Phyllotaxis
the spirals are counted, not admired
Concept

Census — where it appears

A count over every case in a stated range rather than over a chosen sample. Several results here are censuses — every pair of offsets, every fraction in a band — which is what lets a claim be made about all of them rather than about the ones that were tried.

Named by 31 essays across 7 fields — each of them below, with the objects they name alongside it.

A shell section at W = 2.40, D = 0.42. W·D = 1.01, so the whorls are free of each other — an evolute shell, like a ram's horn or a planispiral ammonite. Both are things animals grow.

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.

shells · Raup What the "whorled" bucket contains, at a rise of 0.008. 7 pairs, sharing 7 different factors, and every one of them is k and 2k. The bucket the previous census called whorled is the coarsest pattern the ladder has, repeated k times around the stem — not a residue of odd arrangements.

What "whorled" was hiding

The earlier work'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.

wrong · Jugate census three stems: 1, 2, 3 primordia at a time. 1-jugate at 137.51° counts 2 and 3; 2-jugate at 68.75° counts 2 and 4; 3-jugate at 45.84° counts 3 and 6. Every count shares the factor k, and dividing it out leaves an ordinary lattice: what a k-jugate stem is, exactly, is k copies of an ordinary one wrapped k times round.

Two at a time

Every counter in these essays asks how far it is from element i to element i plus m, and that index is a claim that the elements arrived one at a time. For teasel, for Cephalaria and for a real minority of plants the claim is false — and what those patterns turn out to be is an ordinary lattice, wrapped twice.

lattices · Jugacy What a divergence picked at random gives, at a rise of 0.100. Fibonacci pairs take 59.6% of the circle at this rise, and the share falls as the rise does. The claim that Fibonacci counts are what nature "prefers" needs the preference to come from somewhere, and it is not from the geometry being generous.

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.

wrong · Frequency The 2-jugate forks converge on 68.7539°. Every fork sits at a rational divergence, with denominator 4(m² + mn + n²) — 10/28, 30/76, 74/196 and so on. The limit is 68.7539°, which is 137.5078 divided by 2, and it is at none of them.

Half the golden angle

The forks of the van Iterson tree converge on 137.5078° and sit at none of them. Divide the whole tree by two and the same thing happens at 68.7539° — which is where teasel is, and where a bijugate sunflower counted 42 and 68 has to be.

lattices · Jugate limit One geometry, six boxes: 4.642 per cent to 52.81 per cent forbidden. The share of each box that the coiling geometry excludes, across six boxes that differ only in where their edges were put and in how one axis is sampled. The spread is a factor of 11.4, from 4.642 per cent in the widest box to 52.81 per cent in the tightest, and sampling the same two decades of expansion geometrically rather than uniformly multiplies the answer by 4.59 on its own. The forbidden area is the integral of 1/W and grows as a logarithm while a box grows as a line, so the fraction has no value of its own to quote.

A fraction of nothing

Six boxes differing only in where their edges were drawn give between 4.64 and 52.81 per cent for the same geometry, and sampling one axis geometrically rather than uniformly multiplies the answer by 4.60. The share tends to zero as the box widens, because the region under a hyperbola is a logarithm and a box is a line.

shells · Morphospace 14 specimens separate 14.7% from 50%. The exact binomial power against sample size, for a one-sided test at 5 per cent. It is a staircase rather than a curve because the decision rule is a whole number of specimens: at 14 the cut sits at 5 and the power is 91.0 per cent. A normal approximation smooths that staircase away and reports a different answer.

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 transitions are spaced as a cone's or an ogive's. The sample sizes are small, and that is the uncomfortable part.

wrong · Sample size Fibonacci at a rise of 4.8e-3, asked two ways. Choose a divergence at random and a static lattice at this rise gives a consecutive Fibonacci pair 10.8 per cent of the time. Start coarse at a divergence nobody chose, grow the stem down to the same rise, and it is 100 per cent of 16 runs. The geometry is not generous; continuity is.

