Phyllotaxis
the spirals are counted, not admired
Series

Ablation — the series

27 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Take away the organ eight places back, and the next one goes into the hole. The last 34 organs of a stem at a rise of 0.005, unrolled. The open circle is the organ removed — eight places before the tip. The ring at the top is where the rule puts the next organ with every organ present; the filled mark beside it is where the rule puts it with that one missing. The two are 16.4° apart, against a local spacing of 25°, and the vacancy itself is 22.7° from the undisturbed answer. Nothing else differs between the two runs: same rise, same history, same rule.

    The organ that was taken away

    Every observable this site has is read off an arrangement that was finished before the reading began, and earlier work here showed what that costs. So remove one primordium from a settled stem and place the next one against what is left. The rule has to answer. The rival account cannot, because in it no organ's position was ever computed from its neighbours.

    part 1 · mechanism
  2. The next organ moves for the last 13, and for no others. One row per organ removed, counted back from the tip of a stem at a rise of 0.005 whose counted pair is 8 and 13. Removing any of the last 13 moves the next organ by 2.6° to 167.6°; removing an older one moves it by at most 0.47°, which is under the azimuth grid. The boundary is at 13, and 13 is the larger parastichy number — so the experiment counts the spirals without measuring an angle.

    An experiment a needle could run

    For eight instalments the outstanding item has been a survey — photographs, a protractor, hundreds of specimens — and it has not been done. The intervention is a different kind of ask, and a cheaper one: a needle, one apex, and a yes-or-no per ablation. Here is what it would cost, what it would settle, and the four ways it could come out.

    part 2 · wrong
  3. A cut eight back is never undone. The divergences of a stem whose organ eight places back was removed, against the same stem uncut. It never returns. What it settles into repeats exactly every 8 organs — 47°, 96°, 137°, 273°, 138°, 271°, 230°, 272° — and holds that cycle for the whole 300-organ run, with a mean of 186° and a spread of 83°. A rule that corrects a displacement does not correct a deletion.

    A rule that cannot heal a hole

    The placement rule corrects itself against a displacement — that is what the lag-one correlation of −0.6 has been saying since it was measured. It does not correct itself against a deletion. Which organ is removed decides whether the stem is back on its lattice in twenty-four organs or never, and the boundary between the two is sharp, reproducible and in the middle of the front.

    part 3 · emergence
  4. Both edges of the front heal; the middle of it does not. The same removals, followed for 300 organs each. A cut one to three places back is undone within fifty organs and a cut ten to thirteen places back within sixty. A cut in between is never undone: the divergence sequence settles into an exactly repeating cycle of 4 or 8 angles and holds it for the rest of the run. The rule corrects a displacement and cannot correct a deletion.

    A front with no middle

    Take one organ out of a stem and the pattern sometimes never comes back — but that was measured on a front thirteen organs wide, where five of the thirteen offsets are beyond repair. Repeat it on a front of five and every single ablation heals. The band that cannot be undone is not a number the rule carries; it is what two fixed edges leave over.

    part 4 · emergence
  5. On a rung the response is a run of offsets; near a transition it has a hole. One row per rise, from 0.02 at the top to 0.005 at the bottom, and one column per offset: the organ one place back at the left, 14 places back at the right. A cell is filled where removing that organ moves the next organ by more than 2.5°, and empty where it does not. On a rung the filled cells are a run from one to the larger parastichy number — 8, 12, 13 at the pairs shown on the left. Between a rise of 0.02 and 0.0065 the run ends at 5 and one more cell is filled at 7, with the offsets between them quiet to under a degree. That isolated column is one place inside the larger number of the pair the stem is climbing towards.

