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

Six lattices were not enough

The interaction between the two walls of a slot came back at −25.8° to +132.9° on six lattices, three above zero and three below, with no ordering by rise, by counted pair or by branch. A quantity that looks free on six rows is usually a quantity that has been sampled at six rows.
13 min read 20 figures State the claim, state the testCounted, not admired

Worth reading first: Both walls of the slot · The organ that was taken away · Where a handover sits.

A growing tip on a cylindrical stem sits in a slot. Two organs make its walls: the one that sits a smaller counted number of places back, and the one that sits the larger counted number back. Those two are its chain-neighbours, and between them is the gap the rule puts the next organ into.

Removing either wall alone is a cheap removal everywhere it has been tried. Removing both together is not the sum of removing each, and the difference — the interaction — was the measurement that thread was built to make.

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.
Fig. 1 The two-by-two at one lattice: neither wall removed, each alone, and both. The fourth cell is the measurement and the first three are what it is measured against.

What six lattices said

They said the interaction is large and its sign is unaccountable. It ran from −25.8° to +132.9°: three lattices well above zero, three well below, and nothing in the row — not the rise, not the counted pair, not which branch of the ladder the lattice sat on — that put the two groups on different sides of anything.

That is a real result about a design and a weak one about the world. Six numbers either side of zero is what a quantity with structure looks like when it has been sampled six times, and it is also what a quantity with no structure looks like. Six rows cannot tell those apart, and the sentence that was written down — with no ordering by rise, counted pair or branch — is a list of three columns that were looked at, not a statement that no column would work.

It is the same shape as the shortest hop that turned out to be a coin flip: a rule scored on a handful of rows, coming out at chance, and the honest reading being that the rows were too few rather than that the rule was wrong.

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. 2 The six lattices the design started on, drawn at the interaction. Three above zero, three below, and no column that sorts them.

The candidate that was named

The last time a quantity here looked free, the column doing the work turned out to be position inside the rung — how far down its own rung a lattice had been grown, measured in the logarithm of the rise rather than in the rise itself. It is the variable this site keeps finding underneath things, and it was the obvious one to test.

It is also a variable the six lattices could barely address. Four of them sit on one golden rung, at 34%, 61%, 84% and — for the coarsest — 89% of the way down a different one. Two sit on Lucas rungs. That is not a sweep of a rung; it is four points on one and a scatter elsewhere.

The golden ladder, rung by rung. Each bar is one rung — a run of rises over which a counter returns one pair — drawn from its coarse end on the left to its fine end on the right, with the whole bar scaled to the same width so that positions inside different rungs can be compared. The mark on each bar is the rise at which the two contact steps change places, and it falls between 12 and 35 per cent of the way down on every rung that has one. The dots are the lattices this collection's ablation census was grown at, dropped onto the rungs they belong to. They are not spread across the bars; seven of them sit past the mark.
Fig. 3 The golden branch’s rungs, with the lattices the census has been grown at marked on them. Most of the older design sits on one rung.

The design

Three lattices on every rung of the ladder, at 15%, 50% and 85% of the way down it in the logarithm of the rise. Eight rungs across the two branches gives twenty-four new lattices, and keeping the original six gives thirty.

Three positions is the smallest number that can show a trend rather than a difference. Two points make a line whatever they do; three can fail to, and a rung whose three lattices do not lie on a line is a rung saying something the design would otherwise have had to assume.

The positions are measured in the logarithm of the rise because the ladder is geometric. A rung near the coarse end spans two hundredths in the rise and one near the fine end spans two thousandths, so an absolute step is three different instruments on one ladder and a proportional one is a single instrument.

