Phyllotaxis
the spirals are counted, not admired
Concept

Replication — where it appears

Running the same instrument at the same settings on a second case, to find out whether a result belongs to the design or to the thing it was measured on. Here two bands cut whole disagreed, which put a boundary round a striking finding rather than extending it.

Named by 14 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 Hops give 55, 34, 89, 21 and walls give 34, 55, 21, 89. The leading families of a 900-point head, ranked twice from the same points. On the left, by the median hop the lag makes divided by the local spacing — the measurement a parastichy count is; on the right, by the share of the 1,732 interior walls the lag carries. The lists hold the same four numbers and four of them change place. The lines between are the permutation, and it is why a family read off a wall count is not a family read off a hop length.

Two rankings, one list

An essay in this collection claimed that the four shortest index hops on a seed head and the four largest shares of its cell walls are the same four numbers in the same order, and called the correspondence exact. Measured again from the same points, the two lists hold the same four families and order them differently, and they order them differently in five of the six bands the head can be read in.

tissue · Neighbour definition Which offsets wreck across the Lucas 7/11 band. One row per offset and one column per rise, with a mark where a single removal at that offset wrecks the stem. The set is not the same at every rise: on this band one offset wrecks at only 27 of its 124 rises, in several separate stretches, while others wreck at all of them. So a census taken at one rise of a band and a census taken at another are censuses of different sizes, and every claim of the form "at every offset that wrecks" is quantified over a set the rise decides.

A band with nothing inside it

Five offsets wreck on the Lucas 7/11 band and every one of them keeps the same family at every rise it wrecks at. There are no islands, no transition region and no period to look for, which is what makes the picture from the other band a picture of that band.

cylinder · Rung interior The hops the correction is fitted over, with the new one at the near end. Each cluster of rows placed by the lag it kept and the angle of the hop that lag keeps. The four the correction was fitted over run from 19.5 to 39.1 degrees; the new one sits at 12.78 degrees, a third smaller than any of them. A fifth point beyond the near end of a fitted range is a test of the fit, where a fifth point between two old ones would mostly have been a restatement.

A fifth cluster

A correction to the exchange's size was fitted over four hop clusters and its own file said so. A search turned up a fifth, at a hop smaller than any of the four, and the rule is right on it — which is what a prediction being confirmed looks like when the confirmation is worth having.

cylinder · Ablation Twenty rises at the fine end of both branches, cut at every offset. One row per rise searched, coarse at the top of each block. The bar names the counted pair the stem shows and the numbers on the right are the lags its wrecking cuts leave standing. A pale row is a rise whose settled divergence has left the branch it was started from by more than 20 degrees, which is what happens below the ladder's finest rung — the pairs there are 2 and 4, 8 and 16, 11 and 22, which are not two consecutive terms of any additive sequence. One rise on the Lucas branch keeps a lag of 11 while still on it.

One rise below the census

Ten lattices were cut at every offset and their surviving lags came back as four numbers. One rise further down a rung the census already sweeps, three cuts keep a lag of eleven — which is a fifth number, on a lattice nothing about was unusual except that nobody had cut it.

cylinder · Ablation The two widest bands on the ladder, each cut at every rise. One block per band, one row per offset that wrecks anywhere on it, one column per rise, coarse on the left. A filled cell is a cut that wrecks, and its tone is the family left standing; a pale cell is a cut that recovers. The golden 8/13 band above changes its answer at three of its six offsets, 19 times in all. The Lucas 7/11 band below changes it nowhere: every cut that wrecks on it keeps the 7 family, at every offset and every one of its 124 rises.

The second band, cut whole

One band was cut at every one of its rises and came back with a transition region — a stretch where three offsets change their answer, in short islands with uneven gaps. The obvious question is whether that is a picture of bands or a picture of that band. The other wide band answers it.

cylinder · Rung interior Which offsets wreck across the Lucas 7/11 band. One row per offset and one column per rise, with a mark where a single removal at that offset wrecks the stem. The set is not the same at every rise: on this band one offset wrecks at only 27 of its 124 rises, in several separate stretches, while others wreck at all of them. So a census taken at one rise of a band and a census taken at another are censuses of different sizes, and every claim of the form "at every offset that wrecks" is quantified over a set the rise decides.

