and 0.013 the refined list holds two rises the published one does not, and at exponent 5 the share reaches 0.725 at 0.01732 against 0.550 at the ends of that stretch — 3.13 standard errors above the higher end. It turns in all three readings independently, which is what separates a feature of the curve from a bump in a sample. The ringed point is that maximum.

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.

mechanism · Falloff exponent Where a stem started at each of twenty angles ends up, at a falloff exponent of 3. One column per starting angle and one row per rise, coarse at the top. A filled cell is a run that reached a lattice, its tone the destination it reached; a pale cell is a run that never settles. Runs of one tone across neighbouring columns are basins, and the widest of them spans 7 consecutive angles. The angles are 6.25 to 10 degrees apart, so a basin narrower than that cannot be seen here and a single filled cell says nothing about how wide its basin is.

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.

emergence · Settling The settling share at every rise, sampled two ways. How often a stem reaches a lattice, against the rise, at each of the four falloff exponents. The horizontal rule is the half share the wall is read at. At nine starting angles the four exponents cross it at rises spanning a factor of 1.96 and order themselves 2, 5, 3, 4; at twenty they span a factor of 1.18 and order themselves 5, 4, 2, 3. The ordering reverses and the spread collapses, so the conclusion that the exponents do not separate is confirmed and the numbers that had suggested otherwise were the coarser sampling's own error.

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.

emergence · Exponent How often each group of starting angles reaches a lattice. The share of runs that settle, for the nine angles the table was grown from, for eight angles placed halfway between them, and for three angles below the nine's lowest. The nine settle far more often than either. So it is the refinement rather than the extension that drags the share down, and the settling share this collection reports is biased upwards by the choice of angles rather than by their range.

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.

emergence · Settling Where a stem started at each of twenty angles ends up, at a falloff exponent of 3. One column per starting angle and one row per rise, coarse at the top. A filled cell is a run that reached a lattice, its tone the destination it reached; a pale cell is a run that never settles. Runs of one tone across neighbouring columns are basins, and the widest of them spans 7 consecutive angles. The angles are 6.25 to 10 degrees apart, so a basin narrower than that cannot be seen here and a single filled cell says nothing about how wide its basin is.

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.

emergence · Settling Wrecked endpoints against the settling table's destinations. The upper lane is the 15 divergences the settling table reaches, grown from intact stems started at nine arbitrary angles across four falloff exponents and eight rises, with no cut anywhere in them. The lower lane is where the slot design's 63 wrecked runs finish, read without handedness so that a run ending at 209 degrees is placed at 151. 29 of them sit within 1 degree of a destination and 34 do not. The two measurements share no run and no design, so the agreement is not a construction.

A wrecked run goes somewhere

Where a wrecked stem finishes was called unstable. Half of them finish within a degree of a destination measured from intact stems started at arbitrary angles — two tables that share no run, no design and no question.

cylinder · Both walls How often a stem settles, at three samplings of the starting angle. The share of runs that reach a lattice, over the whole table of four falloff exponents by eight rises. The upper three bars are the pooled share at nine, twenty and forty starting angles; the lower three are the share for each group of angles on its own. The pooled figure falls 40.6 to 32.3 to 30.0 per cent, by 8.3 points and then 2.3, so it is converging. The eleven angles added at twenty and the twenty added at forty settle at 25.6 and 27.7 per cent, which differ by less than their own error.

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.

emergence · Settling The settling share against the rise, at 40 starting angles. One line per falloff exponent: how many of the 40 starting angles still reach a lattice at each rise, coarse on the left. The wall is where a line crosses a half, and the shaded band is the range of rises consistent with that crossing at one standard error — from the finest rise whose share is confidently above a half to the coarsest whose share is confidently below it. Every one of those bands is wider than the factor of 1.62 the four walls differ by, and at exponents 4 and 5 no rise in the table has a share confidently above a half at all.

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.

emergence · Exponent Destinations found at nine, twenty and forty starting angles. How many distinct divergences the settling table reaches at each sampling, split into those sitting on the golden or Lucas sequence and those off both. The on-ladder count is 5 at every sampling, over a fourfold refinement of the starting angle — so the rule's on-ladder targets were enumerated by the first nine runs. The off-ladder count goes 10, 13, 14, and the one the last doubling found is off both ladders.

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.

emergence · Settling The widest basin, at twenty starting angles and at forty. The starting angles of one cell of the table, drawn as ticks, with the run of consecutive angles that all reach one destination filled. At twenty angles it holds 7 angles and spans 43.75 degrees; at forty it holds 14 and spans 47.5. Halving the spacing doubled the count and left the width where it was, which is what a real basin does and what a run of angles produced by the sampling does not. The other half of the same table goes the other way: 170 of 233 runs are a single angle, against 116 of 151.

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.

emergence · Settling

Named alongside it

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

Honest limitsClaim testingSamplingStarting angleFalloff exponentNegative resultBasinMeasurement errorAttractorResolutionDivergenceInstrument setting

All concepts