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

The rung that two organs wreck

On the coarse 3/5 stem both walls of the slot heal when removed alone and wreck when removed together — and the wreck keeps no rigid hop at all. Two of the six pairs in the design end at a destination single removals almost never reach.
12 min read 19 figures State the claim, state the testCounted, not admired

Worth reading first: Both walls of the slot · The organ that was taken away · A stem coarse enough to cut.

At the coarse end of this ladder a single removal does nothing that lasts. Every offset heals on a 3/5 stem, because a front of five organs is its own two edges with no middle and there is nothing in the middle to sever.

That two organs reach it is already known: a sweep of two-organ cuts found four of sixty-four arrangements that never repair. What is new here is which two, and what is left standing afterwards.

Which hops survive one wall, the other, and both. One row per lattice. The last three columns are the lags whose hop the cut stem still holds, unchanged from a control that shares its history — the measurement that identifies what a wrecked stem has become. Removing a single wall always leaves something standing, which is what every single-organ cut in this collection does. Removing both leaves nothing at all on two of six lattices, including the coarse rung that no single removal can wreck. A stem that keeps no rigid hop is not a wrecked lattice with a slip in it; it is a stem that is no longer a lattice.
Fig. 1 The lags each cell of the two-by-two leaves standing, with the two pairs that leave none.

The row

Golden stem at a rise of 0.020, counted 3 and 5. The tip’s two chain-neighbours are the organ three places back and the organ five places back.

Remove the 3-wall alone: the next organ moves 41.7° and the stem heals.

Remove the 5-wall alone: the next organ moves 20.6° and the stem heals.

Remove both: the next organ moves 47.8° and the stem never repairs.

Removing one wall of the slot, then the other, then both. The growing tip sits in a slot between its two chain-neighbours — the organ 3 places back and the organ 5 places back. Each bar is how far the next organ placed moves when those are removed, against a control sharing the same history. Either alone is a cheap removal: 41.7° and 20.6°, both inside the 45° that separates taking a neighbour from taking anything else. Together they move it 47.8°, against 62.3° for the two effects added, so the interaction is -14.5°. The slot is not two independent walls.
Fig. 2 The four cells at the coarse lattice, where the first three heal and the fourth does not.

What makes it worth a paragraph

The displacement barely changes. 41.7° and 20.6° alone, 47.8° together — the interaction is −14.5°, one of the two smallest in the design, and the fourth cell sits only six degrees above the larger single removal.

So the quantity that separates the fourth cell from the other three is not how far the next organ moves. It is whether the arrangement comes back, and those two things are measured on different parts of the same run: the displacement is the first value of the transient and the repair is a property of the three hundred organs above it.

This is the second row in the design where those two come apart, and it comes apart the other way from the first: there, two cells with identical displacements ended differently; here, a cell that ends differently has almost the same displacement.

What removing both walls does that removing each does not. One row per lattice, drawn at the difference between how far the next organ moves when both walls of the slot are removed and how far the two removals move it separately added together. Zero would mean the walls act independently. five of six lattices are more than 10° from it, three above and two below — so the pair is reliably not the sum of its parts and is not reliably larger than it either. On four of them the pair throws the next organ past the 45° that separates a cheap removal from an expensive one, which neither wall alone comes near.
Fig. 3 The interactions, on which this row is one of the two small negative ones.

What it keeps

Nothing.

A wrecked stem normally has exactly one rigid hop: one lag whose angle from organ to organ is unchanged from the control while every other lag moves by tens of degrees. It is the strongest regularity this thread has and it holds on every single-organ wreck in the census.

The lag spectrum of this stem has no such lag. Nothing under half a degree of spread and three degrees of shift, anywhere up to a lag of twenty-four.

One wrecked stem, lag by lag — golden, rise 0.005, organ 7 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 91 degrees. The lag-8 hop swings by 0.00 degrees and sits 0.23 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.
Fig. 4 A lag spectrum with a survivor in it, for comparison with one that has none.

Two of six do that

The other is the Lucas stem at 0.013, counted 4 and 7, where both walls together move the next organ 34.9° and leave no rigid hop either.

So of the six pairs in the design, four end as a wrecked lattice with a slip in it and two end as something with no surviving lag at all. That ratio is worth comparing against the single removals: of the twelve of those, none leaves nothing.

A stem that keeps no rigid hop is a different kind of object from a wrecked lattice. The whole account of what a removal does — one family rigid, the arrangement slipped by a whole number of turns per period — has no purchase on it, because there is no family to be rigid.

