Phyllotaxis
the spirals are counted, not admired
Where the angle comes from

The front deepens down a rung

The offsets that never repair grow from one to five across a single rung, while a counter returns the same pair at every rise. The extra offsets are not a random extension of the ones already there: they are the ones past the smaller counted number, and they are the ones that keep the larger family.
13 min read 24 figures State the claim, state the testCounted, not admiredOne parameter, one family

Worth reading first: The organ that was taken away · Counting the spirals · A head is a set of points.

A removal is felt at all only within the front — the run of most recent organs whose placement was set against the one now missing — and this collection has measured that depth and found it to be the larger of the two counted numbers. Past it, nothing wrecks: cuts made sixteen organs back on a stem whose front is thirteen deep repair without exception.

That is a statement about one stem. Swept along a rung it becomes a statement about a range of stems that a counter cannot tell apart, and then it does some work.

The front is worth holding onto as a physical object rather than a number. It is not a window chosen for convenience: it is the set of organs whose azimuths were set while the removed organ was part of the neighbourhood being minimised over, and it ends where that influence falls below what the arrangement can register. A disturbance that arrives after the rule has chosen moves nothing at all, which is the same fact from the other side.

How many offsets wreck, along the 5/8 rung. The count of offsets that never repair, at each rise on one rung. It runs from 1 at the coarse end to 5 at the fine end, while a counter returns 5 and 8 spirals at every one of them. The front — the run of recent organs at which a removal is felt at all — deepens as the rise falls, so there are simply more places a cut can land and fail to heal. That is the mechanism under the grid: the offsets that appear at the fine end are the ones beyond the smaller contact number, and those are the ones that keep the larger family.
Fig. 1 The count of offsets that never repair, at every rise of one rung. One at the coarse end, five near the fine one, and the same counted pair throughout.

One to five, with the pair held

Twelve rises from 0.018 to 0.007. Every one of them is counted at 5 and 8 spirals. The number of offsets that never repair runs from one to five.

Which family survives, along the 5/8 rung. The family a wrecked stem keeps, at every offset that wrecks and every rise on one rung. A counter shown any of these stems returns 5 and 8 spirals, at all twelve rises, so nothing in the pair distinguishes the columns. The cells say otherwise: at an offset of 4 the stem keeps the 5-family at the coarse end of the rung and the other one at the fine end. The offsets that wreck at all grow from 1 to 5 as the rise falls, because the front deepens, and the extra offsets are the ones that keep the larger number. So the rule stated over the offset alone is a rule with the rise left out of it.
Fig. 2 The same sweep as a grid: offset down the side, rise across the top, and a filled cell wherever the stem never came back.

At the coarse end a single offset wrecks. By the middle of the rung there are two. Past 0.011 there are four or five, and the ones that have appeared are the deeper offsets — six, seven, eight — which simply did not exist as possibilities at the top of the rung because a removal that far back was not felt.

It is worth being careful about what “did not exist” means, because it is not that those cuts were never made. Every offset out to two organs past the front is cut at every rise; the deep ones at the coarse end simply repair. A cut that heals is not a missing measurement, it is a measurement whose answer is no wreck, and the growth in the count is the growth in how many cuts answer otherwise. The census that established this reports the healed offsets for exactly that reason: a table showing only the cuts with an answer would be a picture of the selection rather than of the sweep.

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. 3 Why depth is the right word: the immediate displacement a removal causes falls away with how far back it lands, and where it falls to nothing is where the front ends.
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, 14 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. 4 And how far that reaches at three rises an order apart, which is the same measurement made coarsely.

Which is most of the story, and not all of it

The deepening explains the count without any new mechanism. A front three organs deep offers three places to cut; a front eight deep offers eight. If the rule for which family survives were fixed and correct, the number of wrecked offsets would still grow down a rung.

