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

Two regimes above a hole

Below the repeating pattern there is a transient, and the boundary between them is measurable: the first organ from which every class stays at its own level runs from 7 to 303 organs above the hole on twenty-five of thirty cuts, and five never reach it inside the run.
13 min read 19 figures State the claim, state the testCounted, not admired

Worth reading first: The damage has a period · The organ that was taken away · The survey this site cannot do.

The displacement above a hole is periodic — constant inside each residue class modulo the lag the stem kept — and that is read over the top hundred and twenty organs of a three-hundred-organ continuation. Below that window the profile is doing something else.

This essay measures where the something else ends, because every number this thread has ever read off a displacement profile comes from one side of that boundary or the other, and until now nothing said there were two sides.

How far above the hole the damage becomes a pattern. One row per wrecked cut, drawn at the first organ from which every residue class stays at its own level for the rest of the run. On the 25 rows that reach it at all, it runs from 7 to 303 organs above the removed one; five rows never reach it inside the 300 organs each run is continued for. Below that point the stem is still moving, and the displacement of the first organ after the cut — the quantity that tells a cheap removal from an expensive one — is measured there. Above it, nothing changes again.
Fig. 1 Every wrecked cut, drawn at the first organ from which every class stays at its own level for the rest of the run.

How the boundary is found

Read the class levels from the top of the run, then walk down from the hole. The onset is the first organ from which every subsequent organ sits within ten degrees of its own class’s level, for the rest of the run.

Ten degrees is the same figure that separates a class at the common level from an exception, and it is a gap rather than a threshold in both places. Below the onset organs are tens or hundreds of degrees from their eventual levels; above it they are within a few.

The definition has one property worth noting: it is a first organ after which something holds forever, so a single organ far from its level halfway up the run pushes the onset above it. That makes the number conservative — it is the last failure rather than the first success.

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. 2 A profile with its onset marked, drawn from the hole upwards so that both regimes are in one picture.

The numbers

On the twenty-five cuts that reach an onset at all, it runs from 7 to 303 organs above the removed one.

The distribution is not flat. Most sit between 20 and 70: seven of the twenty-five are under 25, eleven between 25 and 70, and seven above. Three are above 110 and one is at 303, which is beyond the end of the continuation and means the pattern is established only at the very top.

Five cuts never reach an onset inside three hundred organs.

How far above the hole the damage becomes a pattern. One row per wrecked cut, drawn at the first organ from which every residue class stays at its own level for the rest of the run. On the 25 rows that reach it at all, it runs from 7 to 303 organs above the removed one; five rows never reach it inside the 300 organs each run is continued for. Below that point the stem is still moving, and the displacement of the first organ after the cut — the quantity that tells a cheap removal from an expensive one — is measured there. Above it, nothing changes again.
Fig. 3 The same twenty-five with the five that never reach a pattern shown as such rather than omitted.

The transient’s size

Below the onset the largest displacement runs from 8.9° to 179.8° across the census.

The bottom of that range is worth pausing on. A transient whose worst displacement is 8.9° is a stem that barely moved before settling into its new pattern, and the row it belongs to is a cut eight places back on a golden 0.010 stem with an onset of 9 — one organ of disturbance and then the arrangement it will keep.

The top of the range is 179.8°, which is essentially the ceiling: an organ half a turn from where its control put it.

The next organ moves for the last 8, and for no othersOne row per organ removed, counted back from the tip of a stem at a rise of 0.013 whose counted pair is 5 and 8. Removing any of the last 8 moves the next organ by 4.9° to 164.1°; removing an older one moves it by at most 0.70°, which is under the azimuth grid. The boundary is at 8, and 8 is the larger parastichy number — so the experiment counts the spirals without measuring an angle.organ removed, counted back from the tiphow far the next organ moves, in degrees1136.9°286.0°348.3°4164.1°526.2°692.3°7131.0°84.9°— the front ends here90.0°100.7°110.7°120.0°130.7°140.0°150.2°160.2°rise 0.013 · pair 5/8generated from a stated rule, not drawn to look right
Fig. 4 The first organ’s displacement across every offset, which is the transient’s own first value.

