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

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.
13 min read 21 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.

Twenty-four lattices, one two-by-two each, and the interaction between removing the two walls of the slot: thirteen at +85.1° to +135.2° and eleven at −25.1° to −0.5°. Between −0.5° and +85.1° there is nothing.

Eighty-five degrees of empty table is worth more than any threshold anybody could have chosen. It means the two groups would still be two groups wherever the line went, and nobody had to put it anywhere.

The six rows where the second wall is free are not in this count. On those the interaction is the smaller wall’s own cost with a minus sign, produced by cancellation rather than by measurement, and leaving them in fills the negative end of the table with arithmetic.

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. 1 The twenty-four lattices where both walls are really there, in rung order, drawn at the interaction. The gap through the middle is the result.

Every rung agrees with itself

The lattices are not scattered across the gap in some order of their own. They come in rungs, three or four to a rung, and every rung puts all of its lattices on one side. There is no rung with one lattice above the gap and another below it.

That is eight rungs, both branches of the ladder, and a factor of thirteen in the rise between the coarsest and the finest. Whatever decides the sign is a property a whole rung has, and it does not weaken across the rung.

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. 2 One line per rung against position inside it. The lines are flat and none of them crosses the middle.

Which is not what the design expected

The design was built to test position inside the rung, which is the variable this site keeps finding underneath things. It is refused: knowing where in its rung a lattice sits tells nothing about the sign, and the same position on two different rungs gives +135.2° and −16.6°.

What position does is move the interaction three to fourteen degrees down the length of a rung. Against a spread of a hundred and sixty degrees that is a drift.

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. 3 Position inside a rung, the quantity that was refused. Nought at the coarse end, one at the fine end, measured in the logarithm of the rise.

Four candidates, scored

Five accounts were written down before the table was read: position inside the rung, the rise itself, which branch the lattice is on, the smaller counted number and the larger counted number. Each is a rule that puts a row above or below zero, and each is scored on all twenty-four.

Position gets fourteen. The rise gets sixteen. The branch gets thirteen, which is a coin. The smaller counted number gets nineteen. The larger gets twenty-two.

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. 4 The five accounts, scored. Four of them do about as well as guessing and one of them does not.

The rule

Rungs whose larger counted number is 8, 11 or 13 give positive interactions. Rungs whose larger number is 3, 5 or 7 give negative or null ones. That is the golden 5/8, the golden 8/13 and the Lucas 7/11 on one side, and the golden 2/3, the golden 3/5, the Lucas 1/3 and the Lucas 4/7 on the other.

It is not a rule about the branch: it puts a Lucas rung with the golden ones and two golden rungs with the Lucas one. It is not a rule about the rise either: the Lucas 3/4 rung and the golden 3/5 rung overlap in rise and sit on opposite sides.

Every transition as the rise falls. The pair climbs 1/2 → 2/3 → 3/5 → 5/8 → 8/13. Consecutive transitions are 0.382, 0.382, 0.382 of the previous rise — 1/φ² is 0.3820.
Fig. 5 The whole ladder, both branches, so the rungs the rule separates can be seen in the order they occur rather than in the order the table lists them.

The exception

The Lucas 3/4 rung has a larger counted number of 4, so the rule says negative, and its two measured lattices are +90.5° and +85.1°. That is the whole of the rule’s error: two rows out of twenty-four, on one rung, in the same direction.

An exception on one rung is a different thing from a rule that misses at random. It has a name and it was already labelled, for reasons that have nothing to do with this measurement.

The Lucas 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 6 and 40 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; three of them sit past the mark.
Fig. 6 The Lucas branch, where the exception lives. Its second rung is the one this table cannot sort.

Why five candidates and not one

Writing the candidates down before reading the table is what makes twenty-two of twenty-four a score rather than a description. A rule found by looking at the rows and then quoted against them has no denominator: there are always rules that sort twenty-four rows, and the question is how many were available.

Five were written down, and four of them are the columns anybody would reach for first. That the winner is the fifth, and wins by six rows over the runner-up, is the whole of the evidence — and it is the discipline this site applies to a survivor rule, where the account that sounded right scored at chance.