    The response with a hole in it

    Removing an organ is felt out to the larger parastichy number and no further — that is the intervention's headline, and it holds in the middle of a rung. Swept towards a transition the run of felt offsets stops early and one lone offset past it comes alive, with three quiet organs in between. The lone offset is one place inside the count the stem is about to have.

    part 5 · emergence
  6. Two answers 138° apart, and one organ holding the second one up. The repulsion the rule minimises, around the circumference of a stem at a rise of 0.008, at the height the next organ will sit at. It has two low points 138.3° apart: the slot the next organ takes, and the slot the organ after it will take. The runner-up is 13.6% higher. The organ 13 places back carries 14.6% of the energy at the winning slot and twelve places back carries 16.3% at the runner-up — and that is more than the gap, so taking that organ away makes the runner-up win and the next organ appears a whole divergence away. Neither guard is a contact of the organ being placed; twelve is one place inside the larger number of the pair this stem is climbing towards.

    The organ that guards the second slot

    An organ twelve places back is the furthest of any from where the next one goes, and removing it moves the next one by a whole divergence. The reason is that the rule's profile has two low points rather than one, the second is the slot after next, and that organ is holding it up. The comparison between what it holds up and how far behind it is decides the whole thing.

    part 6 · emergence
  7. A second cut moves the next organ, and does not move the boundary. Every pair of organs that can be taken out of a settled stem at a rise of 0.032, where the pattern is 3/5. A row is the offset of the nearer organ removed, counted back from the tip; a column is how many further places back the second one sits. The shade is how far the next organ ends up from where it would have been, from under a degree in the palest cells to a half-turn in the darkest. The run of shaded rows ends at 5, which is the larger parastichy number, and every row below it is blank across the whole width — the largest displacement anywhere past the front is 2.34°. So the nearer of the two organs decides whether the removal is felt at all, and the second one, wherever it is put, cannot make the pattern notice an organ it was not going to notice.

    A cut of two organs

    One organ removed from a stem is felt out to the larger parastichy number and no further, and at the coarsest arrangement the stem always repairs itself — so the one rung where the interesting prediction could be checked had no experiment that could reach it. Two organs can. The second cut brings a parameter with it, and that parameter turns out to be a control.

    part 7 · emergence
  8. One wrecked stem, lag by lag — golden, rise 0.005, organ 4 back. How much each lattice hop moves from organ to organ in a stem that never repaired after a single removal, over its last 120 organs. The lag-one hop is the divergence itself and it swings by 60 degrees. The lag-8 hop swings by 0.09 degrees and sits 0.19 degrees from where the undisturbed stem put it, so that one family of the original lattice is still standing organ by organ. Its multiples inherit the same steadiness and nothing else comes within a factor of twenty. The block of angles this stem repeats has a period of 8, which is the surviving lag and not a coincidence.

    The hop that survived

    A stem that never repairs after an organ is removed settles into an exactly repeating block of angles, and the period of that block is a spiral count of the lattice it was cut from. Nobody could say why. Read the wrecked stem by lags rather than by neighbours and the answer is one line: one family of the original lattice is still standing, organ by organ, and the block is its period.

    part 8 · emergence
  9. A wreck is a whole number of extra turns. For each of the 19 stems that never repair, the slip of its settled divergence multiplied by the lag whose hop survived. Every value lands on a whole number of turns — the horizontal lines — with a largest departure of 2.97 degrees, against divergences that have moved between 0 and 103 degrees. 17 of the 19 close on exactly one turn. So a wrecked stem is the stem it was with one extra turn threaded through every period of the family that survived, which is a dislocation with a stated size rather than damage.

    One turn per survivor

    If a wrecked stem keeps one family of its old lattice exactly, then the angle it settles at is not free. Over the period of the family that survived, the pattern has to come back to where that family left it — which means the whole change in the divergence is a whole number of turns spread over a small whole number of organs. Measured, it is one turn, at seventeen of nineteen.

    part 9 · emergence
  10. Which lag survives, at every lattice and every offset. A row for each of twelve lattices and a column for each offset an organ is removed at, with the lag whose hop survived written in the cell. 30 cells never repair. The lag left standing is one of the two contact families at 29 of them, and at 17 it is the second shortest step on the lattice rather than the shortest, which is what refuses the reading that a rule holds its nearest neighbours. The ringed cells are the five the offset rule gets wrong. Two lattices carry no cells at all because no single removal wrecks them.

