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

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.
12 min read 24 figures State the claim, state the testCounted, not admired

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

A wrecked stem’s displacement profile folds onto the lag it kept, giving one chain per residue. Most chains sit at a common level and a few do not, and the ones that do not are what the exchange is about.

Seventeen of the census’s thirty wrecked cuts come back as one balanced pair: two adjacent chains displaced in opposite directions by the same amount. The exchange’s direction, its location and its size are all quantified over those seventeen.

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. 1 The cuts with no balanced pair, one row per cut, one cell per chain, with the exceptional chains filled.

The other thirteen

They were set aside as three or more exceptional chains, which is a reasonable thing to write and is not quite what the set is.

Eleven of the thirteen do carry three or more exceptions. The other two carry exactly two — l020 cut at offsets 4 and 5 — and they are outside the exchange because their two exceptions are both displaced the same way: at −89.7 and −64.8 degrees on one, and −89.9 and −137.0 on the other.

Both are cuts of the same lattice at neighbouring offsets, which is the ordinary shape of a census row and is why the two are one observation rather than two.

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. 2 The excluded rows whose displacements do not cancel, including the two that carry only two exceptions.

A pair that is not equal and opposite is not a pair

That is the whole of the discrepancy and it is worth naming because it changes a denominator. The exclusion the exchange actually makes is rows with no balanced pair, which is thirteen. The exclusion it was described with is rows with three or more exceptions, which is eleven.

Nothing was computed wrongly. The code has always used the first and the prose has always said the second, and the two differ by two rows that nobody would have found without adding them up.

A period of 5, 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. three of the five 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 1 and the backward one is chain 2, one residue above it, which is the order every row of the census puts them in.
Fig. 3 What a balanced pair looks like in a profile, which is the shape thirteen rows do not have.

What balanced means, exactly

Two chains, adjacent in the residue labelling, displaced in opposite directions by amounts agreeing to within a twentieth of their mean. The bound is loose enough for the azimuth grid and tight enough that an unequal pair fails.

The claim it tests is that a pair of adjacent chains is displaced in opposite directions by the same amount, which is what a chain slipped by one place would do. A swap would give a different signature in the quantity that measures the slip.

How far every organ moved, 5 places back at a rise of 0.013. 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 40 organs it settles into a repeating pattern of five levels, one per residue class modulo 5, which is the lag whose hop this stem kept. two of those levels sit together and three do not.
Fig. 4 A profile folded onto the lag it kept, in which a balanced pair is two classes off the common level.

So what are the thirteen

Rows with three, four, five or six exceptional chains, plus the two with an unbalanced pair. The counts are: four rows with three exceptions, two with four, two with five, three with six, and the two with two.

That is a wider range than three or more suggests. Six exceptional chains out of eight is most of the profile off the common level, which is a different object from a pair with a stray attached.

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. 5 The thirteen rows stacked by the lag they kept, with the number of exceptional chains visible as filled cells.

The question they were set aside with

It was specific: are three exceptions a three-cycle — three chains rotating into one another’s places — or a pair plus a stray? And the test proposed for it was that a three-cycle’s displacements sum to zero.

That is an addition per row and nothing had done it. Thirteen additions, on numbers that have been on disk since the census was built.

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. 6 The excluded rows whose displacements do cancel, which is the half the question was about.

The test as proposed is wrong

Three chains that rotate into one another’s places each move about a third of a turn the same way round, and a third of a turn three times is a whole turn rather than nothing.

That is not a quibble. The census contains exactly one clean three-cycle and the unfolded test calls it the worst row in the table: its displacements sum to −359.6 degrees, which is the largest sum in the census by a factor of two and is one turn to four parts in ten thousand.

Three chains, each moved a third of a turn, on g008 cut at offset 6. The eight chains of a stem that kept a lag of 8, set round the circle, with the three that are displaced marked and their displacements written on. Each is about a third of a turn the same way round, and they sum to -359.6 degrees — one whole turn to within half a degree, on an azimuth grid whose step is a quarter of a degree. A rearrangement that closes by going once round sums to a turn, not to nothing, which is why the test it was set had the wrong number in it.
Fig. 7 The one clean cycle, drawn round its own period, with each chain moved about a third of a turn.

So every sum is folded

Into half a turn either way, with the raw value carried beside it so the distinction stays visible. On twelve of the thirteen rows the two readings agree; on the thirteenth they differ by a whole turn and the folded one is the right one.

That is the correction this reading makes to the question it was given, and it is worth more than the answer, because a test with the wrong number in it would have rejected the one row that answers it.

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. 8 Each row’s raw sum beside its folded one, with the row where the two readings disagree.

What the additions give

Folded, seven of the thirteen sum to within ten and a half degrees of nothing and six sit sixteen and a half to a hundred and seventy-four degrees away.

So the excluded set is not one thing. About half of it is a closed rearrangement — the chains end where chains began — and about half is not, and that split is decided by something specific.

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. 9 How far each excluded row is from closing, ordered smallest first.

