Phyllotaxis
the spirals are counted, not admired
Stems and cones

The end of a wrecked run

The obvious follow-up to a transition in where one organ goes is whether the stem also finishes somewhere different. On this rung the question has no answer: a wrecked run's final divergence takes four values and changes between rises a thousandth apart, three times inside a nine-rise sweep.
12 min read 24 figures State the claim, state the testCounted, not admired

Worth reading first: Both walls of the slot.

The slot design measures the first organ placed after a cut. That is what the whole two-by-two is built on, it is what the free rows are free in, and it is what falls by a hundred and fifty-four degrees at one rise of the golden 5/8 rung.

The natural next question asks about the other end. If the first organ goes somewhere different, does the stem also finish somewhere different?

Where the doubled cut's run finishes, at every rise of the sweep. The divergence the wrecked run ends at, rise by rise. It takes only a handful of values and jumps between them at rises one part in a thousand apart, 4 times inside the 9 rises of the finest sweep alone — where the walls, the block, the intact stem's divergence and the rigid lags are all held. So the end of a wrecked run is not a stable quantity on this rung, and no statement about where the stem finishes is available on either side of the transition. The first organ's displacement is the reproducible half of the same measurement.
Fig. 1 The divergence the doubled cut’s run ends at, at every rise of the swept stretch.

The question is well posed

It is not a fussy one either. A transition in a placement that washed out over the next few hundred organs would be a transient, and one that persisted to the end of the run would be a change in the arrangement the stem settles into.

Those are different things and the collection has the number that would separate them. Every cut run records the divergence it finishes at, and it has done since the design was written.

Two runs of the same rule from unrelated starting angles. Both settle at 137.0°, within 0.5° of the golden angle, from seeds 166° apart.
Fig. 2 A run settling to a divergence, which is the quantity the end of a run is read as.

And it has no answer here

Across the twenty-nine swept positions, the doubled cut’s run ends at 65.25 degrees, somewhere between 137.25 and 137.82, at 182.84, or at 209.56. Four values, and it moves between them at rises one part in a thousand apart.

Inside the nine-rise fine sweep alone it changes three times, over a stretch where the walls, the block, the intact stem’s settled divergence and the rigid lags are all held.

That is the same stretch on which five quantities are identical either side of the transition. A sixth quantity that changes three times inside it is not evidence about the transition; it is evidence about itself.

Where the doubled cut's run finishes, at every rise of the sweep. The divergence the wrecked run ends at, rise by rise. It takes only a handful of values and jumps between them at rises one part in a thousand apart, 4 times inside the 9 rises of the finest sweep alone — where the walls, the block, the intact stem's divergence and the rigid lags are all held. So the end of a wrecked run is not a stable quantity on this rung, and no statement about where the stem finishes is available on either side of the transition. The first organ's displacement is the reproducible half of the same measurement.
Fig. 3 The same reading, where neighbouring rises finish at different values with nothing else moving.

Which is not a defect

A run that has been wrecked has no reason to settle anywhere in particular. The doubled cut wrecks the stem at twenty-five of the twenty-nine positions — the two that recover both sit above 87 per cent of the rung, far below the transition — so the number being read is a wrecked run’s endpoint.

A stem that does not settle onto its branch is a known state in this collection and it is measured rather than excluded. This is the same state arrived at by damage rather than by the rise, and the floor it sits under is a wall rather than a budget — a longer run does not rescue it.

The same table, grown 2.7 times as long. Every rise and every starting angle, grown to 1200 organs and then to 3200. The two middle columns are how many starting angles reached a lattice at each length, and they are the same column: of the 72 pairs of runs, 72 are identical organ for organ and 0 settle at the longer length after failing at the shorter one. Tripling the budget buys nothing anywhere. What the fine rises are short of is not run length: the share of starting angles that reach a lattice at all falls from 7 of 9 to 1, so the arrangements a stem could fall into have mostly stopped existing.
Fig. 4 Runs that settle and runs that do not, which is the distinction a wrecked run’s endpoint falls on the wrong side of.

The two halves of one run behave differently

That is the finding worth keeping. The first organ’s displacement is reproducible, sits exactly on the azimuth grid, is flat across half a rung, and moves once in twenty-nine positions.

The end of the run is none of those things. Same runs, same code, same tolerances, and one quantity is a measurement while the other is a coin.