The column the census never carried. One row per lattice the ablation census was grown at. The bar shows where inside its own rung that rise sat, measured in the logarithm of the rise because the ladder is geometric, with zero the coarse transition and one the fine one. The mark on each bar is that rung's own handover, the rise where the two contact steps change places. Of the ten lattices that ever wreck, eight sit past their handover and one sit before it, with one sitting so close to one that the two steps differ by parts in a thousand. The rise was recorded in every table this collection has published; this fraction was in none of them.
Fig. 4 Position inside a rung, as this site measures it: nought at the coarse end, one at the fine end, in the logarithm of the rise.

Why not the ends

Nought and one are not sampled, and the reason is that a rung’s boundary is where the ladder sweep has already declared the counted pair to be in transition. A lattice grown there has a pair in doubt, and the whole design names two offsets from that pair. Doubt about which organs the walls are would be doubt in every cell of the two-by-two rather than in one of them.

Fifteen and eighty-five per cent are far enough in that the pair is settled at every one of the thirty, which is checked rather than assumed: the walls are counted from the points of the very stem the cuts are made in.

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.
Fig. 5 Why a rung needs more than one rise in it: quantities that look constant at one rise move across the rung they sit in.

Keeping the six

The six original lattices are rows of the new table rather than a thing it replaces. Every number the earlier design reported is recomputed here by the same call and answered from the same store, so the two tables cannot disagree about a row they share.

A design that re-sampled the ladder from scratch would have made every comparison an argument about two designs. This way, if the older six now read differently, the reading has changed and not the arithmetic — and they do not.

The slot interaction at 24 lattices, gathered by rung. One row per lattice, drawn at how much further the next organ moves when both walls of the slot are removed than the two single removals added together account for. Zero would mean the walls act independently. Of the 24 rows that are measurements, 13 are strongly positive and 11 are not, and every rung falls on one side or the other with nothing straddling.
Fig. 6 The thirty lattices in rung order, with the six that were there before among them rather than beside them.

What it cost

Four seconds a lattice, and ninety runs: three cuts and their controls at each of thirty rises. The expensive thing in this thread has never been the cutting; it is the stems the cuts are made in, and those are shared between the cells of a two-by-two by construction.

That is worth saying because the reason the design had six lattices was never cost. It had six because six looked like enough for a table with a sign in it.

Take away the organ five places back, and the next one goes into the hole. The last 30 organs of a stem at a rise of 0.013, unrolled. The open circle is the organ removed — five 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 26.2° apart, against a local spacing of 41°, and the vacancy itself is 36.1° from the undisturbed answer. Nothing else differs between the two runs: same rise, same history, same rule.
Fig. 7 One cut, drawn: the organ removed and the organs placed after it, against a control that shares the history below the hole.

The first reading: the range widened

Thirty lattices run from −104.3° to +135.2°. The new extreme is at the negative end and it is nearly four times the most negative of the six.

A widened range is the least interesting thing a bigger sample can do and it is worth pausing on anyway, because it says the six were not a spread of a distribution. They were part of one, and not the part that reaches furthest.

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. 8 The original six in lattice order rather than sorted, so the range they cover can be compared with the range of thirty.

And the table came apart into two groups

The second reading is the one that matters, and it is not a spread at all. On six of the thirty lattices, removing both walls costs exactly what removing the larger one alone costs — 35.9° and 35.9°, 12.0° and 12.0° — agreeing to the last digit of the grid the azimuths are placed on.

On those rows the smaller wall is free. Taking it away as well changes nothing.

Where taking the second wall as well changes nothing. Each row is one lattice, with three marks: how far the next organ moves when the smaller wall alone is removed, when the larger alone is removed, and when both are. On these lattices the third mark sits on the second, to within two steps of the azimuth grid. The smaller wall is free — taking it away as well changes nothing — and on a row like that the interaction is minus the smaller wall's own cost by construction, which is arithmetic and not a measurement. Three of them are the whole of one rung and the others are the fine ends of two more.
Fig. 9 The six lattices where the third mark sits on the second: removing both walls costs what removing the larger costs.

Which makes their interaction arithmetic

The interaction is both, less the smaller and the larger added. When both equals larger, that is minus the smaller — exactly, by cancellation, with no measurement left in it.