    A survivor has to be a neighbour

    A stem that never repairs after a removal keeps exactly one lattice hop rigid, and nothing predicted which one. Sweep every offset at twelve lattices and the answer narrows sharply: at twenty-nine of thirty the surviving hop is one of the two families a counter returns, and the one exception is a step six times too long to be one.

    part 10 · emergence
  11. Which lag survives, at every lattice and every offset. A row for each of twelve lattices and a column for each offset an organ is removed at, with the lag whose hop survived written in the cell. 30 cells never repair. The lag left standing is one of the two contact families at 29 of them, and at 17 it is the second shortest step on the lattice rather than the shortest, which is what refuses the reading that a rule holds its nearest neighbours. The ringed cells are the five the offset rule gets wrong. Two lattices carry no cells at all because no single removal wrecks them.

    Not the shorter of the two

    If a damaged stem keeps one contact family standing, the obvious guess is that it keeps the nearer one. Across thirty wrecked offsets that is true twelve times and false seventeen, and on one lattice the two steps differ by a quarter of a per cent — where the words shorter and longer are doing no work at all.

    part 11 · emergence
  12. Which lag survives, at every lattice and every offset. A row for each of twelve lattices and a column for each offset an organ is removed at, with the lag whose hop survived written in the cell. 30 cells never repair. The lag left standing is one of the two contact families at 29 of them, and at 17 it is the second shortest step on the lattice rather than the shortest, which is what refuses the reading that a rule holds its nearest neighbours. The ringed cells are the five the offset rule gets wrong. Two lattices carry no cells at all because no single removal wrecks them.

    One offset, two answers

    Which contact family a wrecked stem keeps is decided by where the cut landed, at twenty-five of thirty offsets, by the simplest rule anybody would write down. It is refuted by two runs: the same counted pair, the same offset, two different rises, and two different surviving families.

    part 12 · emergence
  13. Which family survives, along the 5/8 rung. The family a wrecked stem keeps, at every offset that wrecks and every rise on one rung. A counter shown any of these stems returns 5 and 8 spirals, at all twelve rises, so nothing in the pair distinguishes the columns. The cells say otherwise: at an offset of 4 the stem keeps the 5-family at the coarse end of the rung and the other one at the fine end. The offsets that wreck at all grow from 1 to 5 as the rise falls, because the front deepens, and the extra offsets are the ones that keep the larger number. So the rule stated over the offset alone is a rule with the rise left out of it.

    One rung, two answers

    The offset accounts for twenty-five wrecked stems of thirty and is refuted by a single pair of runs that differ in nothing but the rise. Sweep one rung at a thousandth and the refutation stops being an anomaly: the same offset on the same lattice keeps one family at the coarse end and the other at the fine one.

    part 13 · emergence
  14. A prediction and its opposite, scored on the same rows. The reading under test said the family whose member was removed is the one that breaks. Scored across every wrecked offset where the removed organ lies on exactly one contact chain — 9 of 30, the other 21 being silent because the organ lies on neither — it is right no times and its opposite is right nine. A chain that loses a member does not stop existing: the organs above the hole are still spaced at that lag and the rule that placed them is still minimising the same sum, while the other chain has lost the organ its members were positioned against.

    The family that lost a member

    The offset rule restated in the arrangement predicts that the chain whose organ was taken is the chain that breaks. Scored on the nine offsets where the question can be asked, it is right none of the time and its opposite is right all nine.

    part 14 · emergence
  15. How many offsets wreck, along the 5/8 rung. The count of offsets that never repair, at each rise on one rung. It runs from 1 at the coarse end to 5 at the fine end, while a counter returns 5 and 8 spirals at every one of them. The front — the run of recent organs at which a removal is felt at all — deepens as the rise falls, so there are simply more places a cut can land and fail to heal. That is the mechanism under the grid: the offsets that appear at the fine end are the ones beyond the smaller contact number, and those are the ones that keep the larger family.