Nothing about the code distinguishes them. Both are read off the same object at the same moment, and the difference is entirely in how many placements stand between the cut and the number.

The golden 5/8 rung swept at 27 rises, with each removal's cost. How far the first organ placed after a cut moves, at every rise the sweep visits, coarse on the left. Removing both walls costs far more than removing the larger one alone above a rise of 0.00998, and exactly what the larger one costs below it. The change happens in one step of the grid the ladder is named on: 163.59 degrees at 0.00998 and 9.14 degrees at 0.00997, which is a fall of 154.5 degrees for a change of one part in a thousand in the rise.
Fig. 5 The first organ’s displacement across the same rises, which is flat, then flat, and moves once.

Why the first organ is stable

Because it is decided by one thing. The organ after a cut is placed at the minimum of a sum over its neighbours across 1,536 candidate azimuths, and the neighbourhood it is placed against is the stem with two organs missing.

Two rises a thousandth apart give neighbourhoods that differ by a fraction of a degree in every hop, so the minimum is in the same place and the placement is the same grid step. That is why 163.59 degrees repeats exactly across four positions spanning half a rung.

The rule, 26 steps in, at a growth of 0.40. The next primordium goes where the repulsion is least — the marked minimum at 216°. Nothing in the rule refers to any particular angle.
Fig. 6 The sum an organ is placed at the minimum of, which is what makes the first placement reproducible.

And why the end is not

Because it is decided by three hundred of those placements in sequence, each against a neighbourhood the previous ones made. A difference of a grid step early on is a different neighbourhood for the next organ, and so on.

That is the ordinary behaviour of a sequential rule and it is why the ladder itself is measured on the intact stem rather than on damaged ones. A wrecked run is a run whose state has left the basin the intact run sits in.

What the divergence does while the pattern climbs. The stem produces a sequence rather than a constant. Over the second half of the run it stays within 4.7° of 137.51°, and the vertical marks are where the counted pair changed — the wander is largest around them.
Fig. 7 A run’s divergence organ by organ, of which the end is three hundred placements downstream of the start.

The four values are not arbitrary

They are 65.25, about 137.3, 182.84 and 209.56 degrees. Two of those are recognisable: 137.3 is near the golden angle the intact stem settles to, and 209.56 folds to 150.4, which is a destination the settling table reaches at several rises.

So a wrecked run is not going anywhere; it is going to one of a handful of places. That is consistent with the picture the settling work gives and it is not a measurement of anything on this rung, because which of the four it picks is not stable.

Everywhere a cut of one to five organs can send a 5/8 stem. Every settled divergence reached by any arrangement of up to five organs removed from a stem at the 5/8 rung, on one axis. There are six of them and no more. three are slips of the lattice the stem was cut from: each keeps the lag-5 family intact and sits a whole number of turns of it from the next, which is the ladder marked below the axis with rungs 72.0 degrees apart. The other three keep no lag at all and sit near a fraction of a turn, marked above: 175.0 degrees near 1 of 2 turns, 190.0 degrees near 1 of 2 turns, 235.0 degrees near 2 of 3 turns. A stem that is cut either slides along the ladder it was on or leaves it for a lattice with files in it.
Fig. 8 The destinations runs settle to, of which a wrecked run’s endpoints are a subset.

What that rules out

The reading the transition changes where the stem finishes is unavailable, and so is the reading the transition does not change where the stem finishes. Both would be statements about a quantity that changes at rises where nothing else does.

That is a stronger negative than it was looked for and did not change, and it is worth distinguishing. A quantity with no reproducibility supports no claim in either direction.

Where the doubled cut's run finishes, at every rise of the sweep. The divergence the wrecked run ends at, rise by rise. It takes only a handful of values and jumps between them at rises one part in a thousand apart, 4 times inside the 9 rises of the finest sweep alone — where the walls, the block, the intact stem's divergence and the rigid lags are all held. So the end of a wrecked run is not a stable quantity on this rung, and no statement about where the stem finishes is available on either side of the transition. The first organ's displacement is the reproducible half of the same measurement.
Fig. 9 The endpoint across the sweep, which changes at rises where the lattice does not.

What it does to the slot table

Nothing, because the slot table never used the endpoint. Its four cells are displacements of first organs and its interaction is arithmetic on those.

The reason to report this is that the endpoint is in the data — every cell carries it — and a reader with the table in front of them would reach for it. The honest label on that column is that it is not a measurement on this rung.