Which hops survive one wall, the other, and both. One row per lattice. The last three columns are the lags whose hop the cut stem still holds, unchanged from a control that shares its history — the measurement that identifies what a wrecked stem has become. Removing a single wall always leaves something standing, which is what every single-organ cut in this collection does. Removing both leaves nothing at all on two of six lattices, including the coarse rung that no single removal can wreck. A stem that keeps no rigid hop is not a wrecked lattice with a slip in it; it is a stem that is no longer a lattice.
Fig. 5 The same table with more of each spectrum shown, so that a row with an empty column can be seen to be empty rather than truncated.

Where a null survivor has been seen before

In the multi-organ sweeps. Cuts of three, four and five organs produce stems that keep nothing routinely, and the machinery that reads a lag spectrum returns a null for them rather than a number.

So this is not a new phenomenon; it is the smallest cut that produces it, and it produces it with two organs chosen for a reason rather than two organs from a sweep. A cut of two organs at arbitrary offsets is one of sixty-four arrangements; a cut of both walls is one arrangement named by the stem’s own counted numbers.

More organs removed, more stems that never come back. The share of arrangements at the 5/8 rung that never return to the divergence they were cut from, against how many organs the cut removed. One organ wrecks 2 of 8 arrangements and five wreck 63 of 64. The number of arrangements differs from bar to bar because a cut of five organs has more ways of being placed than a cut of one, and it is printed on each bar for that reason. What the dose decides is whether a stem falls off its lattice; where it lands when it does is decided by something else.
Fig. 6 How the share of arrangements that never repair grows with how many organs are removed.

Why the coarse rung is the one that does it

Guessing, and the guess is checkable.

A 3/5 stem’s front is five organs, which means a removal is felt across five and no further. Both walls of the slot are the 3 and the 5, so removing them takes out two of the five organs that matter — forty per cent of the neighbourhood — and leaves three.

On a 5/8 stem the same two removals take two organs out of a front of eight, which is a quarter. On an 8/13 stem, two out of thirteen.

That is the share argument the dose thread already tested and largely refuted for a different question — whether a finer rung mirrors when enough of its front is removed — so it should be offered here with the same scepticism. The share is a plausible account and it is not the one that worked last time it was tried.

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, 16 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. 7 The front’s depth by rise, which is what the two removed walls are a share of.

What the mirror does here

Nothing, and that is worth checking rather than assuming.

At the coarse rung, a wrecked stem’s characteristic destination is its own mirror: the same lattice wound the other way, with its counted pair and its hop order unchanged. Three of the four two-organ wrecks the dose sweep found at 3/5 land there.

This one does not. Its divergence settles somewhere that is not the mirror of 139.7°, and its lag spectrum is empty rather than being the mirror’s full one. So of the small number of two-organ wrecks known at this rung, this one is not the common kind.

That is a single row and it is stated as a single row.

The wrecked stem is the lattice it was cut from, wound the other way. The divergence of each organ placed after two were removed from a settled stem at a rise of 0.032, over the 220 organs following the cut. The upper line is the divergence the undisturbed stem holds, 139.6875°; the lower is 220.3125°, which is 360° minus it and therefore the same lattice with the opposite handedness. The sequence is thrown by the cut, wanders for a few dozen organs, and settles on the second line to four decimal places — 220.3125° against 220.3125° — where it stays. A counter shown the positions afterwards returns 3 and 5, the pair the stem was cut from. Nothing about the pattern has been lost; its chirality has been reversed, which at this rung a single removal cannot do.
Fig. 8 A wrecked coarse stem that has landed on its own mirror, which is what this row does not do.

What the design says and what it does not

It says: on this lattice, two removals that individually do nothing lasting together produce a stem that never repairs and keeps no rigid hop.

It does not say two removals always do that, or that the two walls are special among two-organ cuts, or that the coarse rung is where it happens. All three of those need a sweep and this is a two-by-two on six lattices.

The reason to report it anyway is that the two walls are not an arbitrary pair. They are the pair the placement rule’s own nearest terms belong to, named by the stem rather than by the experimenter, which is what makes a single row worth looking at.

Removing one wall of the slot, then the other, then both. The growing tip sits in a slot between its two chain-neighbours — the organ 4 places back and the organ 7 places back. Each bar is how far the next organ placed moves when those are removed, against a control sharing the same history. Either alone is a cheap removal: 29.5° and 11.3°, both inside the 45° that separates taking a neighbour from taking anything else. Together they move it 34.9°, against 40.8° for the two effects added, so the interaction is -5.9°. The slot is not two independent walls.
Fig. 9 The other row whose pair keeps nothing, on the Lucas branch at a different rise.

