Phyllotaxis
the spirals are counted, not admired
What a plant might be doing

A median that is an exception

Every number in the ablation thread is measured against the value most of a wrecked stem's chains sit at, taken as a median so that a few exceptions cannot move it. On eight of thirty-six rows the median stands on a chain sitting by itself.
14 min read 19 figures State the claim, state the testCounted, not admiredOne parameter, one family

Worth reading first: The damage has a period · The organ that was taken away.

A wrecked stem’s displacement profile is the difference, organ by organ, between the cut run and its control. Folded onto the lag the run keeps rigid, it becomes a set of residue classes, and most of those classes sit at one level with one or two exceptions.

Every number in the ablation thread is measured against that level. The size of an exchange, which chains are exceptional, whether a rearrangement closes — all of them are offsets from it, and the level is taken as the median of the class means.

The eight rows where the two levels disagree. Each wrecked cut's residue classes drawn as points on a circle of angle, one panel per row. A tick is one class's mean displacement. The solid radius is the densest cluster of them, which is the level this file computes; the pale radius is the median of the class means, which is the level everything in the ablation thread has been measured against. On every one of these rows the median stands on a class that is on its own, and on all 28 rows not drawn the two levels agree to within the exception tolerance of 10 degrees.
Fig. 1 The class means of the rows where two ways of computing the level disagree, drawn on their own circle.

Why a median

Because a mean would be dragged by the exceptions, which is exactly what the reading is trying to measure. A profile with six classes at nothing and two at plus and minus a hundred and thirty degrees has a mean somewhere in between and a median at nothing.

That reasoning is right and it has a condition attached that nothing checked: a median is robust while the exceptions are a minority. Half the classes exceptional and the median is on a boundary; more than half and the median is one of them.

Nobody wrote the condition down until a rule fitted to the awkward rows had a candidate artefact attached to it: on a profile with three of four chains exceptional, the median is itself an exception.

A period of 8, and the two classes that are not with the rest. The same wrecked stem, folded on the lag it kept: one row per residue class, each drawn at the mean displacement of its own organs against the level the rest of them share. The bar through each row is the spread inside that class, and the widest of them is 0.87° — so within a class the displacement is a constant. six classes sit at the common level. The two that do not sit at 134.3° and -134.2°, equal and opposite to within 0.0 per cent, and they are neighbouring residues. The stem's own divergence is 137.44°, so an exception is one organ's step.
Fig. 2 An ordinary profile, most of whose classes sit at one level with a pair of exceptions.

What to replace it with

Not a better average. The level is the value the classes that agree agree on, and the honest way to find it is to find the largest set that agrees.

For each class mean in turn, count how many class means sit within the exception tolerance of it round the circle. Keep the candidate with the largest count, breaking ties on the tighter cluster. The level is the circular mean of that cluster’s members.

That assumes no majority. A cluster of two out of eight is reported as a cluster of two out of eight, with the number of rival clusters of the same size beside it, and the reading is refused outright where no two classes agree at all.

The eight rows where the two levels disagree. Each wrecked cut's residue classes drawn as points on a circle of angle, one panel per row. A tick is one class's mean displacement. The solid radius is the densest cluster of them, which is the level this file computes; the pale radius is the median of the class means, which is the level everything in the ablation thread has been measured against. On every one of these rows the median stands on a class that is on its own, and on all 28 rows not drawn the two levels agree to within the exception tolerance of 10 degrees.
Fig. 3 The densest cluster on each row, drawn as a radius, beside the median it replaces.

The tolerance is not a new parameter

The cluster’s width is the same ten degrees that decides which classes are exceptions. That is deliberate and it is what keeps the estimator from having a free parameter the median does not have.

A level is the value a set of classes agree on to within the tolerance that decides who disagrees. Making them one number means there is nothing to tune: the reading is fully determined by a quantity already stated.

The ten degrees is itself a gap rather than a threshold. Across the census the classes at the level sit within 5.6 degrees of it and the exceptions are 12.4 to 160 degrees away, so anything between about six and twelve gives the same table.

