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

A removal that changes nothing

On one of the six lattices, taking away both walls of the slot moves the next organ 11.953125° — which is exactly, to the last digit, what taking away the larger wall alone moves it. The smaller wall's removal contributes nothing at all when the larger one is already gone.
13 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 · The damage has a period.

The two-by-two on the tip’s two chain-neighbours has six rows and an interaction on five of them. This essay is the sixth, where the interaction is exactly the size that makes the smaller removal disappear.

On a golden stem at a rise of 0.008, counted 5 and 8: removing the 5-wall moves the next organ 25.78125°. Removing the 8-wall moves it 11.953125°. Removing both moves it 11.953125°.

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: 25.8° and 12.0°, both inside the 45° that separates taking a neighbour from taking anything else. Together they move it 12.0°, against 37.7° for the two effects added, so the interaction is -25.8°. The slot is not two independent walls.
Fig. 1 The four cells of the design on the one lattice where the fourth is identical to the third.

The two numbers are the same number

Not approximately. The azimuths here are placed on a grid of 1,536 candidate positions, so a displacement is a whole number of grid steps of 0.234375°, and both of these are fifty-one steps. Bit for bit the same value.

That is what a grid buys and it is worth being precise about what it means. Two displacements equal to the last digit are two organs placed in the same grid cell; it does not follow that the underlying minimum is in exactly the same place, only that it is inside the same 0.234° of azimuth.

So the claim is that the smaller wall’s removal moves the next organ by less than a quarter of a degree once the larger wall is already gone, and by 25.78° when it is not.

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 interactions, of which this one is exactly minus the smaller removal’s own value.

What that means about the sum

The interaction on this row is −25.78°, which is precisely minus the smaller removal’s own displacement. Of the six rows it is the most negative and it is the only one whose value is a number from the same table rather than an arbitrary difference.

The other five interactions are −14.5°, −5.9°, +97.3°, +129.4° and +132.9°, and none of them is any of the twelve single-removal displacements. This one is.

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. 3 The neighbouring lattice, at a rise of 0.013, where the same design gives an interaction of +132.9°.

The obvious reading

Once the 8-wall is gone, the 5-wall is not deciding where the next organ goes. Remove it as well and the minimum does not move.

That is a statement about which terms of the placement sum are binding. The rule puts each organ where a sum over its neighbours is least with the weight falling as the cube of distance, so every organ within reach contributes and the question is always which contributions are doing the work. Here the answer appears to be that with the 8-wall present the 5-wall matters, and with it absent the 5-wall’s term is dominated by something else.

The something else is not identified. The design has four cells and cannot decompose a sum.

A stem gathers neighbours linearly; a growing disc barely gathers them at all. On a cylinder of circumference 1 with a rise of 0.02, the nodes within distance d number 2d/0.02 once d exceeds one turn — a fitted exponent of 1.020 and 100 per unit against the 100 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.
Fig. 4 The neighbourhood the rule sums over, in which two terms are removed and the rest are not.

Why it is not the obvious reading

Because it is one row.

Six lattices, one of which does this. The neighbouring lattices in the same design — golden at 0.010 above it and golden at 0.013 above that, both counted 5/8 — give interactions of +129.4° and +132.9°, which is the opposite behaviour by a wide margin. So whatever is happening at 0.008 is not a property of a 5/8 stem.

A reading built on one row of six is worth stating and not worth believing. It is here because a row where a measured quantity comes out exactly equal to another measured quantity is a row that should be shown rather than averaged into a summary.

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. 5 The six rows in order of rise, on which the three lattices sharing a counted pair do not agree.

What else is different about that row

Three things, and none of them obviously does it.

The 8-wall removal on this lattice is the only single removal in the design that heals — the stem repairs and keeps every lag it had. The other eleven either wreck or, on the coarse 3/5 stem, heal in a different way.

The pair on this lattice wrecks, so the fourth cell is a wrecked stem with the same first displacement as a healed one. That is a useful reminder that the first organ’s displacement is a transient measurement and says nothing about what the stem eventually becomes.

And the lags left standing differ: the 8-wall alone leaves everything, since it heals; the pair leaves the 5 and its multiples.

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. 6 What each cell leaves standing, on which this row’s third and fourth cells differ although their displacements do not.

The first organ is not the whole story

That is the general lesson from this row and it is worth more than the row.

Two cuts with identical first displacements produce different stems: one repairs and one does not. So the first organ’s displacement — the reading that separates a cheap removal from an expensive one — measures how hard the next placement is and not how much damage was done.

The two are correlated across the census and they are not the same quantity, and this row is the cleanest available demonstration that they come apart.

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. 7 The first organ’s displacement across every offset at this rise, which is the reading this row shows the limits of.