What would make it a result

A sweep. Take the sixty-four two-organ arrangements at the coarse rung, mark which of them are the two walls, and see whether the four that never repair include it — and whether the ones that keep nothing are the ones nearest the walls.

That sweep exists for the fates and does not exist for the spectra: the dose thread recorded where wrecked stems go and not what they kept, at each size of cut. Adding the lag spectrum is one column.

It is named in the leavings rather than run, because it is a sweep and this round’s was elsewhere.

Where a wrecked stem settles, whatever was taken from it. The settled divergences reached by every arrangement that never repairs, at each size of cut, on a stem whose parastichy pair is 5/8, counted rather than assumed. Each point is one destination and its size is how many arrangements reached it. Cuts of one organ and cuts of five land in the same handful of places; the largest cut invents nothing the smallest did not already reach. The dashed line is the mirror of the divergence the stem was cut from — the place a coarser stem goes when two organs are taken from it — and no arrangement at any size comes within 12 degrees of it.
Fig. 10 Where wrecked stems land by size of cut, which is the sweep this row would have to be placed inside.

What the coarse rung is

The top of the ladder: rises from 0.0700 down to 0.04702 on the golden branch, at which a counter returns 3 and 5, and 0.0700 to 0.06464 on the Lucas branch, at which it returns 1 and 3. This design’s coarse row sits just below the first of those, at 0.020 on the golden branch and counted 3/5.

Its distinguishing property is that its front is short. A removal is felt across the larger counted number and no further, so a 3/5 stem’s front is five organs — which is short enough that every organ in it is one of the front’s two edges, and the edges heal while middles do not.

That is why single removals do nothing lasting there, and it is why two removals are the smallest experiment that can reach it.

What a two-organ cut does at each rise of the 2/3 rung. At every rise the coarse rung is a lattice on, all 36 arrangements of two organs removed, with the ones that never repair counted and split by where they end up. 22 of 324 cuts across the rung reverse the stem's handedness onto the mirror of the divergence they were cut from. 40 fall instead into a cycle whose mean is half a turn, which the lattice they came from has no number for. 4 rises give only the first, 4 give only the second, and at a rise of 0.075 both happen in the same table, which is what says the fate belongs to the cut and not to the rise.
Fig. 11 What happens to a coarse stem at each offset, on which every single removal heals.

What “keeps nothing” is measured as

The absence of a lag, and the absence is measured rather than inferred.

The lag spectrum runs from 1 to 24 and asks two things of each: whether the hop is steady, meaning its spread over the last hundred and twenty organs is under half a degree, and whether it is unmoved, meaning its mean is within three degrees of the control’s. A survivor is the shortest lag passing both.

On this stem none of the twenty-four passes. The steadiest lag has a spread of tens of degrees, which is the same order as the divergence’s own wander in a wrecked stem — so it is not a near miss with a threshold problem in it.

That distinction is worth making explicit because “no survivor” and “a survivor just outside the tolerance” would appear identically in a table.

One wrecked stem, lag by lag — golden, rise 0.013, 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 65 degrees. The lag-5 hop swings by 0.11 degrees and sits 0.09 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 5, which is the surviving lag and not a coincidence.
Fig. 12 A spectrum with a survivor in it, whose one steady lag is orders of magnitude steadier than the rest.

Why a stem with no rigid hop is a different object

Because the whole account of what a removal does rests on there being one.

A wrecked stem keeps one family rigid and slips by a whole number of turns per period, and its displacement profile is constant on the residue classes of that family. Both of those are statements indexed by a surviving lag, and neither can be made about a stem that has none.

So the two pairs in this design that keep nothing are outside the descriptive machinery this thread has built. They are not counter-examples to it — the machinery is about wrecked stems that keep a lag — but they are a reminder that the class of things a removal can produce is larger than the class the machinery describes.

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 description that applies to a stem with a surviving lag, and does not apply to one without.

What the design cannot decide

Whether the two walls are special. Sixty-four two-organ arrangements exist at each rung and this design cuts one of them, chosen for a reason rather than sampled.

So “both walls wreck a coarse stem” and “some two-organ cuts wreck a coarse stem” are both true, and the first does not follow from the second being about the walls. The dose sweep found four of sixty-four arrangements that never repair at the coarse rung; whether the walls are among those four is a lookup that was not done, because the sweep is indexed by offset and gap rather than by which offsets they are.