The exceptional pair, measured in divergences. One row per wrecked cut whose profile has exactly one pair of exceptional classes, drawn at the size of that pair divided by the stem's own settled divergence. Every row sits between 0.882 and 1.076, so the two chains that came apart moved by one organ's step rather than by two or by half of one. The residual is not scatter: rows are grouped by the lag the stem kept, and every lag sits wholly above the line or wholly below it. Why a surviving 5 or 7 overshoots and a surviving 4 or 8 falls short is not answered here.
Fig. 4 The gap the exception tolerance sits in, which is what makes it a separation rather than a choice.

Distances have to be folded

The class means are angles. A class at plus 175 degrees and one at minus 178 are seven degrees apart, and a linear median of a list containing both puts the level near nothing — a value no class is anywhere near.

So every distance in the estimator is folded into a half turn either way, and the cluster’s level is a circular mean rather than an arithmetic one.

That is not a hypothetical repair. Several rows in the census have class means either side of the fold, and the incumbent median takes the arithmetic median of raw values before folding the offsets.

The same sums read raw and read folded. Every excluded cut's displacements added twice: the open mark is the raw total in degrees and the filled mark is the same total folded into half a turn either way. The two agree everywhere but one row, and on that row the raw total is the largest in the table at about minus a whole turn while the folded total is the smallest. Reading the sums raw would have named the one closed cycle here as the worst anomaly in the census.
Fig. 5 The same folding elsewhere in this thread, where an unfolded reading called a clean result the worst in the table.

No new stems

The whole recomputation is arithmetic on numbers already in the cache. Each row of the census returns its class means alongside its offsets, so both readings are available from the same banked table.

Thirty-six rows — the ten lattices of the census plus the one the fine-end search added — read twice, at no cost. That is the cheapest kind of check and it is the kind this thread has been running all round.

It also means the comparison is exact rather than approximate. The two levels are computed from the same numbers, so a difference between them is a difference between two estimators and not between two runs.

That is the discipline the slot table followed when it went from six lattices to twenty-four, and the extended census when it added a lattice rather than replacing the list. A comparison between a reading and a second reading is worth nothing if the two are also two implementations.

Every stem that never repaired, and the lag it kept. The 19 offsets across six lattices at which a single removal leaves a stem that never returns to its divergence. For each one: which organ was removed, the period of the block of angles the stem settles into, the lag whose hop survived the cut unchanged, and how many whole turns the stem gains over one period of that lag. The block and the surviving lag are the same number in every row. The marked row is the one whose survivor is not a parastichy number of the lattice that was cut — a hop 6.8 times the length of a contact hop, which no census would report and which the rule held rigid all the same.
Fig. 6 The census the recomputation is run over, whose class means are already banked.

Where the two agree

Everywhere it matters most. On the twenty rows the exchange is quantified over — the rows whose exceptions are one balanced pair — the two levels agree to within the exception tolerance on every one, and the largest disagreement is 3.13 degrees.

The median disagreement across those twenty is 0.13 degrees, which is half a step of the azimuth grid.

So nothing the exchange reports rests on the estimator. That is the first thing this recomputation had to establish and it is the result that makes the rest reportable rather than alarming — the exchange’s own claims stand at the numbers they were reported at.

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.
Fig. 7 A row the exchange keeps, where the two ways of computing the level agree to within half a grid step.

Where they do not

On eight rows of thirty-six, the level moves by more than the exception tolerance. The moves are 13, 47, 49, 91, 91, 104, 104 and 133 degrees.

Every one of the eight is a row the exchange already sets aside — a row with no balanced pair, which was excluded for a reason having nothing to do with the level.

So the median is right exactly where it had been checked and wrong exactly where nothing had looked. That is not a coincidence: a row with a balanced pair has two exceptions out of five or eight, so its exceptions are a minority by construction and the median’s condition holds.