The offset past the front that is felt anyway. The four cells of the design whose response has a hole in it: a run of felt offsets, a stretch of quiet, and then one isolated offset well outside the front at which a removal moves the next organ by tens of degrees. The open circle on each row is the count coming in at the next rung of that branch's ladder, and the filled point is the isolated offset. It sits one inside the incoming count on every row, including on the Lucas branch, where the incoming counts are 7 and 11 rather than the Fibonacci numbers the rule was found on.
Fig. 5 The relation the count of offsets rests on: how far back a removal is felt, against the number a counter reports.

What it does not explain is which family the new offsets keep, and that is where the arithmetic and the mechanism meet. The offsets that appear at the fine end are the ones beyond the smaller counted number, and the offset rule says those keep the larger family. They do.

Which family survives, along the 5/8 rung. The family a wrecked stem keeps, at every offset that wrecks and every rise on one rung. A counter shown any of these stems returns 5 and 8 spirals, at all twelve rises, so nothing in the pair distinguishes the columns. The cells say otherwise: at an offset of 4 the stem keeps the 5-family at the coarse end of the rung and the other one at the fine end. The offsets that wreck at all grow from 1 to 5 as the rise falls, because the front deepens, and the extra offsets are the ones that keep the larger number. So the rule stated over the offset alone is a rule with the rise left out of it.
Fig. 6 The grid again with the row whose answer changes picked out, which is the part the deepening does not cover.

So the rule’s two clauses are not two halves of one statement. The first clause applies at every rise of the rung, because the shallow offsets exist at every rise. The second clause has nothing to apply to until the front is deep enough to have offsets past the smaller number — which happens partway down the rung and not at the top of it.

That asymmetry is invisible in the census the rule was scored on and it changes what the score means. Twenty-five of thirty reads as a rule that is mostly right; read along a rung it is closer to two rules with different domains, one of which has been tested everywhere and one of which has been tested wherever the sampling happened to land. Neither is refuted here. What is refuted is the idea that the thirty rows were thirty comparable tests of one statement.

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. 7 The census the rule was scored on, one rise per lattice. Every column of the grid above is a single cell here, and the second clause was scored on whichever rise happened to be sampled.

What that does to a score

A rule scored over a census assembled from one rise per rung is being scored on a sample that never varies the thing its second clause depends on. Twenty-five of thirty is a real number and it is not a number about the rule; it is a number about the census.

The rule is not the only published number this sweep unsettles. The depth of the front was itself measured one rise at a time, and it slides too.

The measurement that established it reports that the front runs as deep as the larger of the two counted numbers — thirteen organs on an 8/13 stem, eight on a 5/8 one — and that past it nothing wrecks. On this rung the larger counted number is eight, and it is a constant: a counter returns 5 and 8 at every rise from 0.018 down to 0.007. The depth does not follow it. At the coarse end a single offset never repairs; by 0.012 there are two; only from 0.011 down do offsets as far back as eight wreck at all.

So the front is as deep as the larger counted number is true at the fine end of this rung and false across most of its width, and the procedure that established it could not have shown that. It is the same failure the rest of this essay is about, one step further back, and it is worth naming in those terms rather than filing as a separate correction: a quantity was measured on stems chosen for their counted pairs, came out equal to a number a counter reports, and was written down as though the equality were a law. What it is instead is the value that quantity takes at the rises the census happened to sample.

The honest form of the statement is a bound rather than an equation. The larger counted number is the deepest the front reaches anywhere on a rung, reached near the fine end and not before; elsewhere it is shallower, by a factor of eight at the coarse end of this one. A bound is the weaker claim and it survives the sweep, which is the trade this collection keeps making — and it is also the claim that would have predicted the anomaly rather than been surprised by it.