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. 10 The four cells of the design, each of which carries an endpoint the design does not use.

And to the rest of the thread

Also nothing, but the check is worth stating. The ablation census reads a wrecked stem’s displacement profile, folded onto the lag it kept, over a window at the top of the run. That is a different quantity from a single settled divergence.

A profile is a difference between two runs organ by organ, and it can fall into a stable pattern even where neither run settles to a fixed angle. The census’s own periodicity classification is a reading of that pattern, and its five non-periodic rows are exactly the ones where it does not.

How far every organ moved, 5 places back at a rise of 0.01. One mark per organ above the hole, at the angle it sits from where the same organ sits in a control that shares its history. The collection has read two numbers out of profiles like this one — the largest displacement anywhere, and the first organ's — and never the profile. It is not a bump that decays. After a transient that does not end inside this run it settles into a repeating pattern of five levels, one per residue class modulo 5, which is the lag whose hop this stem kept. one of those levels sit together and four do not.
Fig. 11 A wrecked stem’s displacement profile, which is a difference between two runs rather than one run’s endpoint.

Which is a distinction worth having

A run that never settles can still damage its control in a repeatable way. That sounds paradoxical and it is not: the difference between two runs of the same rule can be steady while both of them wander together.

Whether that is what happens here is not established. It is the reading that reconciles a stable displacement profile with an unstable endpoint, and testing it would mean looking at the control’s endpoint as well.

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. 12 A cut run and its control, whose difference is the profile and whose endpoints are separate quantities.

The control’s endpoint

That check is cheap and it has not been run. The control shares the history below the hole and is continued without a removal, so it is an intact run and should settle to the rung’s own divergence — 137.438 degrees on this stretch.

If it does at every position, then the instability belongs entirely to the cut run and the comparison is between a settled control and an unsettled cut. If it does not, the reading is about the rule at these rises rather than about the damage.

How many organs a stem needs before it is on a lattice. One row per rise, one mark per starting angle, placed at the organ from which every later divergence stays within a degree and a half of the run's own final value. Where a stem settles at all it does so between 0 and 290 organs in, against the 400 every ablation run here grows before it cuts anything. Not one row needs the length it is given. What changes down the table is the count on the right: how many of the nine starting angles reach a lattice at all, which falls from 7 at the coarse rises to 1 at the finest.
Fig. 13 How often runs settle at all, which is the question the control’s endpoint would fall under.

Two rises apart is enough to change it

The strongest form of the instability is worth stating with the numbers. At rises of 0.00997 and 0.00996 the endpoint is 65.25 degrees; at 0.00995 it is 137.25; at 0.00994 it is 65.25 again; at 0.00993 it is 137.26.

Those four rises span four parts in a hundred thousand. Everything else about them is identical to the digits this collection quotes.

What is the same at 0.00998 and at 0.00997. Six quantities read on the two rises the transition sits between. Five of them are identical: the two walls the slot has, the divergence the intact stem settles to, the block the cut opens, which of the three removals wreck the stem, and the lags the doubled cut leaves rigid. The sixth is how far the first organ placed after the doubled cut moves, and it goes from 163.59 to -9.14 degrees. The lattice is the same on both sides; where one organ goes is not.
Fig. 14 The quantities that are identical across the transition, none of which is the endpoint.

What a stable endpoint would have bought

It is worth saying what was hoped for. If the endpoint had been one value above the transition and another below it, the transition would have been a change in the arrangement the stem ends up in — a much larger claim than a change in one placement.

If it had been one value throughout, the transition would have been a transient: a placement that differs and washes out. Either would have been informative and the measurement supports neither.

The golden 5/8 rung swept at 27 rises, with each removal's cost. How far the first organ placed after a cut moves, at every rise the sweep visits, coarse on the left. Removing both walls costs far more than removing the larger one alone above a rise of 0.00998, and exactly what the larger one costs below it. The change happens in one step of the grid the ladder is named on: 163.59 degrees at 0.00998 and 9.14 degrees at 0.00997, which is a fall of 154.5 degrees for a change of one part in a thousand in the rise.
Fig. 15 The first organ’s displacement, which is the half of the run that does support a claim.

The honest form of the result

On the golden 5/8 rung, the doubled cut’s run does not have a reproducible endpoint, so whether the transition persists to the end of the run cannot be asked here.

