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

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.
13 min read 25 figures State the claim, state the testCounted, not admired

Worth reading first: The damage has a period.

Thirteen wrecked cuts have no balanced pair and were added up one at a time. Folded, seven of them 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.

Something sorts them, and it is not what was expected.

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. 1 How far each excluded row is from closing, ordered smallest first, with the lag each cut kept.

The rule

Every row that closes kept a lag of 7 or 8. Every row that does not kept a lag of 4 or 5. Thirteen out of thirteen, on both branches, at every rise in the set.

The lag is the surviving family — the period the displacements repeat at — and it is the quantity the whole ablation thread is indexed by, so it is not an exotic thing to sort by. It is simply not the thing anybody would have reached for.

It is also read off the stem rather than assigned, which matters: a rule that sorted rows by something the design chose would be a rule about the design.

Four accounts of which rearrangements close, scored on 13 cuts. Each candidate rule scored by how many of the excluded cuts it puts on the side the addition does. The lag the cut left standing is right on every one of them; the number of exceptional chains, whether those chains form one run round the circle, and which branch the stem was grown on are each wrong somewhere. The three that lose were written down before the one that wins, which is what makes it a rule that beat rivals rather than the only rule tried.
Fig. 2 Four candidate accounts of which rearrangements close, scored on the excluded rows.

What was reached for first

Three rules, written down before the fourth and scored on the same thirteen rows. All three fail.

The number of exceptional chains is the obvious one: perhaps a row with more exceptions has more chance of cancelling. It is right on eleven of thirteen and puts the one clean three-cycle on the wrong side, since it has three exceptions and closes while a four-exception row does not.

Four accounts of which rearrangements close, scored on 13 cuts. Each candidate rule scored by how many of the excluded cuts it puts on the side the addition does. The lag the cut left standing is right on every one of them; the number of exceptional chains, whether those chains form one run round the circle, and which branch the stem was grown on are each wrong somewhere. The three that lose were written down before the one that wins, which is what makes it a rule that beat rivals rather than the only rule tried.
Fig. 3 The four rules with their scores, of which the count of exceptions comes second.

The second and third

Whether the exceptional chains form one consecutive run round the circle. That is right on five of thirteen, which is worse than a coin.

And the branch — golden or Lucas. That is right on five of thirteen too, and it fails in both directions: each population holds rows from both branches.

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. 4 The excluded rows with their chains and lags, on which the three rival rules were scored.

Why three losers are worth reporting

Because a rule that is right thirteen times out of thirteen means nothing without the rules that are not. Scoring one candidate is a description; scoring four is a comparison, and only the second says the winner beat anything.

That is the same discipline the survivor rule and the exchange’s accounts of its own size are stated under. A file that reported only its best rule would be reporting the search rather than the finding.

Five accounts of the sign, on the 24 lattices that are measurements. Each bar is how many of the lattices an account puts on the right side of zero. The six rows where the second wall is free are left out, because their value is minus the first wall's cost by construction and any rule scores whatever it happens to say about them. Position inside the rung, the rise and the branch all fail. The larger counted number sorts 22 of the 24, and the misses are one rung's worth of rows rather than a scatter.
Fig. 5 Candidate rules scored against each other elsewhere in this collection, which is the habit this follows.

The obvious objection

Thirteen rows sit on seven lattices, several of them cut at neighbouring offsets. So a rule scored over rows is being carried by fewer things than it counts, and a rule about lags could be a rule about which lattices happen to be in the census.

That objection is answerable and it is answered by one lattice.

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 thirteen rows and the lattices they sit on, several of which contribute more than one row.

The lattice that falls both ways

g005 — the golden branch at a rise of 0.005 — contributes three excluded rows. Cut at offset 6 it leaves a stem carrying a lag of 4, and its rearrangement is open at 173.6 degrees. Cut at offsets 7 and 8 it leaves lags of 8, and both close, at 0.2 and 2.6 degrees.

One rise, one seed angle, one stem’s history below the hole — and the split follows which family the cut left standing rather than anything about the lattice it was cut from.

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. 7 The rows that close, two of which come from the same lattice as one that does not.

Which separates the rule from the lattice

That is the whole separation and it rests on one lattice and three cuts. It is thin and it is the right shape: a confound is broken by finding a case where the two candidate causes disagree, and here they do.

What it does not separate is the lag from the number of chains, because they are the same number. A stem that keeps a lag of four has four chains to rearrange, and there is no way to have one without the other.

How far every organ moved, 6 places back at a rise of 0.005. 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 23 organs it settles into a repeating pattern of four levels, one per residue class modulo 4, which is the lag whose hop this stem kept. one of those levels sit together and three do not.
Fig. 8 A profile at a small lag, where four chains carry everything the rearrangement has to do.

What a census with a lag of eleven would say

It would separate them, and one now exists. A lattice one rise below where the census stops keeps a lag of 11 at three offsets, which is a profile with eleven chains in it.

All three of those rows are balanced pairs, so they go to the exchange table rather than to the excluded set and say nothing here. The three cuts at the same lattice that keep a lag of 7 are unpaired, and those do.