A second cut moves the next organ, and does not move the boundary. Every pair of organs that can be taken out of a settled stem at a rise of 0.013, where the pattern is 5/8. A row is the offset of the nearer organ removed, counted back from the tip; a column is how many further places back the second one sits. The shade is how far the next organ ends up from where it would have been, from under a degree in the palest cells to a half-turn in the darkest. The run of shaded rows ends at 8, which is the larger parastichy number, and every row below it is blank across the whole width — the largest displacement anywhere past the front is 1.41°. So the nearer of the two organs decides whether the removal is felt at all, and the second one, wherever it is put, cannot make the pattern notice an organ it was not going to notice.
Fig. 8 The table those thirty rows came from at one rise, with the offset along one axis.
The block is one of the two numbers, and not always the smaller. Every lattice a single organ was removed from, one row each: golden, rise 0.013 carrying 5/8; golden, rise 0.008 carrying 5/8; golden, rise 0.005 carrying 8/13. The two open ticks on each line are that lattice's own parastichy numbers; the filled dots are the blocks the stems that never recovered settled into, one per offset that failed. The claim this table was built to test is that the block is the smaller of the two, which held at the two rises it was first measured at. It does not hold here: 5/8 gives 5 and 8. What survives is weaker and still worth something — every filled dot but 1 sits on one of that row's own ticks, so the orbit carries a count of the lattice it was cut from, and which of the two it carries is decided by the offset rather than by the pattern.
Fig. 9 The same rows in the quantity that first raised the question, which is the period of the motif a wrecked stem repeats.

Concretely: at the coarse end of this rung the second clause is untestable, because no offset past five ever wrecks. Any census that happened to sample 0.018 rather than 0.010 would have found a rule with one clause and no evidence about the other, and would have reported it as a rule that works.

The lattices in the census were not chosen to make this happen, which is why it is worth reporting rather than apologising for. They were chosen to span two branches and a range of pairs — the right criteria for the question which pair — and the rise attached to each was whatever rise produced that pair. Nothing about that procedure is careless. It simply optimises for a different question from the one the rule ended up making a claim about, and the gap between those two questions is where the incompleteness lived.

The general form is worth stating because this collection will meet it again. A sample assembled to vary one quantity holds others fixed by construction, and any rule later fitted to that sample inherits the constancy as an untested assumption. The fix is not a better sample; it is noticing which quantity the rule turned out to depend on, and going back to vary that one on purpose.

The band that never heals is what two fixed edges leave over. Each row is one rung. A stem is grown to 400 organs, one organ is removed, and the stem is followed for 300 more. A filled mark is an offset the stem recovers from, with the number of organs it took printed above it; an open ring is an offset it never recovers from. At the 3/5 rung, a front of 5, every offset heals. At the 5/8 rung, a front of 8, the offsets 4, 5 never heal. At the 8/13 rung, a front of 13, the offsets 4, 6, 7, 8, 9 never heal. What is the same at every rung is the two edges: the first three offsets heal, and so do the last few of the front. What is left in the middle is the band, and it widens because the front does.
Fig. 10 And the coarsest version of the same caution: whether a stem is wreckable at all is a property of where it sits, not of the pair it is counted at.

The depth is not the whole of the rung either

Two things move along a rung, and the deepening is only one of them. The two contact steps also change places, and the two changes do not happen at the same rise: the steps cross at 0.015 and the deep offsets appear at 0.011.

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. 11 The ratio of the second step to the shortest across the same rises. It crosses one at 0.015, four rises above where the deep offsets start appearing.
The divergence slides along 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. 12 And the settled divergence over the same range, which slides through the whole of it without a feature at either place.

That the two are separated by four rises is useful, because it means the sweep can tell them apart in principle. Whatever decides the survivor, it is not the step ordering alone — the ordering reverses at 0.015 and the answers do not change there.

This is the one place in the thread where two candidate explanations have been put on the same axis and separated by measurement rather than by argument. The step ordering was the natural candidate, because the reading it refutes is stated in exactly those terms; and it fails, because a quantity that reverses four rises above the only change in the answers is not the quantity the answers are following. What is left is the depth, which does line up — and a depth that decides an answer is a mechanism rather than a coincidence of arithmetic.