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. 7 Accounts of a survivor scored the same way, on a different table. A rule with a written-down list of rivals is the only kind whose score means anything.

What the sizes look like

On the positive rungs the pair removal moves the next organ 117° to 164°, while each wall alone moves it 1.4° to 49°. On the negative rungs the pair moves it 31° to 78°, and the larger of the two single removals moves it 24° to 34°.

So the difference is not that the negative rungs have a small interaction. It is that on the positive rungs the pair does something categorically different: it throws the organ most of a turn, where neither wall alone moves it a tenth of one.

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. 8 Lattices where the pair costs far more than either wall: the third mark is a long way to the right of both the others.

Which is a threshold in the fate, not in the arithmetic

Forty-five degrees is where this thread separates a cheap removal from an expensive one, and it is a gap in the data rather than a line anybody drew: the two populations measure 9° to 31° and 63° to 168°.

On the positive rungs the pair removal is past that line on every lattice. On the negative rungs it is under it on seven of the eleven. So the rule that sorts the interaction also sorts whether the pair is a cheap removal, which is a second reading arriving at the same partition.

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.
Fig. 9 The two populations of single removals, with the gap between them. The pair removals on the positive rungs land far to the right of both.

And a third reading agrees

What the cut stem keeps is a third instrument. On the positive rungs the pair removal leaves the smaller counted number’s own family standing — the five-hop on a 5/8 lattice, the eight-hop on an 8/13, the seven-hop on a 7/11 — and destroys everything else.

On the negative rungs it leaves either everything or a scatter, and on three of them it leaves the whole ladder of lags intact, which is what a stem that healed looks like.

Three quantities — a displacement, a fate and a set of surviving hops — measured independently and cutting the table the same way. That is the shape this collection trusts most: two routes arriving at one number, with a third behind them.

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. 10 The lags a stem still holds after each removal. The pattern down the last column is the third reading.

A larger number is a smaller step

The obvious thing about 8, 11 and 13 against 3, 5 and 7 is that they are bigger, and the obvious meaning of a bigger counted number is a finer arrangement: more organs round the stem before the pattern repeats, so a smaller angle between neighbours in a chain and a shorter step across the surface.

That is measurable rather than rhetorical. On the positive rungs the smaller wall’s removal costs 14.5° to 25.8°; on the negative rungs it costs 29.5° to 65.9°. So the walls are cheaper to remove where the pattern is finer, and the pair is more expensive.

Which offsets give short hops, at a rise of 0.013. The two lowest points are at 5 and 8, and those are the parastichy numbers. Offset 1 is high because a hop of one node is at least the rise, which is what makes a stem easier to count than a disc.
Fig. 11 The steps across the surface at every lag, at one rise. The two contact numbers are the two shortest, and how short depends on the rung.

Which suggests a mechanism, and does not establish one

A fine arrangement has a narrow slot with two close walls. Removing one leaves the other, and the organ shifts a little. Removing both leaves an opening several organs wide and the organ falls right across it.

A coarse arrangement has a wide slot already. Removing a wall changes it by a fraction of what it already is, and removing both changes it by a bit more. That is a story that fits, and this collection does not accept a story that fits as a result — it is a description of the numbers, not a mechanism behind them.

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.
Fig. 12 The slots a tip can sit in, drawn on a fine lattice where the walls are close together.

What would test it

The story predicts a quantity nobody has measured: the angular width of the slot after each removal. If the account is right, the width after removing both should be several divergence steps on the positive rungs and about one on the negative ones, and the interaction should follow the width rather than the counted number.

That is a measurement on stems already grown, and it would separate the story from the rule — because “the larger counted number is 8 or more” and “the slot opens by more than two steps” agree on this table and need not agree everywhere.