What a grid can and cannot say

Two displacements agreeing to the last digit is a strong-looking statement and it is bounded by the grid.

A finer grid would resolve the two, and they would then differ by something under 0.234°. That is the honest form of the claim: the smaller wall’s contribution, with the larger already gone, is under a quarter of a degree, and a finer grid would give a number rather than a zero.

This site has made the opposite mistake — read structure off a quantity the grid had quantised — so the discipline is to say which side of the resolution a claim sits on. This one sits below it and is stated as a bound.

The flat band, re-measured on a finer grid. A quantity that comes out constant is the first thing an azimuth grid should be suspected of, so the whole band is grown again on a grid of 6144 steps against the 1536 the site uses. The finer grid does resolve structure the coarse one flattened: a shallow minimum 0.0537 degrees deep, with its floor at a rise of 0.0201. What it does not do is separate the ends, which still agree to 0.0195 degrees while carrying opposite step orderings. The matched pair the band is for survives the check that would have broken it.
Fig. 8 One quantity re-measured on a four times finer azimuth grid, which is the check a claim at the resolution needs.

Whether it survives a finer grid

Not tested, and the reason is worth stating rather than hiding.

Re-running this row at four times the grid costs four cut stems and would answer it. It was not run because the answer does not change anything downstream: whether the smaller wall contributes 0.0° or 0.2° once the larger is gone, the row’s place in the design is the same and the interaction is the same to within a per cent.

It would matter if anything were built on the exactness, and nothing is. The claim carried forward is “under a quarter of a degree”, which the current grid supports.

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 7 places back and the organ 11 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: 18.3° and 2.3°, both inside the 45° that separates taking a neighbour from taking anything else. Together they move it 117.9°, against 20.6° for the two effects added, so the interaction is +97.3°. The slot is not two independent walls.
Fig. 9 The Lucas row at the same rise, where the same pair of removals gives +97.3° instead.

What a null cell is worth

A design with four cells and an interaction of zero in one of them is a design with a control it did not plan for.

The three rows with large positive interactions say two cheap removals can make an expensive one. The two with moderate negative interactions say the two effects partly cancel. This one says they can cancel completely — and having all three behaviours in six rows is what makes the claim “the pair is not the sum of its parts” a statement about the sign being unpredictable rather than about a consistent direction.

Without this row the negatives would be −14.5° and −5.9°, both small, and the natural summary would be “large and positive, or nearly nothing”. With it the range of negatives reaches the size of a whole single removal.

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. 10 The interactions sorted, on which this row is the extreme of one direction.

What would settle it

More lattices. Six is a small design and the interaction’s sign varies within it, so the obvious next step is to run the same four cells at a dozen more rises and see whether the sign tracks anything — the rise, the counted pair, the position inside a rung, the branch.

That is affordable: four runs a lattice, of which one is a control shared with everything else at that rise. Twenty lattices is eighty runs, which is less than a single band’s ablation costs.

It is not run here because the round’s slate was elsewhere, and it is named in the leavings for the round after.

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. 11 How the share of wrecked arrangements grows with the number of organs removed, which is the wider sweep this design is one cell of.

What the other five rows do

Two of them are the opposite of this one. At the golden 0.013 and 0.010 stems the two walls alone move the next organ by 26.3° and 4.9°, and by 25.3° and 8.9°, and the pair moves it 164.1° and 163.6° — so where this row’s smaller removal contributes nothing, theirs contribute more than either part.

Two more are mild. The golden 0.020 stem’s pair moves it 47.8° against a sum of 62.3°, and the Lucas 0.013 stem’s 34.9° against 40.8°: interactions of −14.5° and −5.9°, which is a partial cancellation rather than a complete one.

And the Lucas 0.008 stem is a third positive at +97.3°.

So the six rows are three strong positives, two mild negatives, and this one. It is the extreme of its direction and it is the only row whose value is a number from elsewhere in the same table.

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. 12 The six interactions in order, with this row at one end.

What a smaller wall is

On a 5/8 stem the 5-wall is the organ five places back and the 8-wall is the organ eight places back. Both are chain-neighbours of the tip; the 5 is nearer in the sequence and the 8 is nearer in azimuth on the other side.

Which of the two is geometrically closer to the next placement depends on the rise, and at a rise of 0.008 the two contact steps differ by a few per cent — this lattice sits inside the golden 5/8 rung, past its handover at 0.01558, where the 8-step is the shorter of the two.

So on this row the shorter step’s wall is the one whose removal decides everything, and the longer step’s wall contributes nothing once it is gone. Whether that is the pattern or the coincidence is exactly what six rows cannot say.

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. 13 The ratio between the two contact steps across the 5/8 rung, on which this lattice sits past the crossing.