Which regime each existing reading is in

This is the useful part.

The first organ’s displacement — the reading that separates a cheap removal from an expensive one — is the very first value of the transient. It is measured one organ above the hole, and on every row the onset is at least seven organs higher.

The largest displacement — the reading that failed as a reference organ — was looked for in a window three times the larger counted number wide, which is 15 to 39 organs. Against onsets of 7 to 303 that window is sometimes entirely transient, sometimes entirely pattern, and sometimes straddles the two.

The survivor, the block and the slip are all read over the last hundred and twenty organs, which on every row that reaches an onset is inside the pattern.

One wrecked stem, lag by lag — golden, rise 0.005, organ 6 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 55 degrees. The lag-4 hop swings by 0.12 degrees and sits 0.01 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 4, which is the surviving lag and not a coincidence.
Fig. 5 The lag spectrum, read entirely inside the pattern, which is why it is a clean measurement.

The one that straddles

Only the middle reading is in trouble, and it is the one that failed.

A statistic computed over a window that lies in one regime on some rows and another on others is not one statistic. On a row with an onset of 9 and a window of 24, fifteen of the twenty-four organs are pattern and the maximum is a class level. On a row with an onset of 166 and the same window, all twenty-four are transient and the maximum is a genuine peak.

Last round scored a reading over thirty rows of which some were one kind and some the other, and got an answer that was hard to interpret. That is what a mixed window produces.

How far above the hole the damage becomes a pattern. One row per wrecked cut, drawn at the first organ from which every residue class stays at its own level for the rest of the run. On the 25 rows that reach it at all, it runs from 7 to 303 organs above the removed one; five rows never reach it inside the 300 organs each run is continued for. Below that point the stem is still moving, and the displacement of the first organ after the cut — the quantity that tells a cheap removal from an expensive one — is measured there. Above it, nothing changes again.
Fig. 6 The onsets against the window a reference organ was looked for in, which is shorter than most of them.

The five that never settle

Five cuts have no onset inside three hundred organs: one on the golden 0.026 stem, one on the golden 0.010, one on the Lucas 0.013 and two on the Lucas 0.020.

Four of those five are also among the five whose classes are not constant — which is nearly the same statement twice, since a stem whose classes are not constant has no levels for an onset to be measured against.

The fifth, a cut four places back on the Lucas 0.013 stem, is periodic at the top and does not reach its pattern inside the run. That is the one row where the two measurements disagree, and it is a stem still arriving when the run stops.

How constant the displacement is inside one residue class. One row per wrecked cut in the census, drawn at the widest spread found inside any one residue class when the profile is folded on the lag that stem kept. 25 of 30 rows sit between 0.12 and 6.09 degrees, which on a quantity whose between-class differences run past a hundred and fifty degrees is a constant. The five that do not sit from 10.3° up. There is nothing in between, so the line drawn at 10° could have been drawn anywhere in a wide interval.
Fig. 7 The census with the rows that are not periodic marked, which overlaps the rows with no onset.

Against the settling time of an undamaged stem

There is a comparable number from a different thread. Growing a stem from a seed, it settles onto a lattice within 290 organs and usually inside fifty.

The onsets here run to 303 with most under 70. Those two distributions are strikingly alike, and the resemblance is worth stating carefully, because they are not the same measurement: one is a stem arriving at a lattice from an arbitrary starting angle, the other is a stem arriving at a new arrangement after being disturbed from an old one.

That they take about the same number of organs says the rule’s approach to a lattice is the same process in both cases, which is a claim nothing here tests.

How many organs a stem needs before it is on a lattice. One row per rise, one mark per starting angle, placed at the organ from which every later divergence stays within a degree and a half of the run's own final value. Where a stem settles at all it does so between 0 and 290 organs in, against the 400 every ablation run here grows before it cuts anything. Not one row needs the length it is given. What changes down the table is the count on the right: how many of the nine starting angles reach a lattice at all, which falls from 7 at the coarse rises to 1 at the finest.
Fig. 8 How long an undamaged stem takes to reach a lattice, for comparison with how long a wrecked one takes to reach its pattern.