A stem gathers neighbours linearly; a growing disc barely gathers them at allOn a cylinder of circumference 1 with a rise of 0.013, the nodes within distance d number 2d/0.013 once d exceeds one turn — a fitted exponent of 1.009 and 154 per unit against the 154 the geometry fixes. In the disc model an element of age k sits at radius e^(0.4k), so each doubling of the distance adds the same two or three neighbours rather than twice as many.123-0.50000.50011.50distance from the node, log₁₀neighbours, log₁₀a stemslope 1a disca logarithmrise 0.013 · 9231 nodes · meristem growth 0.4slope 1.009 against slope 1
Fig. 13 The neighbours of an organ with their distances, which is where the slot’s width would be read from.

The rule is a fit and it should be said so

Twenty-four rows, five candidate rules written down in advance, and the best of them getting twenty-two. That is a good score and it is a score over a small table, with a rule whose threshold — larger counted number of 8 or more — sits in a gap between 7 and 8 with no rung in it.

There is no rung whose larger number is 8, 9 or 10 other than the golden 5/8. So the threshold is fitted between two adjacent values and any number from 7.5 to 8 gives the same table. What would test it is a rung with a larger number of 9 or 10, and neither ladder has one.

Every family but two is the sum of two others. Four heads and the contact families each actually has. An arc arrives at every family that is the sum of two smaller ones; the two with no arc arriving are the generators, and on every head they are the two smallest. whorled, 144°: 2, 3, 5 — golden, 137.508°: 8, 13, 21, 34, 55, 89 — Lucas, 99.502°: 11, 18, 29, 47, 76 — rational, 137.5°: 8, 13, 21, 34, 55, 89 — 137.0°: 8, 13, 21, 29, 50, 71, 92, 113. So once a pair is counted the rest is arithmetic, and a third counted family is a prediction rather than a second measurement.
Fig. 14 The two sequences the ladder is built from, which is why the counted numbers available are the ones they are and not others.

Where a third sequence would help

There are additive sequences other than the golden and the Lucas, and stems do settle on them — the settling table reaches several of them. A sequence containing 9 or 10 as a larger counted number would give exactly the rung this rule needs to be tested against.

Nothing in this thread has grown a stem on one deliberately. It is a design with twelve runs in it and it would either break the rule or double the evidence for it.

The 10 sequences the settling table's destinations belong to. Every pair the counter returns is two consecutive terms of a sequence in which each term is the sum of the two before it, which is the ladder's own rule. two of these are the sequences this collection is built on. The rest are not, and they are not rare: the 1, 4, 5, 9, 14 sequence and the 2, 5, 7, 12, 19 sequence each supply several destinations, at every falloff exponent the table is grown at. The right-hand column is the divergences that land on each.
Fig. 15 The additive sequences the settling test reaches, several of which carry counted numbers the ladder does not.

What the negative rungs are doing

Nothing much, and that is worth stating plainly. The eleven negative rows run −25.1° to −0.5°, and four of them are within ten degrees of zero, which is where this thread stops calling an interaction one.

So the honest partition is not positive against negative. It is large and positive against small, with the small ones straddling zero in a way that is consistent with the two walls acting nearly independently on the coarse rungs.

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. 16 The six lattices the design began with, sorted. Three of them are in the small group and reading them as negative interactions was reading noise as a sign.

Which changes what the earlier result said

The first version of this measurement reported an interaction that was “reliably not the sum of its parts and not reliably larger than it either”. The second half of that is now wrong.

On the rungs where there is an interaction at all, it is always larger: thirteen lattices, every one of them positive by eighty-five degrees or more. The negative half of the earlier sentence was two arithmetic rows and a set of near-zeros.

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. 17 The two-by-two at the lattice the earlier thread reported from, which is one of the strongly positive ones.

Two rungs the design nearly missed

The golden 2/3 rung and the Lucas 1/3 rung are the coarsest on either branch, and neither had a lattice in the six the design began with. Both are negative here, and the Lucas 1/3 is negative for the wrong reason — all three of its rows are ones where the second wall is free and the number is arithmetic.

So of the four rungs the rule calls negative, one contributes no measurements at all. The rule is really sorting three negative rungs against four positive ones, which is a smaller table than twenty-four rows sounds, and the essay says so rather than quoting the row count.

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. 18 The golden branch’s rungs with the census marked on them. The coarsest rung carried nothing until this design.