    The front deepens down a rung

    The offsets that never repair grow from one to five across a single rung, while a counter returns the same pair at every rise. The extra offsets are not a random extension of the ones already there: they are the ones past the smaller counted number, and they are the ones that keep the larger family.

    part 15 · emergence
  16. The two contact steps change places inside the 5/8 rung. Measured at every rise on one rung, where a counter returns 5 and 8 spirals throughout. The settled divergence slides from 136.6406 to 137.8672 degrees. The ratio of the two contact steps falls to 1.0131 at a rise of 0.016 and the ordering changes hands at 0.015: above it the shorter step belongs to the 5-family and below it to the 8-family. Neither quantity is available to a counter, which is shown positions and reports a pair, and both of them move while that pair does not.

    The shortest hop was a coin flip

    The reading that a wrecked stem keeps its shortest hop was refuted at twelve of twenty-nine across the census. Re-scored along a single rung, where the counted pair is held and the step ordering reverses, it is right at sixteen of thirty-one — which is not a refutation but an absence of information.

    part 16 · emergence
  17. Every wrecked offset, and whose neighbour was removed. Each row is a stem that never repaired, with the family of the organ that was taken and the family that survived. An organ five places back on a stem counted at 5 and 8 spirals lies on the tip's five-chain, so the question can be asked there; an organ four places back lies on neither chain and it cannot. Of 30 wrecked offsets in the census, 9 remove a member of exactly one family and 21 remove a member of neither. On every one of the 9 the family that lost a member is the family left standing, which is the opposite of what the reading predicted.

    The organ that was nobody's neighbour

    Twenty-one of the thirty wrecked offsets remove an organ that lies on neither contact chain through the tip. The reading that explains the other nine has nothing to say about them, and the honest thing is to say so rather than to widen the definition until it does.

    part 17 · emergence
  18. The ordering changes and the survivor does not. Every offset that wrecks, at every rise of the band, with the family left standing written in the cell. The counted pair is 5 and 8 at all 18 rises and the settled divergence is held to a twentieth of a degree, so the one quantity moving across the columns is which of the two contact steps is the shorter — and it changes hands at the marked rise. The cells do not: the 5 family survives at all 24 wrecked cuts, on both sides. Scored on this band, the reading that a wrecked stem keeps its shortest hop is right 14 times out of 24, for an answer that never changed.

    The ordering was not the actor

    Cut an organ out of every rise of a band where the counted pair and the settled divergence are held and the two contact steps change places, and the family left standing does not change. Fifty-five wrecked cuts on two branches, and the ordering reverses underneath every one of them.

    part 18 · emergence
  19. What a cut moves, organ by organ. A stem counted at 5 and 8 spirals with the organ five places back from the tip removed, compared against a control that shares its history to the last digit. Each mark is one organ placed after the cut and how far its azimuth ended up from where the control put the same organ. The quantity folds at half a turn, so 152 degrees is near the largest displacement there is; and it does not decay with height, which is what a stem that never repairs means. There is therefore no organ that the cut disturbed most in any useful sense, and a reading that needs one has nowhere to stand.

    The organ that moved furthest

    A reading that works on nine of thirty rows needs a reference organ, and the obvious repair is to measure one rather than to choose it. Measured, the disturbance turns out to have no far edge at all, so there is no organ that moved furthest in any sense the reading can use — and the generalisation that does work needs no reference organ.

    part 19 · emergence
  20. What a removal costs the next organ. One mark per wrecked cut in the census: how far the first organ placed after the removal ended up from where the control put it. The rows split by which organ was taken. Removing a direct chain-neighbour of the growing tip — an organ at a multiple of one of the two counted numbers — moves the next organ by between 8.9 and 30.7 degrees. Removing anything else inside the front moves it by between 62.8 and 167.6. Nothing lands between the two groups and the ratio across the gap is 2.05, so the line is a gap rather than a threshold. Taking away a neighbour is the cheap removal, which is the opposite of what the words suggest.