The 20 rows of the exchange table, gathered by the lag they kept. One bar per surviving lag, its length the number of rows the table holds at that lag, with the hop that lag's stems keep written beside it. The hop is nearly constant inside a lag, so a correction fitted over 20 rows is fitted over five hops — which is the denominator that matters and is much smaller than the row count suggests. Adding a lag to the table is worth more than adding rows at a lag already in it.
Fig. 9 The lags the extended census keeps, of which the new one contributes to both threads.

Which is an out-of-sample test

The rule says a lag of 7 closes. Three rows at a lag of 7, on a lattice the rule was not fitted on, chosen for reasons that have nothing to do with this reading.

Two of the three close and one does not. Offsets 5 and 6 close at −9.5 and −10.5 degrees; offset 7 is open at −22.6.

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. 10 The three cuts at the new lattice, with the line the rule was drawn at and where each of them falls.

So the rule is a tendency

Fifteen right out of sixteen over both sets, rather than thirteen out of thirteen. That is still a strong rule and it is not a law, and the difference matters because thirteen out of thirteen is the kind of score that stops anybody testing further.

The row that breaks it is not marginal either. At 22.6 degrees it sits inside the open population — which runs from 16.9 to 173.6 degrees — rather than in the gap the line was drawn in.

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. 11 The in-sample rows with the gap the line sits in, against which the out-of-sample failure is well outside.

Which is the useful kind of failure

If the row that broke the rule had landed in the gap, the problem would have been the threshold: a line drawn in six degrees of empty space would have turned out to be delicate, and the reading would have been about the instrument.

It landed well past the gap. So what failed is the claim rather than the line, and that is the outcome that leaves the reading intact and the rule weaker.

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. 12 Where the failing row sits against the line and against the population it joins.

What the threshold is

Twelve degrees, sitting in a gap between 10.5 and 16.9 — six point three degrees of empty space. That is the thinnest gap this thread has drawn a line in and it is reported as such wherever the count is quoted.

The comparable gaps elsewhere are wider relative to their scale: the classes at the common level are within 5.55 degrees of it and the exceptions are 12.4 to 160.4 away.

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

And the control that says the line is not vacuous

The seventeen rows the exchange does cover close within 4.6 degrees, eleven of them within one. They close by construction, since a balanced pair is two displacements that are equal and opposite.

That is the control an addition needs: an instrument that says closed to everything would say it to those seventeen too, and it does, which is the point. What matters is that it says open to six of the thirteen.

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. 14 The rows the addition calls open, against seventeen control rows that close within a fifth of the line.

Why a small lag might not close

No account is offered and one direction is worth naming. The level the displacements are measured against is the median of the class means, so that a few exceptions cannot drag it.

On a profile with four chains and three of them exceptional, the median is one of the exceptions. So the sums on the small-lag rows are measured against a level that is itself displaced, and whether that is enough to open them is not something thirteen rows can say.

A period of 4, 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 0.12° — so within a class the displacement is a constant. one classes sit at the common level. The three that do not sit at 92.7° and -133.4°, equal and opposite to within 36.0 per cent, and they are neighbouring residues. The stem's own divergence is 137.84°, so an exception is one organ's step.
Fig. 15 The classes of a profile most of whose chains are displaced, where the median level is itself an exception.

Which would be a defect rather than a finding

If that is the account, then small lags do not close is a statement about the statistic rather than about the stems, and the repair is a level that does not assume a majority.

That is worth saying plainly, because it is the reading most likely to be right and it is the least interesting one. Testing it would mean recomputing the levels a different way and seeing whether the six open rows close, which is arithmetic on numbers already on disk.

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 116° of scatter.
Fig. 16 Two statistics over one set of numbers, which is the choice a common level is.

Why it was not tested here

Because changing a definition retrospectively makes every earlier number in the thread incomparable, and because the reading as it stands is honest: the rule sorts the rows, the account of why it does is open, and one of the candidate accounts would make the rule an artefact.

Naming the artefact reading as the leading candidate is more useful than quietly not mentioning it, and it is the first thing to check if anybody takes this further.

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. 17 The rows ordered by how far they are from closing, which is what a different level would rearrange.

What the rule is worth as it stands

Fifteen of sixteen rows, one confound broken by one lattice, one candidate account that would make it an artefact, and three rival rules that lose.

That is a modest thing and it is more than the excluded set had before, which was nothing. It also produced a testable next step, which is what a rule with a candidate artefact attached is for.

Four accounts of which rearrangements close, scored on 13 cuts. Each candidate rule scored by how many of the excluded cuts it puts on the side the addition does. The lag the cut left standing is right on every one of them; the number of exceptional chains, whether those chains form one run round the circle, and which branch the stem was grown on are each wrong somewhere. The three that lose were written down before the one that wins, which is what makes it a rule that beat rivals rather than the only rule tried.
Fig. 18 The four rules and their scores, which is the state of the reading.

What it says about the census’s shape

One structural remark. The census’s rows at small lags come from a particular stretch of the ladder rather than from across it, because which lags a cut can keep is decided by the rungs.