Continuity from a coarse start

At a fine rise, one divergence in seven gives a Fibonacci pair. Grow a stem from a coarse start at a divergence nobody chose, down to that same rise, and sixteen runs out of sixteen end on 8/13. The earlier work's interpretation was that continuity does the work; this is the measurement it never had.

emergence · Continuity The angles against the positions, rise by rise. five rises, five seeded stems each. A filled mark is a run whose angle readout returned the pair the position counter finds in the same stem; an open mark is a refusal. At 0.032 the counter says 3/5 and the angles agree on 0 of 5, refusing 5. At 0.013 the counter says 5/8 and the angles agree on 5 of 5. At 0.01 the counter says 5/8 and the angles agree on 5 of 5. At 0.005 the counter says 8/13 and the angles agree on 5 of 5. At 0.008 the counter says 5/8 and the angles agree on 2 of 5, refusing 3. The two instruments share no code path: one is given a list of angles, the other a list of coordinates.

The angles name the branch

Seed the same rule at the Lucas angle and the readout returns 4 and 7, then 7 and 11 — the pairs the position counter finds, and not Fibonacci numbers. So a list of divergence angles carries not only how many spirals there are but which family of ladders the plant is on.

lattices · Counting Every open question here needs under 34 specimens. The sample size at which each comparison reaches 90 per cent power at a 5 per cent false-positive rate, from the exact binomial rather than a normal approximation. The census question — do plants show consecutive Fibonacci pairs far more often than the geometry does — needs 4: 14.7% is the share of divergence angles giving a consecutive Fibonacci pair at a fine rise; 90% is what a grown history gives.

The survey this site cannot do

Four rounds of asking for a dataset, and it is still not here. What the work here 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.

wrong · Survey spec Four fractions of 55, one width. The half-width of the second-moment dip at four divergence angles with the same denominator, on heads of 1279 organs — 23 in each of 55 rows. 21/55 is the Fibonacci convergent, the fraction every earlier measurement of this law was made at and the one whose neighbourhood is emptiest; 12/55, 23/55, 17/55 are not. The four agree within a factor of 1.28, well inside the factor of two the law is stated to, so the width is a function of the denominator and not of how well the fraction approximates its neighbours. What is left over is ordered by the note beside each bar: the more crowded the neighbourhood, the wider the dip comes out, which is the direction a neighbour's own shoulder would push it.

Four fractions with one denominator

The dip in a head's side-count disorder is as wide as 150·q/n², measured over six fractions — every one of them a Fibonacci convergent, which is the emptiest neighbourhood a denominator ever gets. So the law could be about the denominator or about how well the fraction approximates its neighbours. Four fractions of 55 at one head size settle it in one figure.

tissue · Second statistic Four fractions of 34, one width. The half-width of the second-moment dip at four divergence angles with the same denominator, on heads of 791 organs — 23 in each of 34 rows. 13/34 is the Fibonacci convergent, the fraction every earlier measurement of this law was made at and the one whose neighbourhood is emptiest; 9/34, 15/34, 11/34 are not. The four agree within a factor of 1.14, well inside the factor of two the law is stated to, so the width is a function of the denominator and not of how well the fraction approximates its neighbours. What is left over is ordered by the note beside each bar: the more crowded the neighbourhood, the wider the dip comes out, which is the direction a neighbour's own shoulder would push it.

A width read off a staircase

Two fractions of the fourteen measured return a dip width that moves by a factor of two when the head size changes, where the others hold to three per cent. The cause is not their neighbourhood. It is that the disorder statistic changes only when the tessellation changes, so the curve a half-width is read off is a staircase, and a width narrower than the tread cannot be read at all.

tissue · Second statistic 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 How many specimens the census needs, counted at four different pairs, as the closing error spreads. The exact one-sided binomial sample size, at five per cent and ninety per cent power, separating the geometry's Fibonacci share from a census of grown plants, every specimen scored as read. Counted at 13/21: 4, 4, 5, 6, 14 specimens; counted at 21/34: 4, 5, 9, 17, 43 specimens; counted at 34/55: 5, 10, 27, 54, 276 specimens; counted at 55/89: 8, 32, 104, 449, none specimens, at spreads of 1.8°, 3.6°, 5.4°, 7.2°, 10.8°. With every count right the answer is four.