So the −104.3° that widened the range is the smaller wall’s own cost written with a minus sign in front. It is a cell of the table restated, not a seventh observation, and a table that leaves those rows in has its three most negative entries produced by algebra.

Where taking the second wall costs a great deal. Each row is one lattice, with three marks: how far the next organ moves when the smaller wall alone is removed, when the larger alone is removed, and when both are. Here the third mark is far to the right of both the others, which is the interaction the design was built to find.
Fig. 10 For contrast: lattices where the third mark is far to the right of both the others, which is what an interaction looks like.

Setting them aside is not discarding them

They are a finding of their own, and they get their own account. Three of the six are the whole of one short rung and the other three are the fine ends of two rungs whose coarse ends are not free at all. Whatever puts a lattice into that regime is worth knowing.

What they cannot do is carry a claim about an interaction, because on them the quantity is not measuring one. Leaving them in would be the same error as reading a summary statistic past the point where it still has a quantity — the number keeps being produced long after the thing it was a number for has gone.

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. 11 What the fourth cell leaves standing at each lattice, which is the other way of seeing that some pairs do nothing the larger wall does not already do.

Twenty-four measurements

That leaves twenty-four lattices where both walls are really there. Thirteen are strongly positive, at +85.1° to +135.2°; eleven are negative or nearly zero, at −25.1° to −0.5°. There is nothing between −0.5° and +85.1°: a gap of eighty-five degrees with no row in it.

A gap is worth more than a threshold. It means the two groups would still be two groups wherever anybody drew the line, and nobody had to choose where.

Five accounts of the sign, on the 24 lattices that are measurements. Each bar is how many of the lattices an account puts on the right side of zero. The six rows where the second wall is free are left out, because their value is minus the first wall's cost by construction and any rule scores whatever it happens to say about them. Position inside the rung, the rise and the branch all fail. The larger counted number sorts 22 of the 24, and the misses are one rung's worth of rows rather than a scatter.
Fig. 12 Five accounts of the sign, scored on the twenty-four lattices that are measurements rather than on all thirty.

The preview

Every rung’s lattices fall on the same side as each other. All of them, across both branches and a factor of thirteen in the rise. Which rung a lattice is on decides the sign, and where in the rung it sits does not.

That is the answer to the question the design was built to ask, and it is a negative: the candidate was refused, and the column that does the work is the one the candidate was going to be tested against.

The interaction across each rung, coarse end to fine end. One line per rung, drawn against where in the rung each lattice sits — nought at the coarse end, one at the fine end, measured in the logarithm of the rise. The lines are flat. Inside a rung the interaction moves by 3.3 to 13.6 degrees, against a spread of 240 degrees across the ladder, and it falls from the coarse end to the fine one on 6 of the 7 rungs. Position inside a rung was the candidate this design was built to test and it is not what decides the sign.
Fig. 13 One line per rung, against position inside it. The lines are flat, which is the candidate being refused.

What position does do

It moves the interaction by three to fourteen degrees across a whole rung, and it moves it downwards on six of the seven rungs that carry more than one measured lattice. Against a spread of a hundred and sixty degrees across the ladder that is a drift, not a sorting.

And it decides something else entirely: both of the rungs that go free go free at their fine end. So position sets whether the slot has a second wall in it, and the rung sets what removing both of them costs when it has.

The interaction across each rung, coarse end to fine end. One line per rung, drawn against where in the rung each lattice sits — nought at the coarse end, one at the fine end, measured in the logarithm of the rise. The lines are flat. Inside a rung the interaction moves by 3.3 to 13.6 degrees, against a spread of 240 degrees across the ladder, and it falls from the coarse end to the fine one on 6 of the 7 rungs. Position inside a rung was the candidate this design was built to test and it is not what decides the sign.
Fig. 14 The golden branch alone, where the drift down each rung and the one rung that goes free at its fine end are both visible.