The eight rows where the two levels disagree. Each wrecked cut's residue classes drawn as points on a circle of angle, one panel per row. A tick is one class's mean displacement. The solid radius is the densest cluster of them, which is the level this file computes; the pale radius is the median of the class means, which is the level everything in the ablation thread has been measured against. On every one of these rows the median stands on a class that is on its own, and on all 28 rows not drawn the two levels agree to within the exception tolerance of 10 degrees.
Fig. 8 The eight rows where the level moves, all of them rows the exchange sets aside.

What a moved level does

Changes which classes are exceptions, and therefore every number read off the row.

On six rows the count of exceptions falls: from four to three, from three to two, from five to four twice, from six to four, from five to four again. On no row does it rise.

That direction is the signature the objection predicted, and one row loses enough exceptions to change side entirely. Choosing an exceptional class as the level makes the classes at the true level look exceptional, and cannot do the reverse — so a row that gained an exception would mean the new estimator was picking a level that is not a level.

The 13 cuts the exchange sets aside, chain by chain. One row per wrecked cut with no balanced pair, one cell per chain of the lag it kept, and the chains displaced away from the common level filled. The lag is printed at the left and the sum of the displacements at the right, folded into half a turn either way. seven of the 13 sum to within 12 degrees of nothing and six do not, and every row that closes kept a lag of 7 or 8 while every row that does not kept 4 or 5. The rows are stacked by that lag.
Fig. 9 The rows the exchange sets aside, whose exception counts fall when the level is recomputed.

The clearest case

l008 cut at offset 7. Its seven class means are −71.0, −68.9, −64.9, −15.2, 33.5, 33.5 and 35.9 degrees.

The median of seven values is the fourth, which is −15.2 — a class sitting on its own, with the nearest other class fifty degrees away. Measured against it, six of the seven classes are exceptions.

The densest cluster is either of the two triples: {−71.0, −68.9, −64.9} or {33.5, 33.5, 35.9}. Both hold three of seven, so the reading reports a tie of two, takes the tighter, and lands at 34.3. Against that, four classes are exceptions rather than six.

The eight rows where the two levels disagree. Each wrecked cut's residue classes drawn as points on a circle of angle, one panel per row. A tick is one class's mean displacement. The solid radius is the densest cluster of them, which is the level this file computes; the pale radius is the median of the class means, which is the level everything in the ablation thread has been measured against. On every one of these rows the median stands on a class that is on its own, and on all 28 rows not drawn the two levels agree to within the exception tolerance of 10 degrees.
Fig. 10 The row whose median falls on a class sitting fifty degrees from anything else.

Which raises the question the next essay is about

A row with two clusters of three and one singleton has no level in the sense the reading assumes. Neither triple is a majority of seven; the estimator reports a tie and takes the tighter, and the tie is the honest part of the answer.

That is not a defect in the estimator. It is a fact about the row, and the median hid it by always returning a number.

How many rows are like that is a countable question, and it turns out to line up almost exactly with the exchange’s own exclusion — which is computed from something else.

No majority and no balanced pair, over the whole census. One row per wrecked cut. The bar is the share of that row's residue classes sitting at the level they agree on, and the vertical rule is a half — a bar reaching past it has a majority and a median is safe there. The mark at the right says whether the exchange keeps that row. 15 rows are on both lists of 16 and 16, and the two part on g008/6, which is excluded and has a majority of 5 of 8, and l013/4, whose lag is 4 so that a balanced pair leaves two classes each way and a majority is arithmetically unavailable.
Fig. 11 How much of each row’s profile sits at the level it agrees on, against whether the exchange keeps the row.

Four rival estimators, scored

The incumbent median, the densest cluster, a circular median — the class nearest every other, the short way round — a circular mean, and the point opposite the largest empty arc, which is what a reader of a circular histogram does by eye.

They are scored on the one quantity none of them defines: how many classes end up within the tolerance of the level, summed over the census. A level several classes sit at is a level; a level no class sits at is an artefact of the averaging.

Densest cluster 145 of 243. Circular median 141. Median 138. Circular mean 121. Largest gap 63.