The census wants a low count

Four specimens separate the geometry's Fibonacci share of 14.7 per cent from the ninety per cent a grown history gives — if every count is right. Counted with a closing error spread over 7.2°, the same census needs six specimens counted at 13/21, fifty-four at 34/55 and 449 at 55/89, because the geometry's own pairs are all small enough that no closing error under 11° moves them, while a grown plant counted high loses its Fibonacci reading first. Counted at 55/89 with a spread of 9.83° the census reads plants as less Fibonacci than random angles. The count that pins the divergence best is the one a census should avoid.

wrong · Sample size Which chain the backward exception sits on, over the census. Chains are numbered from the removed organ, so chain 0 is the chain the hole was on and chain 2 is two organs along it. The exchange is at the hole's own chain on 10 of the 17 rows that carry one, against 2.8 rows for a chain drawn at random from each row's own period. That is far more often than anywhere else and it is not every row, so the position is a tendency rather than a rule — and the file says so rather than rounding it up.

Which chains changed places

A wrecked stem's displacement profile is a set of levels, one per chain, with two of them out of line — equal and opposite, on neighbouring chains. Nothing said which two. They are the hole's own chain and the one below it, on ten of the seventeen cuts that carry a pair.

mechanism · Damage shape A period of 8, with the hole's own chain at the top. Each mark is one residue class of the displacement profile, placed round a ring at its own residue, with the chain the removed organ sat on at the top. The radius is how far that class sits from the level the rest of them share. six of the eight classes sit together at the middle ring; two do not, and on this row they are one pair, equal and opposite to within a twentieth. The forward one is chain 7 and the backward one is chain 0, one residue above it, which is the order every row of the census puts them in.

One way round, seventeen times

The two chains that change places in a wrecked stem are adjacent, which is symmetric and says nothing about direction. Label them by lag from the hole and the one displaced forwards is always the lower of the two — on every row of the census, without an exception.

mechanism · Damage shape How many counts the census spends when an announced reading is counted again, against counting 13/21 once. Each specimen is counted with a closing error spread over 7.2°, and a reading whose counts share a factor is counted again, up to the number of counts on the horizontal axis; a specimen whose every count announces itself is set aside. Plotted is the census's cost in counts — kept specimens needed times counts spent per kept specimen — solid when the plants are grown and counted at the stated pair, dashed when they are the geometry's. The flat line is 13/21 counted once and every reading scored: 6 counts. 34/55: 29.5, 24.4, 20.7, 21.1, 21.5, 22.6 counts on grown plants, cheapest at 2; 55/89: 71.3, 58.5, 51.7, 52.5, 53.2, 55.3 counts on grown plants, cheapest at 3.

Counting it again

A reading whose two counts share a factor says the count went wrong, and the specimen is still there to be counted again. Counted afresh, the reading kept is exactly one reading conditioned on not announcing itself — the second chance a silent error gets is matched by the second chance a right reading gets — so a recount changes which specimens a census keeps, not what a kept reading says. At 34/55 with closing errors spread over 7.2° it takes the census from fifteen kept specimens to ten and from about thirty counts to twenty-one, and against scoring every reading it turns 449 counts at 55/89 into 52. It never makes a high count as cheap as counting 13/21 once.

wrong · Sample size Four accounts of one angle, scored on the same 17 rows. Each bar is how far an account of the exchanged pair's size sits from the measurement, as a share of the control's divergence, averaged over the census. A step of the wrecked stem's own divergence is the obvious candidate and the worst of the four. The surviving family's own step is not the unit at all: that angle is a fifth of the exchange. A step of the control's divergence is close, and correcting it by a fifth of the surviving hop takes the worst row from 11.8 per cent to 4.0.

Four accounts of one angle

The exchanged pair in a wrecked stem is about one divergence step, and about is doing twelve per cent of work. Four candidate units were written down and scored on the same seventeen rows: the cut stem's own step, the surviving family's step, the control's step, and the control's corrected.

mechanism · Damage shape What the exchange misses one divergence step by, against the surviving hop. Each mark is one wrecked cut. Across: the angle from an organ to the one its surviving lag places above it, on the control. Down: how much the exchanged pair's size falls short of one divergence step. The relationship is a straight line through the origin with a slope of about 0.2, and its sign is right on every row of the census — the four surviving lags give hops on both sides of zero and the residual follows each of them. The honest limit is the four: the hop hardly moves inside a lag, so the line rests on 11 distinct positions and not on 17.