    Removing a neighbour costs least

    Take away an organ that is a direct chain-neighbour of the growing tip and the next organ moves by under thirty-one degrees. Take away anything else inside the front and it moves by at least sixty-three. Thirty cuts, two groups, a factor of two between them and nothing in the gap.

    part 20 · mechanism
  21. 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.

    part 21 · cylinder
  22. 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.

    part 22 · cylinder
  23. 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.

    The lag that never survives

    A correction to the exchange's size rests on four hop clusters, and a fifth would be the first real test of it. The prediction was written for a golden lattice at a lag of eleven. No golden lattice on this ladder reaches one, and the reason is a fact about the rule rather than about the search.

    part 23 · cylinder
  24. The Lucas 7/11 rung between 0.007 and 0.006, cut at every rise. One row per offset, one column per rise of the search, coarse on the left. A pale cell is a rise at which that offset's cut recovers; a dark cell is a cut that wrecks and keeps a lag of 7; a warm cell is one that keeps 11. The nine intermediate rises of the ten-thousandth grid were cut first and the nine of the hundred-thousandth grid inside the one bracket they opened. The change sits between 0.0064 and 0.00639, one step of the grid the ladder names its rises on, and above it no cut keeps 11 anywhere.

    One step of the grid, again

    A search of the fine end stepped from one rise to another a hundred grid steps away, and the family a cut keeps changed somewhere between. Cutting the rises in between puts the change inside one step, with the lattice identical on both sides.

    part 24 · cylinder
  25. The two shortest hops across the Lucas 7/11 rung, and where they cross. The lengths of the two shortest hops at every one of the 378 rises of the rung, fine on the left, computed from the settled divergence rather than measured off a cut. They cross exactly once, between 0.00804 and 0.00803, which is this rung's handover. The rise at which a cut first keeps the longer family is 164 steps of the grid further down and is marked separately; nothing happens to either length there.

    The hops cross once

    Walking a whole rung at the grid its rises are named on costs a few hundred stems and no cuts at all, and it answers a question nobody had asked: whether a rung has one handover or several. It has one, and the ladder had recorded it in the wrong place.

    part 25 · cylinder
  26. Two rises on the Lucas 7/11 rung, each located to one step of the grid. The whole of the Lucas 7/11 rung, coarse on the left, with the two rises this round located. The two contact steps change places between 0.00804 and 0.00803, found by walking all 378 rises of the rung and evaluating hop lengths, with no cut stems at all. The family a cut leaves standing changes between 0.0064 and 0.00639, found by cutting 89 stems. They are 164 steps of the grid apart, which is 45 per cent of the rung's whole span, so the geometry crossing is not what moves the survivor.

    Two rises far apart

    The whole handover thread rests on the claim that where the two contact steps change places is not where the survivor does. On one rung both rises are now located to a single step of the grid, and they sit forty-five per cent of the rung apart.

    part 26 · cylinder
  27. What each offset keeps, above and below 0.00639. Every offset that wrecks anywhere in the search, with the family it keeps above the transition and below it. Offsets 5, 6, 7, 9 keep the same family at every rise they wreck at, on both sides. What changes at 0.00639 is that offset 9 begins to wreck at all, and what it keeps is the larger of the counted pair. One offset does change its own answer, and it cannot be located, because it does not wreck at the rises in between.

    An offset that arrives

    The rise where a lattice first keeps a new family is located to one step of the grid, and what happens there is not what the question assumed. No cut changes its mind: a cut that was not wrecking starts, and what it keeps is the new family.

    part 27 · cylinder

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