Which offsets give short hops, at a rise of 0.018. 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. 13 The ranking at the coarse end, where the smaller family carries the shorter step.
Which offsets give short hops, at a rise of 0.008. 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. 14 And near the fine end, where the ordering has reversed and the answers at the shallow offsets have not.

Three controls

The front is measured, not assumed. The depth used here is the count of offsets that actually fail to repair, read off the cuts themselves, rather than the larger counted number quoted as a proxy for it. The two agree at the rises where both are available, which is the check worth having.

On a rung the response is a run of offsets; near a transition it has a hole. One row per rise, from 0.024 at the top to 0.006 at the bottom, and one column per offset: the organ one place back at the left, twelve 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, 12 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 same relation read from the response itself rather than from the counted numbers, which is the check that the depth is measured and not quoted.

Nothing here depends on the resolution. A thousandth puts twelve rises inside this rung. Halving the step would double the samples and produce the same monotone growth, because what is being resolved is a slide rather than a feature with a width.

A window that fits inside a rung. Stems that climb the ladder at four rates, read over a window at the fine end. The condition is a ratio: the window has to be shorter than a rung. 250 internodes at 260 per rung is 0.96 rungs and agrees on 3 of 3; 400 internodes at 260 per rung is 1.54 rungs and agrees on 1 of 3; 250 internodes at 520 per rung is 0.48 rungs and agrees on 2 of 3; 400 internodes at 520 per rung is 0.77 rungs and agrees on 3 of 3. Read over the whole stem instead, every rate returns nothing — 0 of 3, 0 of 3 — because the quantity the comb is periodic in changes as the pattern climbs.
Fig. 16 The general form of the caution about reading anything inside one rung, where the pair is fixed and everything else is not.

And the growth is not an artefact of the settling test. Every rise on the rung settles by the same criterion every other run on this site uses, and a stem that failed it would be excluded rather than counted as a wreck. That distinction matters here more than usual: a stem that never settles in the first place and a stem that settles and is then wrecked by a cut look similar in the divergence trace and are entirely different objects, and only the second is what this sweep counts. The coarse ladder has rises of the first kind — a band that sticks on a rational — and they are excluded from every ablation table on this site for exactly this reason.

What the finer grid does to the rises already published. The two rises this site has argued from and the one it published as having no answer, each read at both azimuth grids, five stems apiece. A filled mark agrees with the position counter, a half mark contradicts it, an open mark is a refusal. At 0.013 and 0.005 the readings are identical at both grids, so nothing already written depends on the sample count. At 0.008 they are not: 384 azimuths gives 5/8, 5/8 and 1152 gives 8/13, 8/13, against a counter that says 5/8. That rise was chosen in the earlier work because the three shortest lattice offsets there are within a fifth of each other, and a stem with no answer answering differently on a different grid is the object behaving as it was said to.
Fig. 17 The counter checked against the positions rather than trusted, which is the test a rise has to pass before a cut is made on it.
Every open question here needs under 28 specimens. The sample size at which each comparison reaches 80 per cent power at a 5 per cent false-positive rate, from the exact binomial rather than a normal approximation. The census question — do plants show consecutive Fibonacci pairs far more often than the geometry does — needs 4: 14.7% is the share of divergence angles giving a consecutive Fibonacci pair at a fine rise; 90% is what a grown history gives.
Fig. 18 And the form the caution takes for a claim about plants rather than runs.

What it predicts

If the deepening is what governs how many offsets are available, then a finer rung — with larger counted numbers and a deeper front — should offer more of them, and the count should grow the same way inside it.

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.
Fig. 19 Where those rungs are. This essay sweeps one of them; the ladder has several more that no ablation census has visited at all.

The cost of that test is worth stating, because it is the reason it has not been run. Each rise on a rung is a settled stem plus one cut stem per offset, each grown to settle again and compared against a control sharing its history; a finer rung has more offsets and needs longer stems to settle at all, so the work grows faster than the rung count does. That is an argument for doing it once and banking the numbers, not an argument against doing it.

