Phyllotaxis
the spirals are counted, not admired
Stems and cones

A band with nothing inside it

Five offsets wreck on the Lucas 7/11 band and every one of them keeps the same family at every rise it wrecks at. There are no islands, no transition region and no period to look for, which is what makes the picture from the other band a picture of that band.
14 min read 24 figures State the claim, state the testCounted, not admired

Worth reading first: Where a handover sits · The organ that was taken away.

The interesting picture on this ladder is the one with the speckle in it. One band cut at every rise came back with three offsets changing their answer inside it, in thirteen islands one to three rises wide whose gaps fit no period.

This essay is about the other one, and it is about the picture with nothing in it.

Which offsets wreck across the Lucas 7/11 band. One row per offset and one column per rise, with a mark where a single removal at that offset wrecks the stem. The set is not the same at every rise: on this band one offset wrecks at only 27 of its 124 rises, in several separate stretches, while others wreck at all of them. So a census taken at one rise of a band and a census taken at another are censuses of different sizes, and every claim of the form "at every offset that wrecks" is quantified over a set the rise decides.
Fig. 1 The Lucas 7/11 band at every rise, one row per offset. The tone never changes because the family kept never does.

What is being reported

Five offsets wreck somewhere on the Lucas 7/11 band. Every one of them keeps the 7 family at every rise it wrecks at — 124 rises, 1,612 cut stems, and one answer.

That is a negative, and a negative is worth exactly what its resolution is worth. Stating the resolution is therefore most of the work: the sweep steps by two parts in a thousand in the rise, so a feature two rises wide is two parts in a thousand of the rise wherever on the band it sits, and there is not one.

The other band’s islands are one to three rises wide at that same step, which is what makes the comparison a comparison. A sweep whose resolution could not have resolved the thing it did not find would have reported the same picture and meant nothing by it.

The two widest bands on the ladder, each cut at every rise. One block per band, one row per offset that wrecks anywhere on it, one column per rise, coarse on the left. A filled cell is a cut that wrecks, and its tone is the family left standing; a pale cell is a cut that recovers. The golden 8/13 band above changes its answer at three of its six offsets, 19 times in all. The Lucas 7/11 band below changes it nowhere: every cut that wrecks on it keeps the 7 family, at every offset and every one of its 124 rises.
Fig. 2 Both wide bands, with the one that changes above and the one that does not below.

Why a uniform picture is not obviously boring

The reason to expect something is that a band moves plenty. Across its 124 rises the rise itself changes by a factor of 1.16, the shorter of the two contact steps changes places with the longer one, and which offsets wreck at all changes on more than a tenth of its steps.

So the band is not a stretch where nothing happens. It is a stretch where three quantities are held on purpose — the counted pair, the settled divergence and, by construction, nothing else — and where a fourth was found to move on the other band. Here the fourth does not move while the rest of it does.

A divergence that does not move across the 4/7 band. Measured at every rise of a band on the Lucas branch, where a counter returns 4 and 7 spirals throughout. The settled divergence moves by 0.0195 degrees across the whole band, which is a fraction of the azimuth grid step and a fortieth of the slide across the rung it sits in. The ratio of the two contact steps does move: it falls to 1.0010 and the ordering changes hands at a rise of 0.0225, so above that rise the shorter step belongs to the 4 family and below it to the 7 family. Two of the three quantities that vary along a rung are therefore held here and the third is not, which is what makes the ends of this band a matched pair.
Fig. 3 The two quantities a band holds while the ordering reverses, on the design’s earlier case.

Five offsets, and they are not the same five

Offsets 4 through 8 wreck somewhere on this band. The set is not the same at every rise: offsets 5, 6 and 7 wreck at all 124, offset 8 wrecks at 88 of them and offset 4 at only 27.

That is the same phenomenon the golden band showed — which offsets wreck is a property of the lattice and the rise rather than of the lattice — and here it is stronger. On the golden band no offset’s wrecking broke into more than three stretches; offset 4 here is broken into five.

Which offsets wreck across the golden 8/13 band. One row per offset and one column per rise, with a mark where a single removal at that offset wrecks the stem. The set is not the same at every rise: on this band one offset wrecks at only 22 of its 126 rises, in several separate stretches, while others wreck at all of them. So a census taken at one rise of a band and a census taken at another are censuses of different sizes, and every claim of the form "at every offset that wrecks" is quantified over a set the rise decides.
Fig. 4 The same reading on the golden band, where the wrecking set also moves and never breaks this far.

