Phyllotaxis
the spirals are counted, not admired
Stems and cones

One rise below the census

Ten lattices were cut at every offset and their surviving lags came back as four numbers. One rise further down a rung the census already sweeps, three cuts keep a lag of eleven — which is a fifth number, on a lattice nothing about was unusual except that nobody had cut it.
13 min read 24 figures State the claim, state the testCounted, not admired

Worth reading first: The organ that was taken away · Counting the spirals.

The ablation census cuts ten lattices at every offset out to the front and two past it, and the lags its wrecked cuts keep are 4, 5, 7 and 8. Four numbers over thirty rows.

One of its lattices is the Lucas branch at a rise of 0.008, on the 7/11 rung, and its four wrecked cuts all keep 7. One rise below it, at 0.006, on the same rung, three cuts keep 11.

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. 1 The fine end of the Lucas branch, cut at every offset. The rise at 0.006 keeps two families where the ones above it keep one.

What the lattice is

A stem grown at a rise of 0.006 from the Lucas seed angle. It settles to a divergence of 99.344 degrees, which is 0.158 degrees from the Lucas angle itself, and its counted pair is 7 and 11.

There is nothing unusual about it. It is on a rung the ladder found, it settles onto its branch as firmly as anything in the census, and its pair is the pair the rung is named for. The only thing distinguishing it is that it had never been cut.

That is worth saying because the search was expected to end somewhere strange. A lag the census has never seen sounds like it should live at the edge of what the rule can do, and the edge is thirty to ninety degrees off the branch where nothing settles at all. This one is in the middle of an ordinary rung.

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. 2 Where the census sits on the Lucas branch, and where in the 7/11 rung the new lattice falls.

What its cuts do

Six offsets wreck. Offsets 5, 6 and 7 keep a lag of 7, which is what the census’s lattice at 0.008 does at the same rung. Offsets 8, 9 and 10 keep a lag of 11.

So the lattice answers both ways depending on the offset, which is not itself unusual — five of the census’s ten lattices keep two different families at different offsets. What is unusual is which two.

The split is clean as well. It is not that some high offsets keep 11 and some do not: the three lowest wrecking offsets keep 7 and the three highest keep 11, with no interleaving. On a lattice whose answers are decided by the offset and not entirely by it, a clean split is the more surprising arrangement.

The 20 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 20 rows is fitted over five 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. 3 The lags the extended census keeps, with the new one drawn apart.

Eleven is the larger counted number

That is the part worth pausing on. On every other row of the census the surviving family is the smaller of the counted pair or a sub-multiple of it: an 8/13 lattice keeps 8 or 4, a 5/8 lattice keeps 5, a 4/7 lattice keeps 4 or 7.

Here a 7/11 lattice keeps 11. The larger number survives, which is something no golden rise on this ladder does at any offset — the survivor there ranks second or worse by hop length every time.

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. 4 Where the surviving family sits in the ordered hop lengths, across the census.

And it is the shortest hop

At this lattice the 11-hop is the shortest step on the surface, and the three cuts that keep 11 are keeping the shortest hop. The three that keep 7 are keeping the second-shortest.

That is worth setting against the census’s own reading. The survivor is not the shorter of the two contact steps is stated there as a refutation, and the honest version of it in that file is that neither reading is the account — at many offsets the survivor is the second-shortest and at many others it is the shortest. This lattice does both, at different offsets, on one stem.

One wrecked stem, lag by lag — golden, rise 0.013, organ 4 back. How much each lattice hop moves from organ to organ in a stem that never repaired after a single removal, over its last 120 organs. The lag-one hop is the divergence itself and it swings by 65 degrees. The lag-5 hop swings by 0.11 degrees and sits 0.09 degrees from where the undisturbed stem put it, so that one family of the original lattice is still standing organ by organ. Its multiples inherit the same steadiness and nothing else comes within a factor of twenty. The block of angles this stem repeats has a period of 5, which is the surviving lag and not a coincidence.
Fig. 5 The ordered hop lengths at a lattice, with the two contact families named.

Why nobody had cut it

The census’s ten lattices were chosen so that the same counted pair appears at more than one rise. That is what makes its decisive negative available: 4/7 appears twice and 5/8 four times, so the pair and the offset decide the survivor is a claim with rows on both sides of it rather than a summary of one run.

Spanning the lags was not a design goal because nobody had noticed the lags were spanned by four numbers. The list looked like a consequence of the table rather than a fact worth testing.

Every stem that never repaired, and the lag it kept. The 19 offsets across six lattices at which a single removal leaves a stem that never returns to its divergence. For each one: which organ was removed, the period of the block of angles the stem settles into, the lag whose hop survived the cut unchanged, and how many whole turns the stem gains over one period of that lag. The block and the surviving lag are the same number in every row. The marked row is the one whose survivor is not a parastichy number of the lattice that was cut — a hop 6.8 times the length of a contact hop, which no census would report and which the rule held rigid all the same.
Fig. 6 The census as a table, whose lattices were chosen to repeat pairs rather than to span lags.