The gap the line sits in

Ten and a half degrees on one side and sixteen and nine tenths on the other, so the threshold at twelve sits in a gap of six point three degrees.

That is the thinnest gap this thread has drawn a line in and it is reported as such. The comparable gaps elsewhere are wide: the classes at the common level are within 5.55 degrees of it and the exceptions are 12.4 to 160.4 away, which is a gap of nearly seven on a scale ten times larger.

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. 10 The distances from closing, with the shaded stretch showing where the line sits.

The control, which is the seventeen

A test that says yes to everything says nothing, so the seventeen rows the exchange does cover were added up too. They close within 4.6 degrees, and eleven of the seventeen within one.

They close by construction: a balanced pair is two displacements that are equal and opposite, so their sum is nothing to within the tolerance that defines them. That is exactly why they are the control — the addition returns what it must on rows where the answer is known.

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. 11 The excluded rows that close, against which the seventeen kept rows close within a fifth as much.

What a closed rearrangement means

That the displacements of the chains that moved cancel, once folded. Physically: the organs that went somewhere else went to places other organs vacated, and nothing was left over.

An open one means they did not. Something in the profile has a net displacement that the exceptional chains do not account for, which either means the common level is not where the median put it or that the rearrangement is not confined to the chains flagged.

A period of 5, 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 1.62° — so within a class the displacement is a constant. two classes sit at the common level. The three that do not sit at 67.7° and -149.5°, equal and opposite to within 75.3 per cent, and they are neighbouring residues. The stem's own divergence is 136.78°, so an exception is one organ's step.
Fig. 12 The classes a profile folds into and the common level most of them sit at, which the sum is measured against.

The second of those is checkable

Adding every class rather than only the exceptional ones gives almost the same answer on every row: the two agree to within a degree and a half throughout.

That is what it should do, since a class at the common level contributes nothing by construction. It is worth doing because it rules out the reading that the open rows are open because of many small displacements adding up rather than because of the flagged ones.

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. 13 The rows whose displacements do not cancel, whose non-exceptional classes contribute almost nothing.

What the thirteen are worth

They are a sixth of the census’s thirty rows plus a bit, they were excluded for a good reason, and the reason has nothing to do with them being uninformative.

The exchange is defined on a balanced pair. A row without one has no exchange, so it cannot contribute to a claim about direction, location or size. What it can contribute to is a claim about rearrangements more generally, and nothing had asked for one.

How far every organ moved, 6 places back at a rise of 0.008. 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 14 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. five of those levels sit together and three do not.
Fig. 14 A profile with more than two classes off the common level, which is what an excluded row is.

The habit this is an instance of

A design excludes rows for a stated reason and the excluded set becomes invisible. It is not a mistake — the exclusion is right — and the set goes on sitting in the table with nobody reading it.

The same thing happened to the endpoint column of the slot design, which every row carries and no claim uses. The cheapest audit of any table in this collection is to look at what its design does not read.

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. 15 The set a design set aside, which sat unread while every claim was made about its complement.

What it cost to read them

An addition per row on numbers already computed. Nothing was grown, nothing was swept, and the whole reading is arithmetic over a table that has been on disk since the damage census was built.

That is the cheapest thing in this collection to have produced a finding, and the finding is the one in the essay about the cycle.

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. 16 The same sums with nothing highlighted, which is the whole computation this reading performs.

What is not claimed

That the thirteen are a population worth generalising over. Thirteen rows sit on seven lattices, several of them cut at neighbouring offsets, so the independent observations number seven at best.

Every claim made about them is stated with that denominator attached, and the one rule that does sort them is tested again on a lattice from outside the census rather than left at its in-sample score.

The rule tested on a lattice the census does not hold. The three unpaired cuts on the lattice the fifth-lag search turned up, all of them at a lag of 7, which the rule says must close. Two do and one does not. The line the rule was drawn with sits in the gap between the closed and open populations of the original census, and the row that breaks the rule sits well past it — so what failed is the claim rather than the threshold. Out of sample the rule is right 15 of 16 times rather than 13 of 13.
Fig. 17 The rule tested on a lattice the census does not hold, which is where its in-sample score does not stand.

Where the common level comes from

One detail the additions depend on. The level most chains sit at is taken as the median of the class means rather than their mean, so that a handful of exceptions cannot drag it.

That choice is what makes an exception an exception. With a mean, a row with six large displacements out of eight would have a level pulled towards them and the exceptions would partly cancel themselves out of existence; with a median it stays where the quiet classes are as long as most of them are quiet.

On a row with six exceptions out of eight, most of them are not quiet, and the median is one of the exceptions. That is a real limit of the reading and it applies to three of the thirteen.

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 1.01° — so within a class the displacement is a constant. two classes sit at the common level. The six that do not sit at 132.7° and -45.0°, equal and opposite to within 98.7 per cent, and they are neighbouring residues. The stem's own divergence is 137.84°, so an exception is one organ's step.
Fig. 18 The classes of a profile most of whose chains are displaced, where the median level is itself an exception.