Which is the interesting half of a uniform result

Two things could have made a band come back uniform and only one of them is interesting. It could be that nothing was being tested — that no cut wrecked, or that the same three cuts wrecked at every rise and there was nothing for a change of answer to be a change of.

That is not what happened. The set of wrecking offsets changes on 23 of this band’s 123 steps, so the sweep is measuring something that moves; the family left standing is simply not part of what moves.

The ordering changes and the survivor does not. Every offset that wrecks, at every rise of the band, with the family left standing written in the cell. The counted pair is 5 and 8 at all 18 rises and the settled divergence is held to a twentieth of a degree, so the one quantity moving across the columns is which of the two contact steps is the shorter — and it changes hands at the marked rise. The cells do not: the 5 family survives at all 24 wrecked cuts, on both sides. Scored on this band, the reading that a wrecked stem keeps its shortest hop is right 14 times out of 24, for an answer that never changed.
Fig. 5 The families a band’s cuts keep, on the narrower band where the reading was first taken.

Seven, and never eleven, and never four

The family kept is 7, which is the smaller of the counted pair. It is never 11, the larger, and it is never 4 — which is what the golden band’s cuts reach for when they leave 8.

Four is not an arbitrary alternative there. On an 8/13 lattice the four-hop is a sub-multiple of the smaller counted number, and a survivor has to be a neighbour rather than any lag at all. On a 7/11 lattice there is no such sub-multiple, because seven is prime.

The 17 rows of the exchange table, gathered by the lag they kept. One bar per surviving lag, its length the number of rows the table holds at that lag, with the hop that lag's stems keep written beside it. The hop is nearly constant inside a lag, so a correction fitted over 17 rows is fitted over four hops — which is the denominator that matters and is much smaller than the row count suggests. Adding a lag to the table is worth more than adding rows at a lag already in it.
Fig. 6 Which lags the census’s cuts leave standing, gathered by the lag rather than by the lattice.

That is a difference, and it is not an explanation

It is tempting to make the primality of seven the account, and it is worth resisting for one round of thinking. The claim would be that a band whose smaller counted number has a factor offers a second family to switch to, and a band whose smaller number does not offers none.

The ladder holds too few cases to test it. Its counted pairs are 2/3, 3/5, 5/8, 8/13 on one branch and 1/3, 3/4, 4/7, 7/11 on the other, and only three of those have a composite smaller number. Two of the three are on bands too narrow to cut whole.

The third is the golden 5/8, whose smaller number is composite and whose band is 112 rises. That one is cuttable and would be the test — with the caveat that it shares a branch with the band that has the speckle, so a positive there would not separate the pair account from the branch account.

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. 7 Every rung on both branches with its counted pair, which is the population any account of this would have to be tested over.

What a change of answer would have looked like

It is worth naming the thing that is absent, because the absence is the measurement. A change of answer is a rise at which one offset’s cut stops keeping one family and starts keeping another, with both readings taken the same way on stems that differ only in the rise.

On the golden band there are nineteen of them, thirteen at a single offset. Here there are none, and the sweep would have reported one had it been there — the same code, the same tolerances, the same reading.

The tolerance that decides it is the one that says which lag a stem kept, and it is not a threshold anybody tuned: a lag is kept when the hop at that lag holds across the run, and the readings separate into two populations with a gap between them rather than falling either side of a line.

Offset 8 across the 8/13 band, rise by rise. The family this one offset keeps at each of the band's 126 rises, coarse on the left. It wrecks at 123 of them and keeps the 4-family and the 8-family at different rises. The ticks below mark ten islands — runs of 1 and 1 and 2 and 1 and 2 and 1 and 2 and 1 and 3 and 1 rises where the coarse family comes back inside the fine one. The handover is the taller rule and the change of answer is nowhere near it.
Fig. 8 The offset that carries thirteen of the golden band’s nineteen changes, drawn rise by rise.

The handover is not where anything happens, again

The claim the thread rests on is that the rise at which the two contact steps change places is not the rise at which the survivor does. Here the handover sits at 57 per cent of the band and nothing at all happens at it.

That is support of a weaker kind than the golden band gives, and it should be counted as such. Nineteen changes none of which is at the crossing is evidence; no changes anywhere is consistent with the claim and consistent with a dozen other things.