That is one join and it would place this row inside a distribution instead of beside one.

Where a wrecked stem settles, whatever was taken from it. The settled divergences reached by every arrangement that never repairs, at each size of cut, on a stem whose parastichy pair is 5/8, counted rather than assumed. Each point is one destination and its size is how many arrangements reached it. Cuts of one organ and cuts of five land in the same handful of places; the largest cut invents nothing the smallest did not already reach. The dashed line is the mirror of the divergence the stem was cut from — the place a coarser stem goes when two organs are taken from it — and no arrangement at any size comes within 12 degrees of it.
Fig. 14 Where wrecked stems land, by size of cut, which is the distribution this row belongs in.

The other row that keeps nothing

A Lucas stem at a rise of 0.013, counted 4 and 7. Removing the 4-wall moves the next organ 29.5° and wrecks it; removing the 7-wall moves it 11.3° and wrecks it; removing both moves it 34.9° and wrecks it, keeping nothing.

It is not a coarse stem. It sits in the middle of the Lucas 4/7 rung, its front runs to seven, and both of its single removals already wreck — so unlike the coarse row, the pair is not producing a wreck where there was none. What it is producing is a wreck of a different kind.

Two rows, two routes to the same destination: one where the pair wrecks a stem single removals cannot, and one where the pair wrecks a stem single removals already do, differently. That the destination is the same in both is the part worth carrying, because it says the destination is a property of removing two organs rather than of which stem they were removed from.

Removing one wall of the slot, then the other, then both. The growing tip sits in a slot between its two chain-neighbours — the organ 4 places back and the organ 7 places back. Each bar is how far the next organ placed moves when those are removed, against a control sharing the same history. Either alone is a cheap removal: 29.5° and 11.3°, both inside the 45° that separates taking a neighbour from taking anything else. Together they move it 34.9°, against 40.8° for the two effects added, so the interaction is -5.9°. The slot is not two independent walls.
Fig. 15 The Lucas row, whose three cuts all wreck and whose pair keeps nothing.

What the four rows that do keep something keep

The smaller wall’s own family, on three of the four.

At the golden 0.013 and 0.010 stems the pair keeps the 5 and its multiples; at the Lucas 0.008 stem it keeps the 7 and its multiples; at the golden 0.008 stem it keeps the 5. In every case the surviving family is the smaller of the two counted numbers, which is the family the smaller wall belongs to.

That is four rows and it is worth stating as an observation rather than a rule. The census’s own reading is that the survivor is decided by the offset and not by which family is smaller, and four rows in which the offsets are the two counted numbers cannot separate the two.

Which hops survive one wall, the other, and both. One row per lattice. The last three columns are the lags whose hop the cut stem still holds, unchanged from a control that shares its history — the measurement that identifies what a wrecked stem has become. Removing a single wall always leaves something standing, which is what every single-organ cut in this collection does. Removing both leaves nothing at all on two of six lattices, including the coarse rung that no single removal can wreck. A stem that keeps no rigid hop is not a wrecked lattice with a slip in it; it is a stem that is no longer a lattice.
Fig. 16 What each cell keeps, on which four of the six pairs keep the smaller family.

What it costs to say this carefully

One sentence about what was already known, and it is the sentence that decides whether the row is a result.

Two organs wrecking a coarse stem is not new. What is new is that a named pair does it — the two organs the placement rule’s own nearest terms belong to, read off the stem rather than chosen — and that the wreck keeps nothing.

Without the first half the row is a rediscovery. Without the second it is one of four already-known arrangements. With both it is a specific claim about a specific pair, which is the form a single row can carry.

The mirror belongs to the lattice, not to the dose. How close the closest arrangement came to the mirror of the divergence it was cut from, against the share of the front that was removed. The marked point at 40 per cent is the coarse 3/5 rung with two organs taken, which reaches the mirror exactly. Every other point is a finer rung: five sizes of cut at 5/8 running from 13 to 63 per cent, and three organs at 8/13. Taking a larger share of a larger front than the coarse rung needs gets nowhere near, so the quantity that decides it is not the fraction of the neighbourhood removed.
Fig. 17 The share of arrangements that never repair, by how many organs are removed, which is the known result this row sits beside.

Why a coarse stem is hard to damage

Because there is so little of it within reach.

At a rise of 0.020 the organs are far apart up the stem relative to the circumference, so the placement rule’s neighbourhood holds few of them and the front — the stretch of offsets a removal is felt across — runs to five. Remove one of those five and the four that remain still pin the next placement.