The 36 rows where the two levels disagree. Each wrecked cut's residue classes drawn as points on a circle of angle, one panel per row. A tick is one class's mean displacement. The solid radius is the densest cluster of them, which is the level this file computes; the pale radius is the median of the class means, which is the level everything in the ablation thread has been measured against. On every one of these rows the median stands on a class that is on its own, and on all 0 rows not drawn the two levels agree to within the exception tolerance of 10 degrees.
Fig. 12 The estimators compared on the one quantity none of them defines: how much of each profile sits at the level.

What the scores mean and do not

The margin between the top three is small — 145, 141, 138 out of 243 — so this is not an estimator winning by a distance. What separates them is the failure mode rather than the average.

The densest cluster is never a level with one class or none at it. The median is on three rows and the circular median on two. The circular mean is on nine and the largest-gap reading on twenty-three, which is why the last is in the table only to be dismissed.

A statistic that occasionally returns a value nothing is near is a statistic that occasionally measures everything against an anomaly, and that is the property being selected for rather than a decimal place.

The 20 rows where the two levels disagree. Each wrecked cut's residue classes drawn as points on a circle of angle, one panel per row. A tick is one class's mean displacement. The solid radius is the densest cluster of them, which is the level this file computes; the pale radius is the median of the class means, which is the level everything in the ablation thread has been measured against. On every one of these rows the median stands on a class that is on its own, and on all 16 rows not drawn the two levels agree to within the exception tolerance of 10 degrees.
Fig. 13 The rows where every estimator agrees, which is most of the census and all of the exchange’s own rows.

The refusal

A profile whose classes are spread evenly round the circle has no value several of them sit at. The reading must decline it rather than name the mean of a single class.

Eight class means at forty-five degrees apart is such a profile: no two are within ten degrees of each other, so the densest cluster holds one, and there are eight clusters of that size. Asking for its level is refused.

The median returns a value anyway, with exactly one class at it and no way of saying so. That is the whole of the difference between the two readings, and the check is the site’s own habit of feeding a reading input it must refuse.

The eight rows where the two levels disagree. Each wrecked cut's residue classes drawn as points on a circle of angle, one panel per row. A tick is one class's mean displacement. The solid radius is the densest cluster of them, which is the level this file computes; the pale radius is the median of the class means, which is the level everything in the ablation thread has been measured against. On every one of these rows the median stands on a class that is on its own, and on all 28 rows not drawn the two levels agree to within the exception tolerance of 10 degrees.
Fig. 14 The reading’s own refusal case: a profile with no two classes within the exception tolerance.

Where the level sits in the chain of readings

It is the second step of four, and everything after it depends on it.

First the profile: the cut run minus its control, organ by organ, over the top hundred and twenty organs. Then the fold: the profile cut into residue classes modulo the lag the run keeps rigid, and each class’s mean taken. Then the level. Then the offsets — each class’s mean less the level — from which the exceptions, the balanced pair, the exchange’s size and the sum that says whether a rearrangement closes are all read.

So a level in the wrong place does not corrupt one number. It shifts every offset on the row by the same amount, which leaves differences between classes intact and moves anything measured against nothing.

That is why the exchange survives untouched even on a row whose level moves: a balanced pair is a difference between two classes, and a shift does not change it. What moves is the count of exceptions and every sum over them.

How far every organ moved, 6 places back at a rise of 0.01. One mark per organ above the hole, at the angle it sits from where the same organ sits in a control that shares its history. The collection has read two numbers out of profiles like this one — the largest displacement anywhere, and the first organ's — and never the profile. It is not a bump that decays. After about 18 organs it settles into a repeating pattern of eight levels, one per residue class modulo 8, which is the lag whose hop this stem kept. six of those levels sit together and two do not.
Fig. 15 The profile the whole chain of readings starts from, before it is folded onto its surviving lag.

Which numbers are shift-invariant

Worth listing, because it decides what this recomputation can and cannot change.