What each band holds, and what it moves. One row per band. The counted pair is held by construction and the settled divergence is held to a twentieth of a degree; the quantity a band exists to move is which of the two contact steps is the shorter. five of the six do move it — the ordering changes hands exactly once inside, and at both ends the two steps differ by enough for an ordering to mean anything. The remaining one changes hands three times and its two steps are never more than 0.0 per cent apart, so it holds all three quantities and is a control rather than an experiment.
Fig. 9 How far apart the two contact steps get across each band, and where each crossing sits.

What the picture cannot rule out

The sweep reads the family left standing on a stem grown to a fixed length and read over a fixed window. If a change of answer existed but took longer to show than the run allows, this sweep would not see it.

That is not idle. Doubling the run length moved nineteen of twenty-four onsets by more than twenty organs and changed the periodicity verdict on three rows, so run length is known to matter to readings of this kind. It has not been varied on a band.

One cut's profile over 600 organs, with both onsets marked. One mark per organ above the hole, at the angle it sits from where the same organ sits in a control sharing its history. The two vertical rules are where the pattern is said to start: the left one is what a 300-organ run reports and the right one is what this 600-organ run reports. They are 300 organs apart. The onset is measured against the levels the classes hold at the end of the run, so a profile that drifts slowly is compared against different levels at each length and the reading moves with the length.
Fig. 10 One cut’s displacement profile at two run lengths, which is the comparison a band has never been given.

And what the window cannot rule out either

The same applies to the reading window. The family a cut keeps is read from the top of its run, and the width of that window decides other verdicts in this collection.

Neither has been moved on a band sweep, which is the honest limit of the negative reported here. What can be said is that at the settings every other measurement in the thread uses, this band is uniform.

The area never settles, so the number reported is the window. The equivalent width of the disorder dip — the area of the deficit divided by the dip's own depth — for four fractions with a denominator of 34, against the window it was integrated over. The window is in scaled units, u = δ·n²/q, so a head of any size is sampled at the same places on its own dip; 200 scaled units is about 0.0109° at these head sizes. A width would show as a flat stretch. There is none: every line climbs to 512 scaled units or so and then turns over, because past that the window has reached far enough towards the next rational that it is integrating the neighbour's dip instead of this one's background. The dip has a depth and a scale; it does not have an outer edge, and an integral of something with no outer edge is a statement about where it was stopped.
Fig. 11 A quantity read at two windows, which is the kind of dependence a band sweep has not been checked for.

Why the sweep was worth running anyway

Because the alternative was not knowing. Before it, the honest statement about transition regions was one band has one, with a sample of one; after it, the statement is one band has one and the other wide band does not, with a sample of two.

That is a small sample and it is twice the previous one, and it changed the shape of what could be claimed. A finding that survives a second case gets extended; one that does not gets a boundary drawn round it, and a boundary is the more useful of the two when the thing being bounded is a surprise.

What nine rises find on each band, against what all of them find. Two bars per band: the changes of surviving family a nine-rise design finds, and the changes the full sweep finds. On the Lucas band the two agree exactly, at none and none. On the golden band they agree that something changes and disagree about how much — 5 against 19 — because one step of that design is 16 rises and the band carries features one to three rises wide. A sample was never wrong about whether; it was wrong about how many.
Fig. 12 What a nine-rise design finds on each band against what all the rises find.

The cost of a null result

Thirty-eight minutes and 1,612 cut stems for a picture with nothing in it. That is the economics of a control and it is worth stating plainly, because a collection that only ran the sweeps likely to produce something would be a collection of positives.

The comparison is with the nine-rise design, which would have taken two minutes and returned the same verdict. It would have been the right verdict and it would not have been worth anything, because nine rises finding nothing is what nine rises do on a band with thirteen islands in it.

What nine rises could see of 126. Above, offset 6's answer at every rise of the band. Below, the same row with only the rises a 9-cut design visits, which is one every 16. The design was built for a quantity expected to be constant and it reports the ends and the crossing correctly; what it cannot report is where inside the band the answer changes, or that it changes back. Every island here is 2 or 1 rises wide, against a step of 16, so the sample can only land on one by accident.
Fig. 13 One offset’s answer at every rise, with only the rises a nine-cut design visits marked.

Where the survivor does move on this rung

The band is not the rung. The Lucas 7/11 rung runs from a rise of 0.00947 down to 0.0057 and the band covers 0.00922 to 0.00721 of that — about half.