That is the account the front thread gives and it makes the two-organ result straightforward rather than surprising: take two of five and three remain, which is apparently below whatever the rule needs.

What it does not account for is why the resulting stem keeps no lag. A stem with three organs pinning it could settle into a different lattice, and this one settles into something that is not a lattice by the test this thread applies.

Which rises are a lattice, from 0.045 to 0.125. How much the divergence wanders over the last sixty organs, at each rise up the coarse ladder. eleven of the 17 settle, scattering between 0.0000 and 0.3356 degrees. six do not: from 0.09 to 0.115 the divergence sticks on exactly 135.0000 degrees, which is three eighths of a turn, and wobbles about it by 0.79 to 1.60 degrees. A counter shown either kind returns the same pair, so the counts cannot tell them apart. The line is the threshold a rise has to pass before a cut is made on it, and it sits in the gap rather than among the measurements.
Fig. 18 The coarse end of the ladder, where a stem’s front is short and its neighbourhood holds few organs.

The one line

On the coarse 3/5 stem, removing either wall of the slot heals and removing both does not — with the next organ moving 47.8° against 41.7° and 20.6°, so the difference is not in the displacement. Two of the design’s six pairs leave no rigid hop standing at all, which no single removal in the census does, and neither of them lands on the mirror the coarse rung’s other two-organ wrecks reach.

Which hops survive one wall, the other, and both. One row per lattice. The last three columns are the lags whose hop the cut stem still holds, unchanged from a control that shares its history — the measurement that identifies what a wrecked stem has become. Removing a single wall always leaves something standing, which is what every single-organ cut in this collection does. Removing both leaves nothing at all on two of six lattices, including the coarse rung that no single removal can wreck. A stem that keeps no rigid hop is not a wrecked lattice with a slip in it; it is a stem that is no longer a lattice.
Fig. 19 The design’s six rows with what each cell keeps, on which two of the fourth column are empty.

Where this goes next

Removing one wall of the slot, then the other, then both. The growing tip sits in a slot between its two chain-neighbours — the organ 5 places back and the organ 8 places back. Each bar is how far the next organ placed moves when those are removed, against a control sharing the same history. Either alone is a cheap removal: 26.3° and 4.9°, both inside the 45° that separates taking a neighbour from taking anything else. Together they move it 164.1°, against 31.2° for the two effects added, so the interaction is +132.9°. The slot is not two independent walls. Both walls of the slot The growing tip sits between its two chain-neighbours. Removing either alone is a cheap removal on all six lattices — 2.3° to 41.7°. Removing both together throws the next organ past the expensive line on three of them, and the interaction runs from −25.8° to +132.9°. Which rises are a lattice, from 0.04 to 0.13. How much the divergence wanders over the last sixty organs, at each rise up the coarse ladder. 13 of the 19 settle, scattering between 0.0000 and 0.3356 degrees. six do not: from 0.09 to 0.115 the divergence sticks on exactly 135.0000 degrees, which is three eighths of a turn, and wobbles about it by 0.79 to 1.60 degrees. A counter shown either kind returns the same pair, so the counts cannot tell them apart. The line is the threshold a rise has to pass before a cut is made on it, and it sits in the gap rather than among the measurements. A stem coarse enough to cut Below the 3/5 rung is a 2/3 rung, and it runs from a rise of 0.050 to 0.120. It is not a lattice across all of it: from 0.090 to 0.115 the divergence stops settling and sticks on exactly three eighths of a turn, wobbling by a degree and a half — while a counter goes on reporting 2/3 as though nothing had happened. 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.

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.

  • One level and two exceptions — both name ablation, claim testing, control, lattice offset, measurement, mechanism, nearest neighbour, negative result, prediction, rigid hop
  • A step of one organ — both name ablation, claim testing, lattice offset, measurement, mechanism, negative result, prediction, rigid hop
  • Removing a neighbour costs least — both name ablation, control, lattice offset, measurement, mechanism, nearest neighbour, neighbourhood, prediction
  • The organ that moved furthest — both name ablation, claim testing, control, lattice offset, measurement, nearest neighbour, negative result, rigid hop
  • One offset, two answers — both name ablation, claim testing, control, lattice offset, measurement, negative result, rigid hop
  • The angle is not the actor — both name ablation, claim testing, control, lattice offset, mechanism, negative result, rigid hop

Named objects

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

AblationClaim testingControlDispersion relationThe range of the interactionLattice offsetMeasurementMechanismMirror ambiguityNearest neighbourNegative resultNeighbourhoodPredictionRigid hop