That is a cheap test and it is the obvious next sweep. It also matters for the reading about which family lost a member, which can only be asked at offsets that are multiples of a counted number: a deeper front at 8/13 has both 8 and 13 available where a 5/8 front has 5 and 8, so the same sweep would widen the rows that reading is scored on.

The offsets where the removed organ belonged to one family. Each row is a stem that never repaired, with the family of the organ that was taken and the family that survived. An organ five places back on a stem counted at 5 and 8 spirals lies on the tip's five-chain, so the question can be asked there; an organ four places back lies on neither chain and it cannot. Of 30 wrecked offsets in the census, 9 remove a member of exactly one family and 21 remove a member of neither. On every one of the 9 the family that lost a member is the family left standing, which is the opposite of what the reading predicted.
Fig. 20 The rows that reading currently has, and why more front means more of them.
The same rule, the same rise, two lattices, two fronts. How many organs back a single removal is still felt, on a stem seeded onto the golden lattice and on one seeded onto the Lucas lattice, at seven rises. Everything but the seed is identical at each rise — the rule, the spacing, the heights, the azimuth grid — and the two fronts differ at every one of them. Which branch has the wider front changes hands four times going down the range, so no function of the rise gives the column. The golden branch carries 5/8 at four of these rises, across a factor of two in the rise, and its front is eight at all four.
Fig. 21 The two branches carry fronts of different depth at the same rise, so the prediction is about a rung rather than about a rise.

Where this leaves it

The picture that comes out of this is less tidy than a rule and more useful. There is no single quantity that decides which family a wrecked stem keeps. There is a front whose depth is set by the rise, a set of offsets that exist because of that depth, and a rule over those offsets that has been scored without ever varying the depth.

The block a wrecked stem settles into is the count it was cut from. A stem is cut in the middle of its front and followed for 300 organs. It does not come back to its lattice; what it does instead is repeat a fixed sequence of divergences exactly, to the resolution of the azimuth grid. Each row shows that sequence twice over, one bar per organ, drawn to the same scale. At the 5/8 rung the sequence is five angles long and advances -36.1° per block; At the 8/13 rung the sequence is eight angles long and advances 22.5° per block. Every block is the smaller number of the pair that was cut, so the second attractor carries the first one's count — and every precession is one part in twice the block, which makes each of these a two-jugate arrangement with 10 and 16 rows.
Fig. 22 What all of it decides in the end: the length of the motif a wrecked stem repeats, which is the family that was kept.
Take away the organ four 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 — four 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 164.1° apart, against a local spacing of 41°, and the vacancy itself is 172.7° from the undisturbed answer. Nothing else differs between the two runs: same rise, same history, same rule.
Fig. 23 The intervention, unchanged throughout: one organ removed, with everything about the stem it was removed from varying underneath.
The hops of a 5/8 lattice, shortest first — golden, rise 0.010Every 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, 5 and 8, and they differ in length by a factor of 1.076. The lags left standing after a removal are 5 and 8, sitting at rank 2 and 1 in this order, so the family the rule holds is a short step but not always the shortest one.85133161021181122624629lag, in organs — ordered by the length of its stepstep length, in turns of the stemkept: lag 5, lag 8golden, rise 0.010 · pair 5/8 · offsets that wreck: 4, 5, 6, 7, 8generated from a stated rule, not drawn to look right
Fig. 24 The families being chosen between at one lattice. The slider walks the census, and the marked bars are what was actually left standing.

A count is a robust description and its robustness is its blindness: it is the same number over a range of geometries. That is what makes it worth measuring on a plant, and it is why an account of ablation stated only in counted numbers has been quietly incomplete for as long as this thread has been running.