Below the band, still on the rung, the family changes. At a rise of 0.006 three of six wrecking offsets keep a lag of 11 instead of 7, which is the first lag outside the census’s four anything has been found to keep.

Twenty rises at the fine end of both branches, cut at every offset. One row per rise searched, coarse at the top of each block. The bar names the counted pair the stem shows and the numbers on the right are the lags its wrecking cuts leave standing. A pale row is a rise whose settled divergence has left the branch it was started from by more than 20 degrees, which is what happens below the ladder's finest rung — the pairs there are 2 and 4, 8 and 16, 11 and 22, which are not two consecutive terms of any additive sequence. One rise on the Lucas branch keeps a lag of 11 while still on it.
Fig. 14 The fine end of the Lucas branch. The rise below the band’s fine end keeps a different family.

So a band holds more than it was built to hold

A band is built to hold the counted pair and the settled divergence while the step ordering reverses. On this rung it also holds the surviving family, and the survivor changes as soon as the band ends.

That is not something the design promised and it is not something the golden band shows — there the survivor changes inside the band. So the two bands disagree about whether a band holds the survivor, which is one more way of saying that the interesting result belongs to one of them.

Every rise of the 8/13 band, cut at every offset. One column per rise of the band, coarse on the left and fine on the right, and one row per offset that wrecks anywhere on it. A filled cell is the family the cut stem keeps; a pale cell is an offset that recovers at that rise and has no survivor to report. The band holds 126 rises and 1890 cut stems. three of the six offsets change their answer somewhere inside, three never do, and the vertical rule is the handover — the rise where the two contact steps change places. Not one of the 19 changes is at it.
Fig. 15 The band where the survivor does change inside, drawn without its handover marked.

The uniformity is not the whole band

One more distinction, because the sentence nothing changes on this band is easy to over-read. The pale cells in the picture are rises where an offset recovers rather than wrecks, and there are plenty of them: offset 4 has no answer at 97 of the 124 rises.

So the band is uniform in the family kept where a family is kept at all, and the set of rises where one is kept is different for each offset. Those are two separate readings of the same table and only the first is uniform.

Both edges of the front heal; the middle of it does not. The same removals, followed for 300 organs each. A cut one to three places back is undone within fifty organs and a cut ten to thirteen places back within sixty. A cut in between is never undone: the divergence sequence settles into an exactly repeating cycle of 4 or 8 angles and holds it for the rest of the run. The rule corrects a displacement and cannot correct a deletion.
Fig. 16 Which offsets wreck and which recover at one lattice, which is the reading the pale cells are of.

Reading a table with two states in it

Every cell of the sweep is one of three things: a cut that wrecks and keeps 7, a cut that recovers, or an offset that was never tried. Collapsing the second and third would be the easy mistake, and it would turn this offset recovers here into this offset was not measured here.

The sweep tries every offset out to the front and two past it at every rise, so the third state does not occur inside the band. It is worth saying because nothing behind the front wrecks is a separate result that the design depends on and does not re-derive.

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 0.020 carrying 3/5; golden, rise 0.013 carrying 5/8; golden, rise 0.008 carrying 5/8; golden, rise 0.005 carrying 8/13; Lucas, rise 0.020 carrying 4/7; Lucas, rise 0.013 carrying 4/7. 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: 3/5 gives 5, 5/8 gives 5 and 8, 4/7 gives 4 and 7. 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. 17 How far past the front a cut can be made before every cut recovers, which is what bounds the offsets swept.

What replicates and what does not

Two findings came from the first band and they have come apart. That which offsets wreck moves with the rise replicates, and more strongly here. That the family kept changes inside a band does not replicate at all.

A pair of results that separate under replication is more informative than a pair that both hold, because it says the two were never one finding. They were reported together because they came out of the same table.

Every rise of the 8/13 band, cut at every offset. One column per rise of the band, coarse on the left and fine on the right, and one row per offset that wrecks anywhere on it. A filled cell is the family the cut stem keeps; a pale cell is an offset that recovers at that rise and has no survivor to report. The band holds 126 rises and 1890 cut stems. three of the six offsets change their answer somewhere inside, three never do, and the vertical rule is the handover — the rise where the two contact steps change places. Not one of the 19 changes is at it.
Fig. 18 The table the two findings came out of, with the handover marked.

The picture is the argument

There is not much to say about a uniform picture that the picture does not say, which is why this essay is mostly about what would have shown up in it.