The wrecking set moves again

Which offsets wreck a stem was assumed to be a property of the lattice. On one band it turned out to be a property of the lattice and the rise, changing on nearly a fifth of that band's steps. On the second band it changes more, and one offset's wrecking is broken into five separate stretches.

cylinder · Rung interior All three bands cut at every rise, offset by offset. One row per wrecking offset on each band cut whole, one cell per rise, coarse on the left. A pale cell is a rise at which that offset's cut recovers and has no survivor; a dark cell is a cut that wrecks and keeps one of the band's own counted pair; a warm cell is a cut that keeps a family off the pair. The vertical rule on each row is that band's handover, where its two contact steps change places. The golden 8/13 band changes the family it keeps 19 times, the golden 5/8 twice and the Lucas 7/11 not at all.

The third band, cut whole

Two bands cut at every rise disagreed about whether the family a cut keeps ever changes, and three explanations were available for a difference between two things. The cheapest third band settles which of them survives, and it settles it against the account nobody was betting on.

cylinder · Rung interior Four accounts of which bands speckle, scored on the three cut whole. Each candidate explanation of why one band's cuts change the family they keep and another's do not, against what the three bands cut whole actually do. A tick is an account that puts that band on the side the sweep does. The branch the band sits on is right on all three; the size of the counted pair, the number of wrecking offsets and how much of its rung the band spans are each wrong on two. Three bands can eliminate and cannot confirm, and this eliminates three of the four.

The branch is what is left

Four accounts of why one band's cuts change what they keep and another's do not were written down before a third band was cut. Three of them are now wrong on a band each, and the survivor is the one with no mechanism behind it.

cylinder · Rung interior What nine rises find on each band, against what the whole band holds. The coarse design cuts nine rises of a band, evenly spaced in the logarithm of the rise, and asks whether the family a cut keeps changes anywhere. On the golden 8/13 it finds five of nineteen changes and on the Lucas 7/11 it finds none of none, so it had never been wrong about whether anything changes. On the golden 5/8 there are two changes and it finds neither, both of them at single rises with the offset recovering on either side. Its record on that question is now 2 of 3.

Nine rises were not enough

A coarse sample of a band had never been wrong about whether anything changes inside it, and that record was the argument for trusting a negative from it. The third band cut whole makes it two of three, and the missed feature is one rise wide.

cylinder · Rung interior 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 The Lucas 4/7 band cut at every one of its 86 rises. One row per wrecking offset on each band, one cell per rise, coarse on the left, with each band's width in proportion to the rises it holds. A pale cell is a rise at which that offset's cut recovers and has no survivor to report; a dark cell keeps one of the band's own counted pair and a warm cell keeps a family off it, so a change of answer is where the tone changes. The Lucas 4/7 is the test the branch account most needed: 86 rises, 774 stems, 144 wrecks across two offsets and not one change of the surviving family. Across the one bands drawn, 144 cuts wreck of 774 grown. The dashed rule on each row is that band's own handover, where its two contact steps change places.

The fourth band, cut whole

Three bands cut at every rise left one account of which bands change their answer standing, and the account was the one nobody had a reason to prefer. The band that would have killed it has now been cut, and it did not kill it.

cylinder · Rung interior What nine rises find on six bands, against what those bands hold. The coarse design cuts nine rises of a band, evenly spaced in the logarithm of the rise, plus both ends and the handover, and asks whether the family a cut keeps changes anywhere on it. Scored on six bands it is right about whether on five and wrong on the 5/8, where it finds none of 2. On the one band where it finds anything it finds 5 changes of 19, so its record is a record about whether and never about how many. The figure under each band's name is that design's step on it, in rises.

The coarse design scored

Every claim this thread has made about an uncut band rests on a sample of nine rises, and its record was the argument for trusting it. Six whole bands close that record, and two of its six correct verdicts are correct only because there was nothing on those bands to find.

cylinder · Rung interior How broken the wrecking is on four bands, from 2.43 rises a stretch to 126. Every offset that wrecks on every band that wrecks, placed on a logarithmic axis of rises wrecked per separate stretch — the smaller the number, the more broken the wrecking. A filled mark is an offset that wrecks at every rise of its band and an open one comes and goes. The range runs from 2.43 on g58 offset 3 to 126 on g813, a factor of 52, and two of the four bands have no offset that wrecks everywhere at all. A band with no offset that wrecks at every rise is said so at the right.

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.

Claim testingHonest limitsContact familySamplingHandoverAblationCensus designNegative resultLattice offsetResolutionRiseRung

All concepts