The size of an exchange is half the sum of the two exceptions’ magnitudes, and on a balanced pair that is a difference between two classes. Shift-invariant. The orientation — which chain is displaced forwards and which backwards — is a comparison between two classes. Shift-invariant. Whether two exceptional chains are adjacent is a fact about residues. Shift-invariant.

What is not shift-invariant is which classes count as exceptions at all, since that is a comparison against the level. And therefore neither is the sum over the exceptions, which is what the closure reading adds up.

So the recomputation cannot move the exchange’s own numbers and can move which rows are in it, and it can move every closure sum. Both of those happen.

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.
Fig. 16 The exchange’s own quantities, every one of which is a difference between classes and survives a shift.

Why nobody checked

Because the rows where it matters are the rows nobody was reading. The exchange excluded them at the start, on a criterion — no balanced pair — that had nothing to do with the level, and for several rounds nothing was quantified over them.

When they were finally added up the rule that came out of it was thirteen for thirteen, which is the state a rule is most worth testing in and least likely to be questioned in.

The objection was written down in that round’s own file rather than found later, which is the only reason it was checked at all. A rule with no stated weakness is a rule nobody goes back to.

How far each rearrangement is from closing, against the lag it kept. One bar per excluded cut, its length the distance from the displacements summing to nothing, ordered smallest first, and each labelled with the lag the cut left standing. The seven shortest bars are the cuts that kept a lag of 7 or 8 and the six longest are the cuts that kept 4 or 5, with a gap of 6.3 degrees between the two groups and no row in it. The rule sorts every one of them.
Fig. 17 The rule fitted to the rows the exchange sets aside, whose own file recorded the objection this tests.

What a tie means

Four rows have two or three clusters of the same size competing to be the level, and the reading reports the count rather than hiding it.

On g005 at offsets 7 and 8 there are three clusters of two, out of eight classes. On g013 at offset 5 there are two clusters of two out of five, and on l008 at offset 7 two clusters of three out of seven.

A tie is not an error and it is not resolved by choosing better. It is the statement that the row has no single value most of its chains agree on, and it is exactly the information a median destroys by always returning one number.

No majority and no balanced pair, over the whole census. One row per wrecked cut. The bar is the share of that row's residue classes sitting at the level they agree on, and the vertical rule is a half — a bar reaching past it has a majority and a median is safe there. The mark at the right says whether the exchange keeps that row. 15 rows are on both lists of 16 and 16, and the two part on g008/6, which is excluded and has a majority of 5 of 8, and l013/4, whose lag is 4 so that a balanced pair leaves two classes each way and a majority is arithmetically unavailable.
Fig. 18 The rows whose densest cluster is tied, which the median reports as an ordinary level.

What this does not touch

Every number the exchange reports, which is the largest single body of arithmetic in this thread. Its twenty rows are read against a level the two estimators agree on to within half a grid step, so the size of an exchange, its orientation, where it sits and the correction fitted to it are all unchanged.

Nor does it touch the periodicity classification. That is read from the spread within a class, which is a statistic of one class about its own mean and does not reference the level at all — so recomputing the level cannot move a verdict there. The same is true of the onset, which is read from where each class stops moving.

What it touches is exactly one family of readings: anything that counts exceptions or adds them up. That is the excluded set’s whole subject matter, and it is the reason the recomputation was worth doing at all rather than filed as a curiosity about statistics.

What a level is for

Worth stating plainly, because it is easy to read the common level as though it were a property of the stem. It is not. A wrecked run’s profile is a set of displacements, and the level is the value most of them share; it exists because the displacements cluster, and it is interesting only in so far as they do.

On a profile whose classes sit at four or five distinct values there is no such thing, and the right report is that there is none rather than a number with error bars. That is what the cluster reading says and what the median could not: a median always returns a value, and on those rows it returns one that describes nothing.

So the recomputation is less a correction to a statistic than a change in what the reading is allowed to say. It can now decline, and the rows it declines on are the rows every other file had already set aside for a different reason.