A reader who wants one sentence should look at the two blocks in the hero of the essay that cut this band: the upper one has a speckled middle and the lower one does not, at the same resolution, on the same instrument, thirty-eight minutes apart.

The two widest bands on the ladder, each cut at every rise. One block per band, one row per offset that wrecks anywhere on it, one column per rise, coarse on the left. A filled cell is a cut that wrecks, and its tone is the family left standing; a pale cell is a cut that recovers. The golden 8/13 band above changes its answer at three of its six offsets, 19 times in all. The Lucas 7/11 band below changes it nowhere: every cut that wrecks on it keeps the 7 family, at every offset and every one of its 124 rises.
Fig. 19 Both bands without their handovers marked, which is the comparison stripped to its content.

A second branch is not a second sample

There is one more reason not to call this a replication and then stop. The two bands differ in branch, in counted pair and in width all at once, so a difference between them has three candidate causes and the sweep separates none of them.

That is the ordinary situation when a ladder offers eight rungs and only two of them are wide enough to cut. It is worth writing down rather than left implied, because a reader who takes the speckle is one band’s as the speckle is the golden branch’s has read more into two cases than two cases hold.

One rise, two seeds — the response of each. How far the next organ moves when the organ a given number of places back is removed, at a rise of 0.02, on a stem seeded onto the golden lattice and on one seeded onto the Lucas lattice. The shading is the size of the displacement and the vertical line is where the run of felt offsets ends: 5 on the golden stem, whose lattice is 3/5, and 6 on the Lucas stem, whose lattice is 4/7. Nothing else differs between the two runs. Everything that changes with the rise — the spacing, the heights, the size of the neighbourhood the rule sums over — is the same in both rows.
Fig. 20 A quantity measured on both branches, which is the comparison this sweep is a second instance of.

What the sweep would have to be to say more

An account of when a band carries a transition region needs a population, and the population would have to be bands rather than rises. The ladder has six, one of which is a control, and two of which have now been cut whole.

Cutting the other three is about an hour altogether — 70, 86 and 16 rises — and would take the sample from two to five. That is the cheapest thing this thread could do next and it is worth saying that it is cheap, because the instinct after a surprising result is to build a new instrument rather than to point the old one somewhere else.

Where the two contact steps change places, on every rung. One row per rung of the two branches, drawn from its coarse end to its fine one on a logarithmic axis. The mark on each row is the rise at which the two contact steps change places — the rung's handover — and the shaded stretch is the band around it over which the settled divergence holds still. six of the eight rungs have a handover and every one of those sits between 6 and 40 per cent of the way down its rung, never past the middle. The two rungs without one are the coarsest on each branch, whose crossing is above the range this ladder reaches.
Fig. 21 The six bands with their extents, three of which have never been cut at more than nine rises.

What is already known about the narrow three

Two of the three do not wreck at any offset at their coarse ends, because a coarse enough stem cannot be wrecked by a single removal at all. So a sweep of the 3/5 band and the Lucas 3/4 band would be measuring something different from a sweep of the two wide ones.

The Lucas 3/4 is a control for a separate reason as well: its two contact steps stay within four parts in a thousand of each other across the whole band, so the ordering changes hands three times inside it and neither end is separated enough for an ordering to mean anything.

Two ways of predicting how wide a band is. A band ends where the settled divergence has moved 0.05° from its value at the handover, so the width should follow from how fast the divergence changes there. Reading that rate as the rung's average slope predicts widths that are wrong by factors of 0.20 to 5.92 — wrong in both directions, so no constant rescues it. Reading it as a curvature about a stationary point gives 0.41 to 1.08, with five of the six inside a third. The difference between the two is the difference between a curve and its average, and a band is exactly where the two are least alike.
Fig. 22 Each band’s extent against two predictions of it, which is where the six differ most.

What a reader should carry

That a band can be cut at every one of its rises and come back with nothing inside it, and that this is a result rather than a failure.

And that the value of a null depends entirely on its resolution. Nine rises finding nothing on this band would have been worth very little; 124 rises finding nothing is what makes the other band’s speckle a property of that band.

Where the two contact steps change places, on every rung. One row per rung of the two branches, drawn from its coarse end to its fine one on a logarithmic axis. The mark on each row is the rise at which the two contact steps change places — the rung's handover — and the shaded stretch is the band around it over which the settled divergence holds still. six of the eight rungs have a handover and every one of those sits between 6 and 40 per cent of the way down its rung, never past the middle. The two rungs without one are the coarsest on each branch, whose crossing is above the range this ladder reaches.
Fig. 23 Every band on the ladder with its extent, and the two that are wide enough to cut whole.