That sentence is longer than either of the two it replaces and it is the one the data supports. Reporting a quantity’s instability is a result; reporting a number from it would not have been.

Where the doubled cut's run finishes, at every rise of the sweep. The divergence the wrecked run ends at, rise by rise. It takes only a handful of values and jumps between them at rises one part in a thousand apart, 4 times inside the 9 rises of the finest sweep alone — where the walls, the block, the intact stem's divergence and the rigid lags are all held. So the end of a wrecked run is not a stable quantity on this rung, and no statement about where the stem finishes is available on either side of the transition. The first organ's displacement is the reproducible half of the same measurement.
Fig. 16 The instability itself, which is what is being reported in place of a number.

Where the question could be asked

On a rung where the doubled cut does not wreck. Two of the twenty-nine positions recover — at 88 and 91 per cent of the rung — and on those the run returns to what the control does, so the endpoint is the rung’s own divergence by construction.

That is not helpful, because a recovering cut has no transition to be on either side of. A rung where the pair removal wrecks below a transition and recovers above it would be, and whether one exists is not known.

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. 17 The slot table’s rows, among which a rung with the right shape would have to be found.

What this is an instance of

A quantity that is recorded, looks like a measurement, and is not. The collection has found several: a block that was the grid rounding a constant, an onset that was a run length, a residual that was a window.

This one is the least subtle of them, because the instability is visible the moment the quantity is plotted against anything. It went unreported for as long as nobody plotted it.

Both vary; only one of them varies enough to find. Each organ's step exponents, divided by its own mean so the two are comparable. The ogive's run over 15 per cent of their mean across 5 rings. The convex head's run over 1.15 per cent across 5 — inside the band a 15 per cent error on each ring position leaves, so no ruler separates it from a flat disc.
Fig. 18 What a measurement can and cannot report about itself, which is the distinction all of these fall on.

How many placements stand between the two

That is the quantity worth naming, because it is what separates the stable half of the run from the unstable half. The first organ after the cut is one placement downstream of a neighbourhood that differs from its neighbour rise’s by a fraction of a degree.

The endpoint is three hundred placements downstream, each against a neighbourhood the previous ones built. A rule of that shape amplifies a grid step into a different destination, and the number of steps it takes to do so is what nobody has measured.

Measuring it would mean asking, at each rise, how far along the run the two neighbouring rises’ placements first diverge. That is arithmetic on runs already grown and it would turn this instability from an obstacle into a reading.

What the divergence does while the pattern climbs. The stem produces a sequence rather than a constant. Over the second half of the run it stays within 3.8° of 137.30°, and the vertical marks are where the counted pair changed — the wander is largest around them.
Fig. 19 A run’s divergence organ by organ, along which two neighbouring rises would separate somewhere.

Which would be the interesting measurement

If two rises a thousandth apart stay together for two hundred organs and then separate, that is a horizon and it has a length. If they separate at the tenth organ, the endpoint is essentially unrelated to the rise and the four values are a lottery over four basins.

Both are testable and neither has been tested. It is the best question this reading leaves and it costs nothing beyond what is already grown.

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. 20 A cut run against its control, which is the comparison a horizon length would be measured in.

What this says about reading a table

The practical lesson is about columns rather than about rungs. A design records what it happens to record, and a column that the design does not use has never been checked for being a measurement at all.

The slot table has such a column and now it has a label. Any table in this collection may have others, and the cheapest way to find out is to plot a column against the thing the table is swept over and see whether it does anything.

Agreement between two windows happens only on a slow enough shoot. Five stems at each of four rates and five disturbances, each read through two overlapping windows of 250 internodes. A filled mark is agreement — both windows reported the same pair; a half mark is a disagreement; a small mark is one window reporting and one refusing; an open mark is silence. Agreement appears 0 times in 25, 1 times in 25, 14 times in 25, 14 times in 25 at 130, 250, 400, 700 nodes per rung, and the two rates it is almost absent from are the two at which a rung is no longer than the window.
Fig. 21 A table’s columns scored against each other, which is a use only the columns a design relies on get.

Two rungs, and only one has been asked

Everything here is about the golden 5/8 rung, because that is the rung the transition is on. Whether a wrecked run’s endpoint is unstable on every rung or on this one is not known, and it is the sort of thing that could easily differ.

The slot table holds twenty-four rows across eight rungs and every one of them carries an endpoint. Plotting all twenty-four against the rise they sit at would take a minute and would say whether this is a property of wrecked runs or of a stretch of one rung.

