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

The twenty-first row

Recomputing the level moves one row out of the set the exchange sets aside and into the exchange itself. Its hop is four times larger than any the correction was fitted over, and the correction fails on it in the one way it had never failed.
15 min read 20 figures State the claim, state the testCounted, not admiredForm is not mechanism

Worth reading first: The damage has a period.

The exchange is what a wrecked stem does to two adjacent chains of organs: it displaces one forwards and the other backwards by the same amount. Its size has an account — one step of the control’s divergence, less about a fifth of the surviving hop’s own angle — and the account is right in sign on every row it was fitted over.

Recomputing the common level moves exactly one row into the exchange and none out of it. The row breaks the account.

g005 at offset 6, read against two levels. One wrecked cut's four residue classes, drawn twice. On the left the level is the median of the four class means, which falls on the class at -5.5 degrees sitting on its own; three classes are then exceptions and the row is set aside as having no balanced pair, summing to -173.6 degrees. On the right the level is the pair of classes that agree, at -138.6 degrees; two classes are then exceptions, they are adjacent and equal and opposite at 133.1 and 134.2 degrees, and the row is an exchange. The exchange table goes from 20 rows to 21.
Fig. 1 The row read against two levels: three exceptions and no pair under one, a balanced adjacent pair under the other.

The row

g005 cut at offset 6, on the golden branch at a rise of 0.005, keeping a lag of four. Four residue classes, whose means are 87.2, −138.9, −138.4 and −5.5 degrees.

The median of four values is the second when sorted, which lands on −5.5 — a class sitting on its own, with the nearest other class more than eighty degrees away. Against that level three of the four classes are exceptions and the row has no balanced pair.

The densest cluster is the two classes at about −138.6. Against that level, two classes are exceptions, they are adjacent, and they are −134.2 and +133.1 degrees — equal and opposite to within eight parts in a thousand.

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. 2 The four class means and the two candidate levels, one of them a class sitting on its own.

So it is an exchange

By the exchange’s own definition, which is exactly two exceptional chains, displaced in opposite directions by the same amount to within a twentieth. Under the recomputed level this row satisfies it.

The two chains are adjacent — chains 1 and 2 out of four — which is the shape a chain slipped by one place would produce and is what every other row in the table shows.

The chain displaced forwards is one residue below the chain displaced backwards, which is the exchange’s orientation claim, and it now holds on twenty-one rows rather than twenty.

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. 3 The two chains that change places in a wrecked profile, which is what the orientation claim is about.

What the account predicts there

The hop is the angle from an organ to the one four places above it on the control, which at a settled divergence of 137.84 degrees is four times that, folded: −168.63 degrees.

That is very nearly a half turn, and it is negative. The account is one divergence step less a fifth of the hop, with the hop signed — so on a negative hop the correction adds.

Predicted size: 137.84 − 0.2 × (−168.63) = 171.57 degrees.

What the exchange misses one divergence step by, against the surviving hop. Each mark is one wrecked cut. Across: the angle from an organ to the one its surviving lag places above it, on the control. Down: how much the exchanged pair's size falls short of one divergence step. The relationship is a straight line through the origin with a slope of about 0.2, and its sign is right on every row of the census — the four surviving lags give hops on both sides of zero and the residual follows each of them. The honest limit is the four: the hop hardly moves inside a lag, so the line rests on 11 distinct positions and not on 17.
Fig. 4 The exchange’s shortfall against the surviving hop, with the fitted line the account is drawn from.

What is measured

133.65 degrees, which is 4.19 degrees below one uncorrected divergence step rather than 33.7 above it.

As a share of the divergence, the corrected account is wrong by 27.5 per cent. Its worst row before this was 4.0 per cent. One uncorrected divergence step — the account with no correction at all — is wrong by 3.0 per cent.

So on this row the rule is wrong by seven times its own worst error, and doing nothing is nine times better than doing the correction.

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, on its worst row. 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. 5 The candidate accounts of the exchange’s size, and how badly each does on its worst row.

And the sign is wrong

That is the part that matters. The account’s strongest claim is not its coefficient; it is a claim about direction: the residual runs against the hop, on every row, with the hop’s own angle falling on both sides of zero across the census.

Twenty rows, twenty times the residual and the hop on opposite sides of nothing. That was quoted as one chance in a million under a coin and it was the reason to believe the rule was about the hop at all.

Here the hop is −168.63 and the residual is −4.19. Both negative. It is the first row in the census where they run the same way.

What the exchange misses one divergence step by, against the surviving hop. Each mark is one wrecked cut. Across: the angle from an organ to the one its surviving lag places above it, on the control. Down: how much the exchanged pair's size falls short of one divergence step. The relationship is a straight line through the origin with a slope of about 0.2, and its sign is right on every row of the census — the four surviving lags give hops on both sides of zero and the residual follows each of them. The honest limit is the four: the hop hardly moves inside a lag, so the line rests on 11 distinct positions and not on 17.
Fig. 6 The residual against the hop with no line through it, where every row before this one has them on opposite sides.