The instrument did not change

Every cell is a displacement against a control sharing its history to the last digit, on a stem whose heights are prescribed by the rise, with the walls counted from the points of the stem the cuts are made in. Thirty lattices is the same experiment run more often.

That matters because the conclusion reverses a sentence. A reversal produced by a changed instrument is a comparison of instruments; this one is a comparison of sample sizes.

One rule at p = 1, cut three ways. loop cut at 3/√h: 8/13 at 137.62° with 0.58° of scatter. exponential cut-off, 3: 8/13 at 137.58° with 0.50° of scatter. no cut at all: no lattice, 44° of scatter. The first two agree to 0.03° — the prediction held — and the third is what the same rule does when nothing cuts it.
Fig. 19 The check that a continued run is the rule and not a second implementation of it, which is what makes two runs comparable at all.

What is not explained

Why a counted number should decide anything about an interaction between two removals. The rule sorts twenty-two of twenty-four rows and offers no reason, and the story about slot widths is a story until the widths are measured.

This collection has a standing position on that: a rule that sorts a table is worth recording and is not an explanation, and saying which is which is most of the discipline.

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. 20 The accounts in a fixed order rather than by score, so the four that fail are as legible as the one that does not.

What a reader should not take from this

That a finer arrangement is more fragile. The interaction is about a pair of removals and says nothing about how often a stem gets one: which offsets wreck a stem at all is a separate census with a separate answer, and the fine rungs are not more prone to wrecking than the coarse ones.

Nor that the larger counted number is doing anything causal. It is a label on a rung, and the rung carries a rise, a divergence, a slot width and a step length, all of which move together down the ladder. Every one of those would sort this table as well as the counted number does, and the counted number is only the cheapest of them to write down.

The hops of a 8/13 lattice, shortest first — golden, rise 0.005. Every lattice hop at this rise, ordered by how long its step is across the surface of the stem, with the two that a wrecked stem here leaves standing picked out. The two shortest are the contact families the counter returns, 8 and 13, and they differ in length by a factor of 1.088. The lags left standing after a removal are 4 and 8, sitting at rank 37 and 2 in this order, so the family the rule holds is a short step but not always the shortest one.
Fig. 21 The lags ordered by step length at one lattice, which is one of the several quantities that move together as the ladder is descended.

The one line

On the twenty-four lattices where the slot has two walls, the interaction between removing them is decided by the rung and not by where on the rung the lattice sits: positive and large where the larger counted number is 8, 11 or 13, small or negative where it is 3, 5 or 7.

Twenty-two rows of twenty-four, with the two misses on one rung and in one direction.

Where this goes next

The slot interaction at 30 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. The pale rows are the ones where removing the second wall costs nothing at all, so their value is minus the first wall's own cost and is arithmetic rather than a measurement. 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. 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. 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. 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. The exception was already labelled The larger counted number sorts twenty-two of twenty-four lattices by the sign of their slot interaction. Both misses are on the Lucas 3/4 rung — the one rung a different measurement had already singled out, for reasons with nothing to do with this one.

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.

  • Every rise of a band — both name ablation, claim testing, control, lattice offset, matched design, measurement, negative result, parastichy pair, rise, rung
  • Seven rises and two seeds — both name ablation, control, fibonacci, ladder, lucas numbers, matched design, measurement, parastichy pair, rise, rung
  • Three offsets, three crossings — both name ablation, claim testing, control, lattice offset, matched design, measurement, negative result, parastichy pair, rise, rung
  • One offset, two answers — both name ablation, claim testing, control, lattice offset, measurement, negative result, parastichy pair, rise, rung
  • The front deepens down a rung — both name ablation, claim testing, control, lattice offset, measurement, negative result, parastichy pair, rise, rung
  • The hole on the other branch — both name ablation, fibonacci, ladder, lattice offset, lucas numbers, matched design, measurement, parastichy pair, rung

Named objects

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

AblationClaim testingControlFibonacciGeometric ladderThe range of the interactionLadderLattice offsetLucas numbersMatched designMeasurementNearest neighbourNegative resultParastichy pairPredictionRiseRung