What six could not have shown

Two of the original six are in the free regime and were counted as negative interactions. One more sits on a rung whose measured lattices are all near zero anyway. So of the six, three carried the positive signal and three carried something else, and the something else was two different things.

That is exactly the shape of a sample that produces “no ordering by anything”: it mixes regimes. The ordering was there and the design could not see it, which is a different failure from the ordering not existing.

The readings, and where in their rungs they fail. Each bar is one candidate account of which family a wrecked stem keeps, scored across every wrecked cut in the census. Under each bar are the positions inside their own rungs of the cuts it gets wrong, as percentages from the coarse end. The best of them is right 25 times of 30, and the positions of its failures are the point: two of them are the single lattice grown at the far fine end of its rung, which is also the only census row past three quarters of the way down. Nothing here rescues a reading. What it shows is that the table these readings were scored on varies a quantity nobody chose, over a range nobody stated.
Fig. 15 The accounts of a survivor that miss, scored the same way. A rule that sorts most of a table and misses on one group is the usual shape of an underpowered design.

What the bigger design still cannot do

It cannot say why a rung’s larger counted number should decide the sign of an interaction between two removals. It has a rule that sorts twenty-two of the twenty-four rows and an exception on the twenty-third and twenty-fourth, and no account of the mechanism behind either.

It also cannot rule out that three positions on a rung are three positions. A quantity that moves at the fifth decimal place of the rise would look flat here exactly as position looked flat in the six — and this site has already found one that does, on a band whose answer changes at a rise of its own.

The honest form of the claim is therefore about a resolution as well as about a table: at three positions a rung, the interaction is set by the rung and drifts by under fifteen degrees inside it. A finer sweep of one rung would test that, and it has not been run.

Where a rung's handover sits, over six rungs. The rise at which the two contact steps change places, as a fraction of the way down each rung from its coarse end, over every rung on both branches that has one. All six fall in the coarse half and none in the middle: the positions measure 6, 12, 15, 33, 35, 40 per cent. That is a fact about the arithmetic of the lattice rather than about any experiment run on it, because the crossing is where the two step lengths are equal and both are functions of the rise and the divergence alone.
Fig. 16 How the ladder’s own features distribute across a rung, which is the other reason three positions is a coarse instrument.

The instrument is the same instrument

Every cell of every two-by-two is a displacement of one organ against a control that shares its history to the last digit, on a stem whose heights are prescribed by the rise. Nothing here grows a stem at an angle and asks whether it looks right, and nothing compares two runs that do not share a past.

That is what makes thirty lattices comparable with six rather than a second experiment. The design got wider; it did not change.

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. 17 The displacement a single removal produces, offset by offset, which is the quantity every cell of the design is built from.

Two lattices where the pair wrecks what neither wall does

There is a third thing in the table, and it is the strongest form the interaction takes. On two of the thirty, either single removal heals — the stem recovers its divergence and its counted pair — and removing both wrecks it.

The earlier design had one such lattice. Thirty has two, at the fine end of the golden 3/5 rung and the fine end of the Lucas 3/4, and the second of them is also one of the free rows, which is a combination nothing predicted.

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.
Fig. 18 Which offsets wreck a stem and which it heals, across the census. A pair of removals that wrecks where each alone heals is off the bottom of this picture.

What a wider table is for

Not precision. The interaction at any one lattice was already measured to the grid the azimuths sit on, and thirty lattices do not make one of them more exact.

What it buys is the ability to be wrong in a stateable way. With six rows, “no ordering by anything” is a sentence about the sample. With thirty and a gap of eighty-five degrees in the middle of them, a rule that sorts the table can be written down, scored, and found to miss on one rung — and the rung it misses on can be named.