Why one row is worth an essay

Because it is a null with a number attached, and those are rarer than nulls.

“Removing the smaller wall does nothing” is a claim that could be made about any row whose interaction happens to be small, and it would be unfalsifiable at any finite resolution. Here the interaction is not small: it is −25.78°, which is exactly the smaller removal’s own displacement, so the claim is that a specific quantity cancels a specific other quantity to the last digit of the grid.

That is a much narrower statement than “no effect”, and narrow statements are what this collection is for. It is still one row of six and it is still not believed.

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: 25.8° and 12.0°, both inside the 45° that separates taking a neighbour from taking anything else. Together they move it 12.0°, against 37.7° for the two effects added, so the interaction is -25.8°. The slot is not two independent walls.
Fig. 14 The row, with the third and fourth bars drawn at the same length.

Reading it against the front

A removal is felt across a front — the stretch of offsets at which taking an organ out has a lasting effect — and on a 5/8 stem the front runs to eight. Both walls of this stem’s slot are inside it.

That rules out the simplest account of the null, which would be that the 5-wall is too far away to matter. It is not: removed on its own it moves the next organ 25.78° and wrecks the stem. What is being claimed is conditional — it matters except when the 8-wall is already gone.

A conditional effect is what an interaction is, so the row is not anomalous in kind. It is anomalous in size, being the only one of the six where the condition removes the effect entirely.

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, 13 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. 15 The front by rise, on which both of this stem’s walls sit well inside it.

What it says about the cheap-removal reading

The reading is that taking a chain-neighbour of the tip costs less than taking anything else, and this row is consistent with it: 25.78° and 11.95° are both cheap removals, and the pair at 11.95° is cheap too.

What the row adds is that the cheapness of the pair is not a third data point. It is the second one again, so a table listing six pairs as six measurements of what a double removal costs would be listing five.

That is the kind of arithmetic a summary hides. Averaging the six fourth cells gives 92°, which is a number no cell holds and which is built from one value counted twice.

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. 16 What a removal costs the next organ across the census, which is the reading these six rows sit inside.

What a reader should do with a row like this

Notice it and not build on it.

The two things it is evidence for are both weak. That the two walls’ contributions can cancel completely is one row of six; that the smaller wall stops mattering once the larger is gone is a reading of that one row. Neither is asserted anywhere in the library, and the check that covers this design asserts only that the pair is not the sum of its parts, which this row satisfies as strongly as any.

What it is good for is the general point it makes without needing to be true: the first organ’s displacement and what the stem becomes are different measurements, taken from different parts of the same run, and they can agree exactly while the outcomes differ.

The next organ moves for the last 13, and for no othersOne 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.organ removed, counted back from the tiphow far the next organ moves, in degrees1138.0°284.4°353.4°4167.6°529.3°6101.7°7120.7°816.4°9165.2°1056.7°1181.1°12140.6°132.6°— the front ends here140.0°150.0°160.5°rise 0.005 · pair 8/13generated from a stated rule, not drawn to look right
Fig. 17 The displacement across offsets at this rise, on which two values coinciding is not rare.

Where this row sits on the ladder

At a rise of 0.008, which is inside the golden 5/8 rung — that rung runs from 0.01791 down to 0.00689 — and past its handover at 0.01558, which is the rise at which its two contact steps change places.

So this lattice is on the fine side of its rung’s crossing, in the stretch the ablation census turned out to be almost entirely sampling: twenty-five of thirty of its wrecked cuts sit past their rung’s handover.

Which means the row is not unusual in its position. It is one of the many stems grown at a rise a person would pick for a 5/8 lattice, and the two neighbouring lattices in this design — 0.010 and 0.013 — are on the same side of the same crossing and behave completely differently.

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. 18 Where each census lattice sits inside its own rung, on which this one is unremarkable.

The one line

On a golden stem at a rise of 0.008, removing the 8-wall moves the next organ 11.953125° and removing both walls moves it 11.953125° — the same value to the last digit of the azimuth grid, so the 5-wall contributes under a quarter of a degree once the 8-wall is gone and 25.78° when it is not. It is one row of six, the two neighbouring 5/8 lattices do the opposite, and it is shown rather than averaged because a measured quantity coming out exactly equal to another measured quantity is 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 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: 25.8° and 12.0°, both inside the 45° that separates taking a neighbour from taking anything else. Together they move it 12.0°, against 37.7° for the two effects added, so the interaction is -25.8°. The slot is not two independent walls.
Fig. 19 The row again, with the third and fourth cells drawn at the same length.

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°. 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. 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. 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.

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.

Named objects

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

AblationArtefactClaim testingControlDiscretisationThe range of the interactionMeasurementMechanismNearest neighbourNegative resultNeighbourhoodThe placement ruleResolutionSample size