Why the hop is so large

Because four times 137.84 degrees is 551.4, which folds to −168.6. A lag of four on a golden lattice puts an organ nearly opposite the one four places below it.

The census’s other hops run from 12.78 to 39.14 degrees in magnitude, because their lags are 5, 7, 8 and 11 and those multiply the divergence to something near a whole number of turns. Four does not.

So this row is not a slightly larger version of the others. It is at a hop four times the largest and thirteen times the smallest, in a regime the rule has never been asked about.

The hops the correction is fitted over, with the new one at the near end. Each cluster of rows placed by the lag it kept and the angle of the hop that lag keeps. The four the correction was fitted over run from 19.5 to 39.1 degrees; the new one sits at 12.78 degrees, a third smaller than any of them. A fifth point beyond the near end of a fitted range is a test of the fit, where a fifth point between two old ones would mostly have been a restatement.
Fig. 7 The hop clusters the correction is fitted over, against which this row’s hop is off the scale.

Which is what the rule’s own file said

The correction is quantified over twenty rows and its honest denominator is five, because the hop is nearly constant inside a lag and the census keeps five lags. A rule fitted over five clusters spanning 12.78 to 39.14 degrees is a rule about that range.

The round that added the fifth cluster said so explicitly and treated a point beyond the near end of the range as the interesting test. This is a point far beyond the far end, and the rule does not survive it.

That is a refutation in the useful direction. The account was not wrong about the rows it described; it was wrong about how far it reaches, and the file that stated its denominator was right to state it.

The correction's coefficient at each of the five lags. How far the exchange falls short of one step of the control's divergence, as a share of the surviving hop's own angle, at every row of the table. The stated rule is a fifth and the least-squares fit over the whole table is 0.184. The new lag's three rows sit at 0.164, 0.163, 0.162, inside the spread the four older lags already covered rather than beyond it.
Fig. 8 The correction’s coefficient at each lag, with the stated fifth and the fitted value drawn as rules.

What it does to the fit

The least-squares coefficient over the whole table moves from 0.203 to 0.049, which is not a third digit changing. A single row at a hop four times the largest dominates a least-squares fit by construction, so the number is not meaningful either before or after.

The right response is not to refit. It is to say that the fit was over five clusters spanning a factor of three, that a sixth cluster at a factor of thirteen breaks it, and that the coefficient is a description of the first five.

Refitting over all six would produce a rule that describes none of them.

What the exchange misses one divergence step by, against the surviving hop. Each mark is one wrecked cut. Across: the angle from an organ to the one its surviving lag places above it, on the control. Down: how much the exchanged pair's size falls short of one divergence step. The relationship is a straight line through the origin with a slope of about 0.2, and its sign is right on every row of the census — the four surviving lags give hops on both sides of zero and the residual follows each of them. The honest limit is the four: the hop hardly moves inside a lag, so the line rests on 11 distinct positions and not on 17.
Fig. 9 The fit with and without the new row, which moves a coefficient by a factor of four.

What survives

The two claims that are not about size. The orientation — forwards chain one residue below backwards chain — holds on twenty-one of twenty-one. The location — the backward-displaced chain is the hole’s own — was thirteen of twenty and is now thirteen of twenty-one, because the new row puts it elsewhere.

Those are the claims about which chains and which way round, and they are shift-invariant: they are comparisons between classes, so recomputing the level cannot move them.

What breaks is the claim about how much, which is the only one measured against a level.

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.
Fig. 10 Which chain carries the backward displacement across the census, including the new row.

Whether the row should be in the table at all

It satisfies the exchange’s stated criterion, so yes. But it is worth stating what makes it unlike the other twenty.

Its level is a cluster of two out of four, which is not a majority and cannot be — a lag of four with a balanced pair leaves two chains at the level. Every other row in the exchange has a majority.

So the row is admitted by a definition and is at the edge of what the reading can support. That is exactly where a rule should be tested, and it is also a reason to report it separately rather than dissolved into a table of twenty-one.

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 The row’s cluster size against the rest of the census, where every other exchange row has a majority.

The size is not small

133.65 degrees, against a control divergence of 137.84. The other twenty rows’ sizes run from about 97 to 145 degrees, so this one is comfortably inside the range.

That rules out the reading in which the row is a marginal detection — two chains barely displaced, admitted by a loose tolerance. Both chains are thrown more than 130 degrees and they are equal and opposite to eight parts in a thousand.

What is unusual is the hop, not the exchange. The two chains do exactly what the account says two chains do; what they do not do is fall short of a divergence step by a fifth of a hop that is nearly a half turn.