So lag and position on the ladder are correlated in this census, and a third confound is live: the rule could be about the coarse half of the ladder rather than about the lag. The g005 split does not break that one, since all three of its rows sit at one rise.

The golden ladder, rung by rung. Each bar is one rung — a run of rises over which a counter returns one pair — drawn from its coarse end on the left to its fine end on the right, with the whole bar scaled to the same width so that positions inside different rungs can be compared. The mark on each bar is the rise at which the two contact steps change places, and it falls between 12 and 35 per cent of the way down on every rung that has one. The dots are the lattices this collection's ablation census was grown at, dropped onto the rungs they belong to. They are not spread across the bars; seven of them sit past the mark.
Fig. 19 Where the census’s lattices sit on the ladder, which is what the lag is correlated with.

What would settle the artefact question

Recomputing each row’s common level without assuming a majority, and re-adding. If the six open rows close under a level that is not the median, the rule is about the statistic; if they stay open, it is about the stems.

That is one function and no new stems, so it is minutes. The reason it is not done here is that it would change a definition the rest of the thread uses, and a definition changed inside one reading is a definition two readings disagree about.

The right shape for it is a separate reading that recomputes both and reports the pair, which is what the reading window was given when the same question arose about a spread.

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. 20 The classes of a profile and the level they are measured against, which is what a second statistic would move.

Two more things the rule could be about

Neither is separable here and both are worth naming so that a later reading knows what to break.

The rise: small lags in this census come from coarser lattices, so the rule could be about coarse stems. And the fraction of the profile that is exceptional: at a lag of four, three exceptions is three quarters of the chains, and at a lag of eight it is three eighths.

The second is the same confound as the lag itself wearing different clothes, and it is why a census with a lag of eleven in its excluded set would be worth more here than three more rows at a lag of five.

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. 21 The excluded rows by lag, in which the fraction of chains flagged falls as the lag rises.

What a perfect in-sample score is

Worth a paragraph on its own, because thirteen out of thirteen is a number that reads as strength and is closer to a warning. A rule read off a set of rows and then scored on the same rows can only lose to a rule that is worse at describing them.

The three rivals here were written down before the winner and lost, which makes the comparison worth something. It does not make the winner tested, because all four were scored on the rows that produced them.

The distinction is between a rule that beat rivals and a rule that survived a case it had not seen, and only the second is a test. This one is now both, which is why the essay can say fifteen of sixteen instead of thirteen of thirteen.

The readings, and where in their rungs they fail. Each bar is one candidate account of which family a wrecked stem keeps, scored across every wrecked cut in the census. Under each bar are the positions inside their own rungs of the cuts it gets wrong, as percentages from the coarse end. The best of them is right 25 times of 30, and the positions of its failures are the point: two of them are the single lattice grown at the far fine end of its rung, which is also the only census row past three quarters of the way down. Nothing here rescues a reading. What it shows is that the table these readings were scored on varies a quantity nobody chose, over a range nobody stated.
Fig. 22 A rule’s misses reported beside its hits, which is how a score is kept honest in this collection.

Where the out-of-sample case came from

Not from looking for one. A search of the fine end of both branches was run to find a lattice whose cuts keep a lag outside the census’s four, for an entirely different thread.

It returned one lattice with six wrecked cuts on it. Three answer the question that search asked; the other three are unpaired, keep a lag of 7, and land here.

Nobody designed the second half, and that is the ordinary return on measuring a whole object rather than the part a question needs. It is the same reason the census cuts every offset rather than the ones expected to wreck.

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.
Fig. 23 The search that produced the out-of-sample rows, run for a different thread entirely.

What a reader should carry

That the lag a cut leaves standing sorts the excluded rows thirteen times out of thirteen in sample, that three rival accounts lose to it, and that one lattice’s own cuts falling both ways is what stops it being a statement about lattices.

And that out of sample it is fifteen of sixteen. A rule with a perfect in-sample score is a rule that has not been tested, and the lattice that tested this one was found by a different thread looking for something else.

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. 24 The out-of-sample rows, two of which the rule gets right and one of which it does not.

What the picture at the top shows

Thirteen bars, one per excluded cut, ordered by how far that row’s displacements are from summing to nothing. The label on each names the lattice and the offset, and the note beside it names the lag the cut kept.

The seven shortest bars are lags of 7 and 8 and the six longest are lags of 4 and 5. The shaded stretch at the left is the line the rows are called closed inside, and there is nothing in the gap between the two groups.

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. 25 The thirteen rows once more, sorted by their distance from closing and labelled by lag.

The one line

Every excluded row that closes kept a lag of 7 or 8 and every row that does not kept 4 or 5 — thirteen out of thirteen, against three rival accounts that score eleven, five and five — and one lattice’s own three cuts fall on both sides in step with their lags.

Out of sample it is fifteen of sixteen, the row that breaks it sits well outside the gap the line was drawn in, and the leading candidate account would make the whole rule a property of how the common level is computed.

Where this goes next

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. 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. 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.

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.

Claim testingConfoundingContact familyDamage profileExceptional chainExchangeFalsifiabilityHonest limitsOut of sampleResidue classRule scoringSample size