A fifth of the hop

The exchanged pair misses one divergence step by up to twelve per cent, and the miss is not scatter: every row keeping a lag of 5 or 7 overshoots and every row keeping a 4 or an 8 falls short. Subtract a fifth of the surviving hop's own angle and the worst row is four per cent.

mechanism · Damage shape Where the pattern starts, measured at two run lengths. One row per wrecked cut. The small mark is the onset a 300-organ run reports and the ring is what a 600-organ run reports; a row with only one mark reports an onset at only one length. 19 of the 24 rows that report both move by more than twenty organs, and the largest move is from 135 to 435. The range the thread has been quoting, 7 to 303 organs, becomes 28 to 473.

Twice the run

Five wrecked cuts never reached a pattern inside three hundred organs and one reached it at three hundred and three, which is a number asking to be checked. Run every cut in the census twice as far and three of the thirty change their answer.

mechanism · Run length Where the pattern starts, measured at two run lengths. One row per wrecked cut. The small mark is the onset a 300-organ run reports and the ring is what a 600-organ run reports; a row with only one mark reports an onset at only one length. 19 of the 24 rows that report both move by more than twenty organs, and the largest move is from 135 to 435. The range the thread has been quoting, 7 to 303 organs, becomes 28 to 473.

An onset at the end of the run

One cut reported that its pattern began 299 organs into a 300-organ run. Given twice the room it reports no pattern at all. The reading was the run stopping, not the disturbance ending, and the definition guarantees one at the last organ of every run.

mechanism · Run length The 3 cuts the two run lengths disagree about. Each block is one wrecked cut, with its widest within-class spread drawn at both run lengths and the 10 degrees that separates periodic from not marked by the rule. Two of these become periodic when the run is doubled, at spreads falling from about seventy degrees to about eight. One goes the other way, from six degrees to a hundred and seventy — and that one is the row an entirely separate reading of the same census independently reports as its worst fit.

Three rows change sides

Twenty-five of thirty wrecked cuts have a periodic displacement profile over three hundred organs and twenty-six do over six hundred. The count barely moves and the membership does: two rows join, one leaves, and the gap the threshold sits in narrows from 1.69 to 1.27.

mechanism · Run length How steady each class is, over 300 organs and over 600. Each mark is one wrecked cut, placed across at the widest spread found inside any one of its residue classes over the shorter run and up at the same reading over the longer one. A mark on the diagonal is a row the two lengths agree about. The rules are the 10 degrees that separates a profile called periodic from one that is not: three rows fall in different quadrants at the two lengths, two of them becoming periodic and one ceasing to be. The gap between the two groups narrows from 1.69 times to 1.27.

A window nobody aligned

Every reading this thread takes of a wrecked stem is taken inside a window, and there are three of them: a run of three hundred organs, a window of a hundred and twenty at its top, and a search window of fifteen to thirty-nine. None was aligned to anything, and one of them turned out to decide its own answers.

mechanism · Run length

The offsets that never change

Three of the six offsets that wreck anywhere on the band keep the same family at every rise they wreck at — 98, 22 and 81 rises of the 126. And which offsets wreck at all is a function of the rise, which no reading of a band had drawn.

cylinder · Rung interior

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.

emergence · Survey spec

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.

emergence · Survey spec

A counter on the settling table

The settling table has reported an angle for every run that reaches a lattice, and nobody has ever counted one. A hundred and seventeen settled runs, regrown and counted: every one of them has a parastichy pair, and sixty-five of them land on sequences the ladder does not carry.

emergence · Settling

Ten sequences, two of them the ladder's

Every parastichy pair the settling table produces is two consecutive terms of a sequence in which each term is the sum of the two before it. Ten such sequences account for all fifteen destinations — nine, once one of the ten turns out to be a reading rather than a ladder — and the two the collection is built on are neither the largest nor the smallest.

emergence · Off-ladder

A list that was a rounding

Three destinations only a steep falloff reaches, read at two grid steps. Thirteen, read at a tenth of a degree. And four arrangements, read by what the counter returns rather than by the angle — a different four, with one the angle reading hides.

emergence · Off-ladder

A wrecking set with a range

Which offsets wreck a stem was taken to be a property of the lattice, and every band cut whole has found it to be a property of the rise instead. Six bands turn that replication into a measured range, and the range is a factor of fifty-one.

cylinder · Rung interior

Named alongside it

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

Honest limitsMeasurementClaim testingNegative resultParastichy pairResolutionAblationControlRiseSummary statisticDivergence angleFibonacci

All concepts