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. 12 A wrecked profile with a balanced pair, of the kind this row turns out to be.

A reading the account could take

That a hop near a half turn is not a hop in the sense the rule means. The correction is motivated by the two exchanged chains being a hop apart as well as a step apart, so the relaxation between them carries a share of the hop’s angle.

At a hop of −168.6 degrees the two chains are nearly opposite, and a fifth of the angle between two nearly opposite things is not obviously the same quantity as a fifth of a twenty-degree gap.

That is a plausible restriction and it is not testable here. It would need several rows at large hops, and the ladder offers one — this one.

Which offsets give short hops, at a rise of 0.005The two lowest points are at 8 and 13, and those are the parastichy numbers. Offset 1 is high because a hop of one node is at least the rise, which is what makes a stem easier to count than a disc.00.50011.5051015index offsetmedian hop between node i and node i+m2324 nodes, 18 offsets triedshortest at 2 and 3
Fig. 13 The hop lengths on this row’s own lattice, of which the surviving lag’s is nearly a half turn.

What is not claimed

That the exchange’s account is wrong about the rows it was fitted on. It is right in sign on all twenty of them, right in order of magnitude on the fifth cluster it predicted, and better than every rival tried on the same rows.

What is claimed is that its reach was never established and is now bounded from one side: it holds over hops from 12.78 to 39.14 degrees and fails at 168.63.

Between 39 and 168 there is nothing, because the ladder’s lags do not produce a hop there. That gap is a fact about the object rather than about the work, and it means the boundary cannot be located.

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. 14 The lags the census keeps, whose hops leave a gap between thirty-nine and a hundred and sixty-eight degrees.

What a twenty-first row is worth

More than the twentieth. The nineteenth and twentieth rows were the fifth hop cluster, which extended the fitted range downwards by a third and confirmed the rule.

This one extends it upwards by a factor of four and refutes it. Both are single additions to a table of seventeen, and the asymmetry in what they bought is entirely about where in the range they landed.

That is the general lesson and it was already written down: a point between two existing ones is mostly a restatement, and a point beyond the end of a fitted range is where a rule and its neighbours come apart.

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. 15 The accounts scored on average, where the corrected rule’s advantage over doing nothing is small.

How the row was found

Not by looking for it. The recomputation was run to test an objection to a different rule — whether the level a rearrangement’s chains are measured against is itself an exception — and it reads all thirty-six rows of the extended census the same way.

One of the thirty-six turned out to have a balanced pair under the new level. That is the whole of how it arrived: no search, no criterion tuned, no row singled out.

Which matters for how the result is read. A row found by looking for rows that break a rule is a row selected for breaking it; a row that arrives as a side effect of a different check is not. The second is worth more and it is what happened here.

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. 16 The rows the exchange sets aside, one of which turns out to have a balanced pair under a recomputed level.

The lattice it comes from

g005 is the golden branch at a rise of 0.005, which is below the ladder’s finest rung and one of the census’s own ten lattices. Its cuts keep lags of 4 and 8, and this is its only row at a lag of four.

The same lattice contributes four other rows to the census: one at offset 4 keeping 8, which is in the exchange, and three at offsets 7, 8 and 9. Two of those three are the rows with three tied clusters of two — the weakest profiles in the census.

So this lattice is where the awkward rows live, and the one that changes side comes from it. That is a caution rather than a disqualification: the row satisfies every stated criterion, and its lattice’s other rows failing different criteria is not evidence about it.

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. 17 The census’s ten lattices, of which one contributes most of the awkward rows.

Why a lag of four is where this had to happen

Because the hop is the lag times the divergence, folded. On a golden lattice at 137.5 degrees, the lags that give a small hop are the ones near a multiple of a full turn: 5 gives 47.5, 8 gives 20.1, 13 gives 7.6, 21 gives 2.9.

Four gives 190, which folds to −170. Three gives 52.5. Two gives 275, folding to −85.

So the small-hop lags on a golden lattice are the Fibonacci numbers, which is not a coincidence — it is the same fact that makes them the contact families. A lag off that sequence has a large hop by construction.

The census’s other lags are 5, 7, 8 and 11, all of them on one ladder or the other. Four is the only non-member, and it is the only large hop.

Which offsets give short hops, at a rise of 0.005. The two lowest points are at 8 and 13, and those are the parastichy numbers. Offset 1 is high because a hop of one node is at least the rise, which is what makes a stem easier to count than a disc.
Fig. 18 The hop lengths by lag on this lattice, where the ladder’s own numbers give small hops and four does not.

Which suggests the boundary is not about size

If the rule fails at 168 degrees because 168 is large, the boundary is somewhere between 39 and 168 and cannot be located. If it fails because the surviving lag is off the ladder, the boundary is categorical and there is one case of it.

Nothing here separates those. The one row with a large hop is also the one row with an off-ladder lag, and there is no lattice on this ladder that would produce one without the other.