Why 300 organs was enough, and nearly was not

Every cut here is continued for three hundred organs, which was chosen several rounds ago as comfortably longer than a recovery takes.

Against onsets it is not comfortable. One row’s onset is at 303 — outside the run — and three more are above 110. If the continuation had been two hundred organs, six or seven rows rather than five would have had no measurable onset, and the top of the range would have been an artefact of the run length rather than a measurement.

At three hundred the range is measured on twenty-five of thirty rows. It would be worth knowing whether the five that fail do so because they need longer or because they never settle, and answering that costs a longer continuation on five rows.

The same table, grown 2.7 times as long. Every rise and every starting angle, grown to 1200 organs and then to 3200. The two middle columns are how many starting angles reached a lattice at each length, and they are the same column: of the 72 pairs of runs, 72 are identical organ for organ and 0 settle at the longer length after failing at the shorter one. Tripling the budget buys nothing anywhere. What the fine rises are short of is not run length: the share of starting angles that reach a lattice at all falls from 7 of 9 to 1, so the arrangements a stem could fall into have mostly stopped existing.
Fig. 9 The comparison that answers a question of this shape elsewhere: the same runs at two lengths.

What the transient is

The stem finding its new arrangement. Below the onset the organs are being placed against a neighbourhood that still contains the disturbance; above it, against a neighbourhood that is the new pattern repeating.

That is a description rather than a mechanism, and the reason it is worth having is that it makes the front and the pattern separate objects. The front — the stretch of stem a removal is felt across — is a property of the transient. The surviving lag is a property of the pattern. They have been measured on the same runs and read as one story.

On a rung the response is a run of offsets; near a transition it has a hole. One row per rise, from 0.032 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 — 5, 8, 13 at the pairs shown on the left. Every rise shown here is in the middle of its rung, so every row is a run and nothing past it is felt — what happens between the rungs is a separate matter.
Fig. 10 The front, which is a measurement made entirely inside the transient.

The onset is not the front

Worth stating plainly because the two are both “how far a removal reaches”.

The front is the range of offsets at which a removal has any lasting effect — cut further back than it and the stem repairs. It runs to the larger counted number and it is a property of where the cut is made.

The onset is how far above the hole the stem takes to settle into what it will be. It runs from 7 to 303 and it is a property of what happens after the cut.

A stem can have a front of 13 and an onset of 166. The two numbers are about different axes and this collection has not previously had names for both.

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.
Fig. 11 The front, measured as which offsets have a lasting effect, which is the other axis.

What a longer run would settle

Two things.

Whether the five rows without an onset ever reach one. That is the same question the settling thread asked about the fine end of the ladder and answered by growing the same runs three times as long: if the table is identical, the limit is a wall rather than a budget.

And whether the row at 303 is really at 303 or at something larger. An onset measured at the very end of a run is a lower bound, and it is the largest number in the range.

Neither changes anything else here, which is why it is a leaving rather than a gap.

How far above the hole the damage becomes a pattern. One row per wrecked cut, drawn at the first organ from which every residue class stays at its own level for the rest of the run. On the 25 rows that reach it at all, it runs from 7 to 303 organs above the removed one; five rows never reach it inside the 300 organs each run is continued for. Below that point the stem is still moving, and the displacement of the first organ after the cut — the quantity that tells a cheap removal from an expensive one — is measured there. Above it, nothing changes again.
Fig. 12 The onsets with the run length marked, on which one row sits at the end of the run.

What it means for the next reading

Any new quantity read off a displacement profile has to say which regime it comes from, and the answer is now available for each row rather than assumed.

For a reading about the cost of a removal, the transient is the right place and the first organ is the natural sample. For a reading about what the stem became, the pattern is the right place and the class levels are the natural sample.

For a reading about neither — a maximum, an average, a decay rate — the honest first question is whether the window it is taken over is in one regime on every row. Last round’s was not, and nothing said so.