What the picture at the top shows

Five rows and 124 columns, coarse on the left, with a mark wherever a single removal at that offset wrecks the stem. Three rows are solid all the way across; one begins a third of the way in; one is broken into five stretches with long gaps between them.

Every filled cell is the same tone, and that is the finding. On the other band the same picture has two tones in its lower rows, arranged in islands.

Which offsets wreck across the Lucas 7/11 band. One row per offset and one column per rise, with a mark where a single removal at that offset wrecks the stem. The set is not the same at every rise: on this band one offset wrecks at only 27 of its 124 rises, in several separate stretches, while others wreck at all of them. So a census taken at one rise of a band and a census taken at another are censuses of different sizes, and every claim of the form "at every offset that wrecks" is quantified over a set the rise decides.
Fig. 24 The band once more. The uniform tone is the result, and the broken row is the thing that does vary.

The one line

Five offsets, 124 rises, 1,612 cut stems, and one family kept in every cell that holds one: the Lucas 7/11 band has no transition region, no islands and nothing to fit a period to.

Which offsets wreck moves across it, and more than on any band measured before; which family they keep does not move at all.

Where this goes next

The two widest bands on the ladder, each cut at every rise. One block per band, one row per offset that wrecks anywhere on it, one column per rise, coarse on the left. A filled cell is a cut that wrecks, and its tone is the family left standing; a pale cell is a cut that recovers. The golden 8/13 band above changes its answer at three of its six offsets, 19 times in all. The Lucas 7/11 band below changes it nowhere: every cut that wrecks on it keeps the 7 family, at every offset and every one of its 124 rises. The second band, cut whole One band was cut at every one of its rises and came back with a transition region — a stretch where three offsets change their answer, in short islands with uneven gaps. The obvious question is whether that is a picture of bands or a picture of that band. The other wide band answers it. A period fitted to the speckle, at every period it could have. Each mark is one candidate period, drawn at the share of rises it gets right when it is given its best phase and its best family in each residue class — the most generous reading of periodic there is. The flat rule is what saying nothing gets: name the commonest family and stop. The best period scores 76 per cent against 76 for no period at all, a gain of 0 points over 123 rises, so the alternation the coarse design reported is not a period being sampled badly. The alternation is not a period Nine sampled rises gave 8, 4, 8, 4 at one offset of one band, and a period was the obvious thing to look for. At full resolution it is thirteen islands one to three rises wide, with gaps of 1, 2, 3, 6, 7, 8, 9, 16, 31, 44 and 48 — and a fitted period buys exactly nothing. Offset 4 across the 8/13 band, rise by rise. The family this one offset keeps at each of the band's 126 rises, coarse on the left. It wrecks at 98 of them and keeps the 8-family throughout. The ticks below mark no island: the answer changes once and stays changed. The handover is the taller rule and the change of answer is nowhere near it. The offsets that never change Three of the six offsets that wreck anywhere on the band keep the same family at every rise they wreck at — 98, 22 and 81 rises of the 126. And which offsets wreck at all is a function of the rise, which no reading of a band had drawn.

What links here

Computed from the collection, not written here: the essays that point at this one.

Shares its objects with

Essays that name at least two of the same things, and that neither author linked.

  • Three offsets, three crossings — both name ablation, claim testing, control, handover, honest limits, lattice offset, negative result, resolution, rise, rung, sampling
  • The side the census sat on — both name ablation, claim testing, control, handover, honest limits, lattice offset, negative result, rise, rung, sampling
  • The wrecking set moves again — both name ablation, claim testing, contact family, handover, honest limits, lattice offset, replication, rise, rung, sampling
  • Six lattices were not enough — both name ablation, claim testing, control, honest limits, lattice offset, negative result, rise, rung, sampling
  • The exception was already labelled — both name ablation, claim testing, control, handover, honest limits, lattice offset, negative result, rise, rung
  • The ordering on six bands — both name ablation, claim testing, control, handover, lattice offset, negative result, resolution, rise, rung

Named objects

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

AblationClaim testingContact familyControlHandoverHonest limitsLattice offsetNegative resultReplicationResolutionRiseRungSampling