That is worth writing down as an open question with the reason it cannot be closed, rather than as a caveat. The two readings make different predictions about a hypothetical row and no such row is available.

The 17 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 17 rows is fitted over four 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. 19 The lags the census keeps, of which one is off both ladders and carries the large hop.

What a rule fitted to five points is worth

Worth saying in general, because this thread keeps producing them. A correction with one free coefficient fitted over five clusters is a line through five numbers, and five numbers can be fitted by a great many things. What such a fit can establish is a sign and an order of magnitude; what it cannot establish is a functional form, and the exchange’s own file says so where it scores the rule.

The five clusters span 12.78 to 39.14 degrees, a factor of three. Over a factor of three a proportional shortfall and a constant one are distinguishable in principle and not comfortably — which is why the round that added the fifth cluster treated a point a third below the previous minimum as worth having: it discriminated mildly in favour of the proportional reading, and mildly was the honest word for it.

A point at four times the top of the range is a different kind of test, and it is the kind a five-point fit is least likely to survive. Nothing about that is a criticism of the fit. It is the ordinary arithmetic of extrapolation, and the useful thing this round adds is the measurement rather than the surprise.

What would rescue the correction is a mechanism that predicts where it stops. A fifth of the hop is motivated by the two exchanged chains being a hop apart as well as a step apart, and nothing in that motivation says the relaxation stays proportional when the hop approaches a half turn — where the two chains are nearly opposite and the word between stops meaning much. That is a plausible restriction, it is not testable on this ladder, and it is written down here so that a ladder which does offer the test can be pointed at it.

Where the twenty-first row leaves the thread

With a correction that describes twenty rows well, fails on the twenty-first, and has a stated range. That is a better position than a rule with no known boundary, and it is a worse one than the round before this reported.

The three claims the exchange makes are now in three different states. The orientation holds on twenty-one of twenty-one and is the strongest thing in the file. The location at the hole is thirteen of twenty-one, reported as a tendency and unchanged. The size is accounted for over a factor of three in the hop and refuted at four times the top of it.

Reporting them separately is the point. A file that summarised itself as the exchange is understood would be carrying one refuted claim inside two that hold.

What is claimed

That recomputing the common level moves g005 at offset 6 into the exchange table, giving twenty-one rows rather than twenty, and moves none out.

That the row’s surviving lag is four, its hop is −168.63 degrees — four times the largest the correction was fitted over — and its measured size is 133.65 against a predicted 171.57, an error of 27.5 per cent of the divergence where the rule’s worst row was 4.0.

And that it is the first row in the census whose residual runs with its hop rather than against it, which is the claim the correction rested on and the one it had never lost.

g005 at offset 6, read against two levels. One wrecked cut's four residue classes, drawn twice. On the left the level is the median of the four class means, which falls on the class at -5.5 degrees sitting on its own; three classes are then exceptions and the row is set aside as having no balanced pair, summing to -173.6 degrees. On the right the level is the pair of classes that agree, at -138.6 degrees; two classes are then exceptions, they are adjacent and equal and opposite at 133.1 and 134.2 degrees, and the row is an exchange. The exchange table goes from 20 rows to 21.
Fig. 20 The row that changes side, and the two readings of the same four numbers that put it on either.

Where this goes next

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. 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. What the exchange misses one divergence step by, against the surviving hop. Each mark is one wrecked cut. Across: the angle from an organ to the one its surviving lag places above it, on the control. Down: how much the exchanged pair's size falls short of one divergence step. The relationship is a straight line through the origin with a slope of about 0.2, and its sign is right on every row of the census — the four surviving lags give hops on both sides of zero and the residual follows each of them. The honest limit is the four: the hop hardly moves inside a lag, so the line rests on 11 distinct positions and not on 17. A fifth of the hop The exchanged pair misses one divergence step by up to twelve per cent, and the miss is not scatter: every row keeping a lag of 5 or 7 overshoots and every row keeping a 4 or an 8 falls short. Subtract a fifth of the surviving hop's own angle and the worst row is four per cent. The hops the correction is fitted over, with the new one at the near end. Each cluster of rows placed by the lag it kept and the angle of the hop that lag keeps. The four the correction was fitted over run from 19.5 to 39.1 degrees; the new one sits at 12.78 degrees, a third smaller than any of them. A fifth point beyond the near end of a fitted range is a test of the fit, where a fifth point between two old ones would mostly have been a restatement. A fifth cluster A correction to the exchange's size was fitted over four hop clusters and its own file said so. A search turned up a fifth, at a hop smaller than any of the four, and the rule is right on it — which is what a prediction being confirmed looks like when the confirmation is worth having.

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 testingDivergenceExceptional chainExchangeFalsifiabilityFittingHonest limitsHop lengthModel scopeNegative resultOut of sampleResidual