Five accounts of the sign, on the 24 lattices that are measurements. Each bar is how many of the lattices an account puts on the right side of zero. The six rows where the second wall is free are left out, because their value is minus the first wall's cost by construction and any rule scores whatever it happens to say about them. Position inside the rung, the rise and the branch all fail. The larger counted number sorts 11 of the 24, and the misses are one rung's worth of rows rather than a scatter.
Fig. 19 The same five accounts in a fixed order rather than sorted by score, so the ones that fail can be read as easily as the one that does not.

Why the walls are counted rather than named

The two offsets are read off the very stem the cuts are made in, not from a table of what a lattice at that rise ought to have. Those two would agree nearly always, and the reason for insisting is that the whole design consists of naming two offsets: an offset read from the wrong stem names an organ that is nobody’s neighbour, while every other number in the row stays perfectly plausible.

That failure has already happened once in this thread, when a stem grown at the wrong starting angle returned a plausible answer to a question nobody had asked. It cost a table that had to be thrown away, and the check that would have caught it is the one now run on every row.

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. 20 The census the walls are counted against, lattice by lattice, so that a named offset can be checked against the stem it was named on.

The one line

Six lattices reported an interaction with no ordering in it. Thirty report two regimes and a rule: six rows where the second wall is free and the number is arithmetic, and twenty-four where every rung falls on one side of zero and the side is set by the rung’s larger counted number.

The candidate the design was built to test — position inside the rung — moves the interaction by a few degrees and decides nothing about its sign. It does decide whether there is a second wall to remove.

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°. Five accounts of the sign, on the 24 lattices that are measurements. Each bar is how many of the lattices an account puts on the right side of zero. The six rows where the second wall is free are left out, because their value is minus the first wall's cost by construction and any rule scores whatever it happens to say about them. Position inside the rung, the rise and the branch all fail. The larger counted number sorts 22 of the 24, and the misses are one rung's worth of rows rather than a scatter. The rung decides the sign Twenty-four lattices where both walls of the slot are really there. Thirteen give a strongly positive interaction, at 85° to 135°; eleven give a negative or null one, at −25° to −0.5°. Nothing lies between. Every rung's lattices fall on the same side as each other. Where taking the second wall as well changes nothing. Each row is one lattice, with three marks: how far the next organ moves when the smaller wall alone is removed, when the larger alone is removed, and when both are. On these lattices the third mark sits on the second, to within two steps of the azimuth grid. The smaller wall is free — taking it away as well changes nothing — and on a row like that the interaction is minus the smaller wall's own cost by construction, which is arithmetic and not a measurement. Three of them are the whole of one rung and the others are the fine ends of two more. When the second wall is free On six of thirty lattices, removing both walls of the slot costs exactly what removing the larger one alone costs — 35.9° and 35.9°, 12.0° and 12.0°, agreeing to the last digit of the grid the azimuths sit on. The smaller wall is not a wall on those rows.

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.

  • Three offsets, three crossings — both name ablation, claim testing, control, honest limits, lattice offset, matched design, measurement, negative result, parastichy pair, rise, rung, sampling, underdetermination
  • The exception was already labelled — both name ablation, claim testing, control, honest limits, the range of the interaction, lattice offset, matched design, measurement, negative result, parastichy pair, rise, rung
  • One offset, two answers — both name ablation, claim testing, control, honest limits, lattice offset, measurement, negative result, parastichy pair, rise, rung, underdetermination
  • The offsets that never change — both name ablation, claim testing, control, honest limits, lattice offset, measurement, negative result, parastichy pair, rise, rung, sampling
  • The side the census sat on — both name ablation, claim testing, control, honest limits, lattice offset, negative result, parastichy pair, rise, rung, sampling, underdetermination
  • One rung, two answers — both name ablation, control, honest limits, lattice offset, measurement, negative result, parastichy pair, rise, rung, underdetermination

Named objects

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

AblationClaim testingControlHonest limitsThe range of the interactionLattice offsetMatched designMeasurementNearest neighbourNegative resultParastichy pairRiseRungSample sizeSamplingUnderdetermination