What that does to those three rows

It does not obviously break them. Two of the three close and one does not, which is the same split the other ten show, and their sums are among the smallest in the table.

But it is worth flagging rather than buried, because a sum measured against a level that is itself displaced is a sum with an offset in it. On a row with six of eight chains flagged, what common level means is doing more work than the phrase admits.

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. 19 The rows that close, three of which have most of their chains off the level the rest are measured against.

And what a better level would be

The mode of the class means, or the level that most of them cluster at by some measure that does not assume a majority. Nothing was changed, for the reason that runs through this collection: altering a definition retrospectively makes every earlier number incomparable.

It is written down as the thing to try if the excluded rows ever carry a claim heavier than the ones made here.

Both statistics, on the same stems, at a rise of 0.005. Five seeded stems at each disturbance, held at a fixed rise. Bars are how many returned the pair the position counter finds; open portions are refusals. The pair comes out from 0.1 to 0.25, and across that whole range the lag-one correlation of the same sequences is -0.33, -0.58, -0.59 — decisive, negative and flat. There is no trade between the two: one stem supplies both. Below the window the sequence has locked onto the sampling grid and is a cycle rather than a sample; above it there is no lattice left, at 117° of scatter.
Fig. 20 Two statistics over one set of numbers, which is the choice a common level is.

The two unbalanced rows, on their own

They deserve a paragraph, since they are the reason the count is thirteen. Both are l020 — the Lucas branch at a rise of 0.020 — cut at offsets 4 and 5, and both keep a lag of 4.

Offset 4 displaces two chains by −89.7 and −64.8 degrees; offset 5 displaces two by −89.9 and −137.0. In both cases the two chains move the same way round, so nothing cancels and the sums are −154.4 and +133.1 degrees folded.

Two chains moving the same way is not a rearrangement that closes and it is not obviously anything else either. With a lag of four there are only four chains, so two of them moving together is half the profile.

How far every organ moved, 4 places back at a rise of 0.02. 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 30 organs it settles into a repeating pattern of five levels, one per residue class modulo 5, which is the lag whose hop this stem kept. three of those levels sit together and two do not.
Fig. 21 One of the two rows whose exceptions are not equal and opposite, whose two displaced chains go the same way.

Which is where a small lag bites

A profile at a lag of four has four classes, and a profile at a lag of eight has eight. The same number of exceptional chains is a very different fraction of the two.

Three exceptions out of four leaves one quiet class to define a level from; three out of eight leaves five. So the excluded rows at small lags are systematically harder to read than the ones at large lags, and that difference turns out to sort them — though whether it is the lag or the number of chains is not separable here, since they are the same number.

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. 22 The excluded rows stacked by lag, where the rows with fewest chains have most of them flagged.

What a reader should carry

That the exchange’s excluded set is thirteen rows rather than eleven, because two of them carry a pair that is not equal and opposite rather than three or more exceptions.

And that adding up thirteen rows of numbers already on disk produced a split — seven closed and six open — and a correction to the test that was proposed for them, which had the wrong number in it.

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. 23 The thirteen rows ordered by how far they are from closing, which is the split the additions found.

What the picture at the top shows

Thirteen rows, one per excluded cut, stacked by the lag each kept. Each row has one cell per chain of that lag, and a filled cell is a chain displaced ten degrees or more from the level the rest sit at.

The number on the right is the sum of that row’s displacements, folded into half a turn either way. The rows near the top have small sums and the rows near the bottom have large ones, and the ordering by lag is why.

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. 24 The thirteen rows once more, with their chains, their lags and their sums.

The one line

The exchange sets aside thirteen of the census’s thirty wrecked cuts, not eleven: two of them carry exactly two exceptional chains displaced the same way, which is a pair only in the sense of a count.

Added up — one addition each, on numbers already on disk — seven of the thirteen close to within ten and a half degrees and six sit sixteen and a half to a hundred and seventy-four degrees away.

Where this goes next

Three chains, each moved a third of a turn, on g008 cut at offset 6. The eight chains of a stem that kept a lag of 8, set round the circle, with the three that are displaced marked and their displacements written on. Each is about a third of a turn the same way round, and they sum to -359.6 degrees — one whole turn to within half a degree, on an azimuth grid whose step is a quarter of a degree. A rearrangement that closes by going once round sums to a turn, not to nothing, which is why the test it was set had the wrong number in it. A cycle sums to a whole turn The test proposed for whether three displaced chains are a three-cycle was that their displacements sum to zero. Three chains rotating into one another's places each move about a third of a turn the same way round, and a third of a turn three times is a whole turn — which the unfolded test calls the worst row in the census. 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. The lag decides whether it closes Seven of the thirteen excluded rows have displacements that cancel and six do not. Every row that closes kept a lag of seven or eight and every row that does not kept four or five, thirteen times out of thirteen — and then a lattice nobody had cut broke it. 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.

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.

AblationBalanced pairCensus designClaim testingDamage profileExceptional chainExchangeHonest limitsMeasurement errorNegative resultResidue classSelection effect