A period of 8, and the two classes that are not with the rest. The same wrecked stem, folded on the lag it kept: one row per residue class, each drawn at the mean displacement of its own organs against the level the rest of them share. The bar through each row is the spread inside that class, and the widest of them is 0.87° — so within a class the displacement is a constant. six classes sit at the common level. The two that do not sit at 134.3° and -134.2°, equal and opposite to within 0.0 per cent, and they are neighbouring residues. The stem's own divergence is 137.44°, so an exception is one organ's step.
Fig. 13 The class levels, which is the pattern’s own natural sample.

What sets the onset

Not the offset, on the evidence available. On the golden 0.010 stem the five wrecked offsets give onsets of 303, none, 18, 0 and 1 — which is the whole range in one lattice. On the Lucas 0.008 stem the four give 29, 28, 166 and 8.

Not the rise either, in any simple way. The golden stems at 0.026, 0.020, 0.016, 0.013, 0.010, 0.008 and 0.005 give onsets spread across the range at every one of them.

So the onset is a property of the individual cut rather than of the lattice or of the offset, which makes it a quantity to be measured per row rather than predicted. That is unusual in this thread, where most quantities are functions of two coordinates, and it is worth stating as a negative before somebody looks for a trend in it.

The next organ moves for the last 8, and for no others. One row per organ removed, counted back from the tip of a stem at a rise of 0.013 whose counted pair is 5 and 8. Removing any of the last 8 moves the next organ by 4.9° to 164.1°; removing an older one moves it by at most 0.70°, which is under the azimuth grid. The boundary is at 8, and 8 is the larger parastichy number — so the experiment counts the spirals without measuring an angle.
Fig. 14 The offsets a stem wrecks at, against which the onsets show no ordering.

The row at 303

One cut — four places back on the golden 0.010 stem — has an onset of 303, which is the last organ of the continuation.

That is a lower bound rather than a measurement. The pattern is established at the very top of the run and nothing says it would not have been established at 400 or at 900; what the number says is that the stem was still moving at organ 302.

It is the largest value in the range and it is quoted with the range, so “7 to 303” is a range whose upper end is not a measurement. The honest form is “7 to at least 303”, and the reason the essay does not use it throughout is that it would have to be repeated every time the range appears.

How far above the hole the damage becomes a pattern. One row per wrecked cut, drawn at the first organ from which every residue class stays at its own level for the rest of the run. On the 25 rows that reach it at all, it runs from 7 to 303 organs above the removed one; five rows never reach it inside the 300 organs each run is continued for. Below that point the stem is still moving, and the displacement of the first organ after the cut — the quantity that tells a cheap removal from an expensive one — is measured there. Above it, nothing changes again.
Fig. 15 The onsets with the run’s own length marked, on which the largest sits at the end.

What a second length would answer

The same comparison the settling thread made: grow the same cuts twice as long and see whether the five rows without an onset acquire one and whether the row at 303 moves.

If the table is identical, the five are stems that never settle into a pattern and the onset range’s top is real. If the five acquire onsets at 400 or 600, the boundary between “a transient” and “no pattern” is a run-length artefact and the count of twenty-five is a count of what fits in three hundred organs.

It costs thirty runs of six hundred organs, which is under an hour. It is a leaving rather than a gap because nothing in this round rests on which answer it gives.

The same table, grown 2.7 times as long. Every rise and every starting angle, grown to 1200 organs and then to 3200. The two middle columns are how many starting angles reached a lattice at each length, and they are the same column: of the 72 pairs of runs, 72 are identical organ for organ and 0 settle at the longer length after failing at the shorter one. Tripling the budget buys nothing anywhere. What the fine rises are short of is not run length: the share of starting angles that reach a lattice at all falls from 7 of 9 to 1, so the arrangements a stem could fall into have mostly stopped existing.
Fig. 16 The two-length comparison as it is made elsewhere, whose whole content is whether the longer table differs.

Two regimes and one instrument

The reason this matters beyond bookkeeping is that the same instrument reads both regimes and reports one number.