The order the checks were made in

Worth recording because it is what makes the result usable. The objection was written down first, in the file that reported the rule it threatened. The estimator was then chosen for a property stated in advance — no free parameter, and a refusal where no cluster exists — rather than for what it would do to any number. Only then was it run.

That order is what separates a correction from a rationalisation, and it is cheap to follow when the objection is written at the time. The alternative shape — a rule fails, a statistic is adjusted, the rule recovers — is the same three events in an order that would make the outcome worth nothing.

Nothing about the sequence guarantees the answer is right. What it does is make the answer checkable by somebody who was not there.

What is claimed

That the common level, recomputed as the densest cluster of class means rather than their median, agrees with the incumbent on every row the exchange is quantified over, worst 3.13 degrees and median 0.13.

That it disagrees by more than the exception tolerance on eight of thirty-six rows, all of them rows the exchange sets aside, with moves of 13 to 133 degrees.

And that the count of exceptional chains falls on six rows and rises on none, which is the direction predicted by a median standing on an exception rather than beside one.

The eight rows where the two levels disagree. Each wrecked cut's residue classes drawn as points on a circle of angle, one panel per row. A tick is one class's mean displacement. The solid radius is the densest cluster of them, which is the level this file computes; the pale radius is the median of the class means, which is the level everything in the ablation thread has been measured against. On every one of these rows the median stands on a class that is on its own, and on all 28 rows not drawn the two levels agree to within the exception tolerance of 10 degrees.
Fig. 19 The eight rows the recomputation moves, and how far each moves.

Where this goes next

A period of 8, and the two classes that are not with the rest. The same wrecked stem, folded on the lag it kept: one row per residue class, each drawn at the mean displacement of its own organs against the level the rest of them share. The bar through each row is the spread inside that class, and the widest of them is 0.87° — so within a class the displacement is a constant. six classes sit at the common level. The two that do not sit at 134.3° and -134.2°, equal and opposite to within 0.0 per cent, and they are neighbouring residues. The stem's own divergence is 137.44°, so an exception is one organ's step. One level and two exceptions Inside a wrecked stem's period most residue classes sit at one level and a couple do not. On seventeen of the thirty cuts the exceptions are exactly two, equal and opposite to within five per cent — and on all seventeen they are neighbouring residues, which was not looked for. No majority and no balanced pair, over the whole census. One row per wrecked cut. The bar is the share of that row's residue classes sitting at the level they agree on, and the vertical rule is a half — a bar reaching past it has a majority and a median is safe there. The mark at the right says whether the exchange keeps that row. 15 rows are on both lists of 16 and 16, and the two part on g008/6, which is excluded and has a majority of 5 of 8, and l013/4, whose lag is 4 so that a balanced pair leaves two classes each way and a majority is arithmetically unavailable. No majority and no pair Whether a wrecked stem's chains agree on a level and whether its exceptions form a balanced pair are computed from different things. Sixteen rows have neither, sixteen have neither, and fifteen rows are in both — and the two they part on are the two most interesting rows in the census. The 13 cuts the exchange sets aside, chain by chain. One row per wrecked cut with no balanced pair, one cell per chain of the lag it kept, and the chains displaced away from the common level filled. The lag is printed at the left and the sum of the displacements at the right, folded into half a turn either way. seven of the 13 sum to within 12 degrees of nothing and six do not, and every row that closes kept a lag of 7 or 8 while every row that does not kept 4 or 5. The rows are stacked by that lag. The rows nobody added up Seventeen of the census's thirty wrecked cuts come back as one balanced pair of displaced chains, and every claim about the exchange is quantified over those seventeen. The other thirteen were set aside as having three or more exceptions and never looked at again. They are one addition each.

What links here

Computed from the collection, not written here: the essays that point at this one.

Shares its objects with

Essays that name at least two of the same things, and that neither author linked.

Named objects

A flat tag is an object no other essay names yet.

ArtefactClaim testingDamage profileExceptional chainHonest limitsInstrument settingMeasurementNegative resultRefusalResidue classRobust statisticSummary statistic