There is a version of this essay that overstates it, and it is worth naming so that this one does not become it. The claim here is not that the counted pair is uninformative about ablation: it fixes which two families are available at all, and that restriction is the strongest result the thread has. The claim is narrower — that the pair does not fix which of the two survives, because the quantity that does is one the pair holds constant over. A description can be exactly right about what it describes and silent about the question being asked of it, and telling those two apart is most of what measuring anything is for.

Where this goes next

Which family survives, along the 5/8 rung. The family a wrecked stem keeps, at every offset that wrecks and every rise on one rung. A counter shown any of these stems returns 5 and 8 spirals, at all twelve rises, so nothing in the pair distinguishes the columns. The cells say otherwise: at an offset of 4 the stem keeps the 5-family at the coarse end of the rung and the other one at the fine end. The offsets that wreck at all grow from 1 to 5 as the rise falls, because the front deepens, and the extra offsets are the ones that keep the larger number. So the rule stated over the offset alone is a rule with the rise left out of it. One rung, two answers The offset accounts for twenty-five wrecked stems of thirty and is refuted by a single pair of runs that differ in nothing but the rise. Sweep one rung at a thousandth and the refutation stops being an anomaly: the same offset on the same lattice keeps one family at the coarse end and the other at the fine one. On a rung the response is a run of offsets; near a transition it has a hole. One row per rise, from 0.02 at the top to 0.005 at the bottom, and one column per offset: the organ one place back at the left, 14 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 — 8, 12, 13 at the pairs shown on the left. Between a rise of 0.02 and 0.0065 the run ends at 5 and one more cell is filled at 7, with the offsets between them quiet to under a degree. That isolated column is one place inside the larger number of the pair the stem is climbing towards. The response with a hole in it Removing an organ is felt out to the larger parastichy number and no further — that is the intervention's headline, and it holds in the middle of a rung. Swept towards a transition the run of felt offsets stops early and one lone offset past it comes alive, with three quiet organs in between. The lone offset is one place inside the count the stem is about to have. The band that never heals is what two fixed edges leave over. Each row is one rung. A stem is grown to 400 organs, one organ is removed, and the stem is followed for 300 more. A filled mark is an offset the stem recovers from, with the number of organs it took printed above it; an open ring is an offset it never recovers from. At the 3/5 rung, a front of 5, every offset heals. At the 5/8 rung, a front of 8, the offsets 4, 5 never heal. At the 8/13 rung, a front of 13, the offsets 4, 6, 7, 8, 9 never heal. What is the same at every rung is the two edges: the first three offsets heal, and so do the last few of the front. What is left in the middle is the band, and it widens because the front does. A front with no middle Take one organ out of a stem and the pattern sometimes never comes back — but that was measured on a front thirteen organs wide, where five of the thirteen offsets are beyond repair. Repeat it on a front of five and every single ablation heals. The band that cannot be undone is not a number the rule carries; it is what two fixed edges leave over.

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.

  • The organ that was nobody's neighbour — both name ablation, claim testing, control, honest limits, lattice, lattice offset, measurement, negative result, parastichy pair, the placement rule, rigid hop, rung
  • A stem too fine to settle — both name claim testing, control, honest limits, lattice, measurement, negative result, parastichy pair, the placement rule, rise, rung
  • The band was not the sampling — both name claim testing, control, honest limits, lattice, measurement, negative result, parastichy pair, the placement rule, rise, rung
  • The hop that survived — both name ablation, honest limits, lattice, lattice offset, measurement, parastichy pair, the placement rule, rigid hop, rise, rung
  • Two accounts of one number — both name ablation, claim testing, honest limits, lattice, measurement, negative result, parastichy pair, the placement rule, rise, rung
  • One turn per survivor — both name ablation, honest limits, lattice, lattice offset, measurement, parastichy pair, the placement rule, rigid hop, rise

Named objects

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

AblationClaim testingControlHonest limitsLatticeLattice offsetMeasurementNegative resultNodes per rungParastichy pairThe placement ruleRigid hopRiseRung