A displacement profile is a sequence of angles. Take its maximum and the answer comes from wherever the largest angle happens to be; take its mean and the answer is a mixture; take its value at a stated lag and the answer depends on which side of the onset that lag falls.

Every one of those is a defensible statistic and none of them is well defined without the boundary. Now that the boundary is measured per row, a future reading can say which regime it is in — and the three readings this thread already has can be sorted, which is what this essay is for.

What a cut moves, organ by organ. A stem counted at 5 and 8 spirals with the organ six 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 143 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.
Fig. 17 A profile read over sixty organs, which on most rows spans both regimes.

Why the boundary is conservative

The onset is defined as the first organ after which every later organ stays within ten degrees of its own class’s level for the rest of the run. A single organ that strays halfway up pushes the onset above it.

That is deliberate and it costs something. A definition taking the first organ after which the condition holds for, say, sixty organs would give smaller numbers and would call a stem settled that later leaves. The conservative version cannot do that, and in exchange it reports 303 for a stem whose profile is flat from organ forty except for one excursion.

Which of those two happens on which row is not separated here, so a large onset means either a long transient or a late excursion. Telling them apart is a matter of reading the profile, and the profiles are drawn.

How far every organ moved, 4 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 3 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. 18 A profile from the hole upwards, on which a long transient and a late excursion look different.

The one line

A wrecked stem has a transient and then a pattern. The boundary sits 7 to 303 organs above the hole on twenty-five of the census’s thirty cuts and is not reached at all on five; the transient’s worst displacement is 8.9° to 179.8°. The first organ’s displacement is a transient measurement, the survivor and the slip are pattern measurements, and the reading that failed was taken over a window that is sometimes one and sometimes the other.

How far above the hole the damage becomes a pattern. One row per wrecked cut, drawn at the first organ from which every residue class stays at its own level for the rest of the run. On the 25 rows that reach it at all, it runs from 7 to 303 organs above the removed one; five rows never reach it inside the 300 organs each run is continued for. Below that point the stem is still moving, and the displacement of the first organ after the cut — the quantity that tells a cheap removal from an expensive one — is measured there. Above it, nothing changes again.
Fig. 19 The boundary, measured on every wrecked cut in the census.

Where this goes next

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. The damage has a period Every wrecked stem in the census has had two numbers read out of its displacement profile and the profile itself read out of none of them. Folded on the lag the stem kept, twenty-five of the thirty are constant inside each residue class to between 0.12° and 6.09°. 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. How constant the displacement is inside one residue class. One row per wrecked cut in the census, drawn at the widest spread found inside any one residue class when the profile is folded on the lag that stem kept. 25 of 30 rows sit between 0.12 and 6.09 degrees, which on a quantity whose between-class differences run past a hundred and fifty degrees is a constant. The five that do not sit from 10.3° up. There is nothing in between, so the line drawn at 10° could have been drawn anywhere in a wide interval. The plateau was a prediction Last round's search for a reference organ found that the largest displacement above a hole is a plateau rather than a peak, and reported it as a failure. A profile constant on each of k residue classes has exactly k levels, so its maximum is attained by a whole class — a ninth to a quarter of every window, forever.

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.

  • The clock a share cannot see — both name claim testing, control, honest limits, measurement, negative result, sample size, settling time, summary statistic, transient
  • One level and two exceptions — both name ablation, claim testing, control, description versus mechanism, measurement, mechanism, negative result, rigid hop
  • A period the grid invented — both name ablation, claim testing, honest limits, measurement, negative result, resolution, summary statistic
  • One offset, two answers — both name ablation, claim testing, control, honest limits, measurement, negative result, rigid hop
  • One rise per rung is a sample — both name claim testing, control, honest limits, measurement, negative result, sample size, summary statistic
  • The angle is not the actor — both name ablation, claim testing, control, description versus mechanism, mechanism, negative result, rigid hop

Named objects

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

AblationClaim testingControlDescription versus mechanismHonest limitsMeasurementMechanismNegative resultResolutionRigid hopSample sizeSettling timeSummary statisticTransient