That is the cheapest of the three open questions this reading leaves and it is the one most likely to change how the result is stated.

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. 22 The slot table’s twenty-four rows across eight rungs, every one of which carries an unread endpoint.

What a reader should carry

That the slot design measures one organ, and that the other end of the same run is not a measurement on this rung — four values, changing between rises a thousandth apart, three times inside a nine-rise sweep.

And that this is why the transition is described as a change in a placement rather than as a change in the stem. The narrower description is not modesty; it is the only one the data carries.

The golden 5/8 rung swept at 27 rises, with each removal's cost. How far the first organ placed after a cut moves, at every rise the sweep visits, coarse on the left. Removing both walls costs far more than removing the larger one alone above a rise of 0.00998, and exactly what the larger one costs below it. The change happens in one step of the grid the ladder is named on: 163.59 degrees at 0.00998 and 9.14 degrees at 0.00997, which is a fall of 154.5 degrees for a change of one part in a thousand in the rise.
Fig. 23 The single removal’s cost across the same rises, which is stable in the way the endpoint is not.

What the picture at the top shows

The divergence the doubled cut’s run finishes at, rise by rise, coarse on the left, with a horizontal gridline at each of the values it takes.

The line jumps between two of those values several times in the left half of the plot, and the vertical rule — the transition, where the first organ’s displacement falls by a hundred and fifty-four degrees — passes through a stretch where the endpoint is doing nothing in particular.

Where the doubled cut's run finishes, at every rise of the sweep. The divergence the wrecked run ends at, rise by rise. It takes only a handful of values and jumps between them at rises one part in a thousand apart, 4 times inside the 9 rises of the finest sweep alone — where the walls, the block, the intact stem's divergence and the rigid lags are all held. So the end of a wrecked run is not a stable quantity on this rung, and no statement about where the stem finishes is available on either side of the transition. The first organ's displacement is the reproducible half of the same measurement.
Fig. 24 The endpoint once more, with the transition marked and passing through nothing.

The one line

The doubled cut’s run ends at 65.25, about 137.3, 182.84 or 209.56 degrees across twenty-nine rises of the golden 5/8 rung, changing between values at rises one part in a thousand apart and three times inside the nine-rise fine sweep.

So does the stem finish somewhere different? has no answer here in either direction, and the transition stays what it is: a change in where one organ goes.

Where this goes next

What is the same at 0.00998 and at 0.00997. Six quantities read on the two rises the transition sits between. Five of them are identical: the two walls the slot has, the divergence the intact stem settles to, the block the cut opens, which of the three removals wreck the stem, and the lags the doubled cut leaves rigid. The sixth is how far the first organ placed after the doubled cut moves, and it goes from 163.59 to -9.14 degrees. The lattice is the same on both sides; where one organ goes is not. A transition and not a slope The question was whether a fourth cell's cost declines smoothly to nothing or falls in one step. It falls in one step, and the answer decides whether a word in the collection names something or is a threshold on a continuum. The golden 5/8 rung swept at 27 rises, with each removal's cost. How far the first organ placed after a cut moves, at every rise the sweep visits, coarse on the left. Removing both walls costs far more than removing the larger one alone above a rise of 0.00998, and exactly what the larger one costs below it. The change happens in one step of the grid the ladder is named on: 163.59 degrees at 0.00998 and 9.14 degrees at 0.00997, which is a fall of 154.5 degrees for a change of one part in a thousand in the rise. Where a slot loses a wall Two rungs were reported to go free at 84 and 85 per cent of themselves — the second removal stops costing anything over the larger one alone. Three samples a rung cannot say whether that is a transition or a slope, and twenty-nine more say it is a transition one grid step wide. Every transition as the rise falls. The pair climbs 1/2 → 2/3 → 3/5 → 5/8 → 8/13. Consecutive transitions are 0.381, 0.382, 0.382 of the previous rise — 1/φ² is 0.3820. A stem too fine to settle Below a rise of about four thousandths the counter stops returning contact families and starts returning pairs like 2/13 and 13/24. Lengthening the stem does not fix it. That is a ceiling on every sweep this collection runs up the ladder, and it has never been written down.

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.

AblationBoth wallsClaim testingDisplacementHonest limitsIdentifiabilityInstrument settingNegative resultReproducibilitySettled divergenceSlotTransient