Where on the rung it sits

The Lucas 7/11 rung runs from a rise of 0.00947 down to 0.0057. The census’s lattice sits at 33 per cent of it and the new one at 76 per cent.

Between them the family the cut leaves standing changes. The search stepped from 0.0070 to 0.0060 in one move, so the rise at which it changes is located to within a sixth of the rung and no better.

That is coarse by the standards of this thread, which has located a transition to one part in a thousand elsewhere. The difference is that this was a search rather than a sweep: it was looking for whether such a lattice exists at all, and a search that resolved every step finely enough to place a transition would have cost ten times what it did.

The ratio is a U across every rung, and its floor is the number that was reported. The ratio of the second comb to the main comb, on five stems at each of 15 rises spanning two rungs, against the ladder's own coordinate for where each rise sits inside its rung. Both rungs give the same shape: a floor of 0.71 and 0.79 about two thirds of the way up, climbing towards the transition at either end. The dashed line is a transported disturbance with no rule in it at 1.29, which does not vary with the rise at all — a kinematic lattice's angle sequence has no rise in it. Where the rule's curve crosses that line the two accounts are indistinguishable.
Fig. 7 A quantity read across one rung, which is the resolution the search worked at.

And where the band sits

The band around that rung’s handover covers rises from 0.00922 down to 0.00721, and it was cut at every one of its 124 rises. Every cut on it keeps 7.

So the band holds the survivor fixed across its whole width and the survivor changes twenty rises below its fine end. That is a coincidence of construction rather than a finding — a band is grown outwards from its handover until the settled divergence moves a twentieth of a degree, and this one happens to stop above the change.

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. 8 The band, whose every cut keeps one family across all 124 of its rises.

What the change is not

It is not a change of counted pair. The pair is 7 and 11 at 0.0070 and 7 and 11 at 0.0060, and it is 7 and 11 everywhere on the rung by definition of a rung.

It is not a change of branch either: 99.141 degrees at 0.0070 and 99.344 at 0.0060, both within a quarter of a degree of the Lucas angle. So whatever moves between those two rises, it is not the lattice’s identity.

The settled divergence down the Lucas branch. Every rise from 0.07 down to 0.0057, plotted against the divergence the rule settles on, with each rung drawn in its own stroke and the branch's limit angle marked. The curve does not slide: it turns one times in four rungs, climbing across one and falling across the next, so a value it takes on one rung it takes again on another. That is what makes a matched pair possible — two rises, different counted pairs, one angle — and it is the whole reason the design exists on this branch. The widest excursions from the limit angle, coarse rung first, are 5.264°, 2.920°, 2.295°, 0.427°.
Fig. 9 The settled divergence down a branch, which is what a rung holds while everything else moves.

What it might be

No account is offered and the honest reason is that the search has one case. Two observations are available: the offsets that keep 11 are the three highest that wreck here, and the 11-hop is the shortest step at this rise and not at the coarser ones.

The second is checkable and would be the thing to check. If the 11-hop becomes shortest between 0.0070 and 0.0060, then the survivor follows the ordering of the hops on this rung, which is a claim the census’s own decisive negative would then have to be read against.

Which offsets give short hops, at a rise of 0.05The two lowest points are at 2 and 3, 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.0123102030index offsetmedian hop between node i and node i+m23260 nodes, 34 offsets triedshortest at 2 and 3
Fig. 10 How hop lengths change with the rise, which is the quantity that would have to cross.

Nine rises would settle it

The rise at which the 11-hop overtakes the 7-hop is arithmetic on the geometry and costs nothing. Whether the surviving family changes at that rise is nine cut stems at nine rises, about five minutes.

That is the cheapest open question in this thread and it is worth naming precisely so that it does not get lost: does the survivor change at the rise where the hop ordering does? If it does, this rung has a second kind of handover in it.

The 13/21 rung, at two azimuth grids. Five stems at each of five disturbances, all at a rise of 0.0019, read through a lag window of 45 from a seed of 40 nodes. A filled mark is a stem that returned 13/21, which is what the position counter finds in it; an open mark is a refusal. The only difference between the two rows is how many azimuths the placement rule samples when it takes its minimum: 384, which every run on this site has used since the earlier work, against 1152. At the coarse grid the rule locks onto its own sample points and the readout refuses 22 of 25 stems; at the fine one it reads all 25. The ceiling was a parameter of the program.
Fig. 11 The same kind of quantity read at a finer step, which is what nine rises between two would look like.

Three rows, and they are one measurement

The three cuts that keep 11 are at offsets 8, 9 and 10 of the same lattice, and their exchanges are 97.25, 97.25 and 97.28 degrees.

That is one number three times. The offsets differ and nothing else does — same rise, same seed angle, same history below the hole — so the three rows are three cuts and one observation, and any arithmetic quantified over rows should say so.

The correction's coefficient at each of the five lags. How far the exchange falls short of one step of the control's divergence, as a share of the surviving hop's own angle, at every row of the table. The stated rule is a fifth and the least-squares fit over the whole table is 0.184. The new lag's three rows sit at 0.164, 0.163, 0.162, inside the spread the four older lags already covered rather than beyond it.
Fig. 12 The correction’s coefficient at every row of the extended table, with the three new rows together.

Which is the same caveat the exchange makes about itself

The exchange’s own file says the honest denominator is four hops rather than seventeen rows, because the hop is nearly constant inside a lag. That argument applies to its extension unchanged.

So the right way to report this is that the denominator has gone from four to five, not that the table has gone from seventeen rows to twenty. Both sentences are true and only one of them is about evidence.

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. 13 The old table gathered by lag, where four clusters carry seventeen rows.

How it was added to the census

By passing the census a longer list of lattices, not by editing the list. The extended table is the old ten plus one, so every number the old table reported is answered from the same cache rather than measured again.

That is the same discipline as keeping the six original lattices inside the larger slot table. A comparison between a table and a table with one more row in it is worth nothing if the two are also two implementations.

A wreck is a whole number of extra turns. For each of the 19 stems that never repair, the slip of its settled divergence multiplied by the lag whose hop survived. Every value lands on a whole number of turns — the horizontal lines — with a largest departure of 2.97 degrees, against divergences that have moved between 0 and 103 degrees. 17 of the 19 close on exactly one turn. So a wrecked stem is the stem it was with one extra turn threaded through every period of the family that survived, which is a dislocation with a stated size rather than damage.
Fig. 14 The census read a second way, which is the reading the extra lattice slots into unchanged.

What the three new rows are like

All three are balanced pairs: two adjacent chains displaced in opposite directions by the same amount, which is the shape seventeen of the census’s thirty rows have and thirteen do not.

So all three enter the exchange table rather than the set it sets aside. That was not guaranteed and it is what makes the fifth cluster usable: a lattice whose cuts all came back unpaired would have found a new lag and tested nothing.

A period of 8, with the hole's own chain at the top. Each mark is one residue class of the displacement profile, placed round a ring at its own residue, with the chain the removed organ sat on at the top. The radius is how far that class sits from the level the rest of them share. six of the eight classes sit together at the middle ring; two do not, and on this row they are one pair, equal and opposite to within a twentieth. The forward one is chain 7 and the backward one is chain 0, one residue above it, which is the order every row of the census puts them in.
Fig. 15 What a balanced pair looks like in a displacement profile, which is the shape all three new rows have.

And what the other three rows are like

The other three cuts at this lattice — offsets 5, 6 and 7, keeping a lag of 7 — are not balanced pairs. They carry four, four and five exceptional chains.

They therefore belong to the excluded set, and they arrive as an out-of-sample test of a rule about that set which was fitted without them. That is a piece of luck: one search produced a positive for one thread and a test for another.

The 13 cuts the exchange sets aside, chain by chain. One row per wrecked cut with no balanced pair, one cell per chain of the lag it kept, and the chains displaced away from the common level filled. The lag is printed at the left and the sum of the displacements at the right, folded into half a turn either way. seven of the 13 sum to within 12 degrees of nothing and six do not, and every row that closes kept a lag of 7 or 8 while every row that does not kept 4 or 5. The rows are stacked by that lag.
Fig. 16 The cuts the exchange sets aside, which the new lattice adds three more of.

What the search covered

Twenty rises, ten on each branch, from inside the two rungs that carry an 11 or a 13 down past the ladder’s finest rise. Four of the Lucas rises are on a rung and six are below it.

Below the rung the settled divergence leaves the branch — 129.4, 129.5, 165.6, 165.7, 129.7 and 129.7 degrees, thirty to sixty-six degrees from the Lucas angle — on pairs like 3/11, 11/13 and 11/14. Those are not lattices this collection would call lattices.

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. 17 Both branches at the fine end, with the rises that have left their branch drawn pale.

So the positive is one lattice, and that is not nothing

Twenty rises returned one usable case. That is a thin harvest for eight minutes of stems and it is the right harvest for a search whose target is rare.

The alternative reading — that a search returning one case has found nothing much — is wrong here for a specific reason. The case is the only one on this ladder, so a search that found two would have been finding the same thing twice.

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. 18 Both branches’ rungs, of which two carry a number large enough for this question.

What it changes

Three claims of the exchange are re-scored on twenty rows rather than seventeen and all three survive. The correction’s fitted coefficient moves from 0.188 to 0.184, which is a change in the third digit.

That is the least dramatic outcome available and it is the one worth having. A rule that moved when a new cluster arrived would have been a fit; a rule that does not is a rule with one more observation behind it.

The other two claims are the direction the exchange runs, which is now right on twenty rows rather than seventeen, and where it sits, which goes from ten of seventeen at the hole to thirteen of twenty.

The hops the correction is fitted over, with the new one at the near end. Each cluster of rows placed by the lag it kept and the angle of the hop that lag keeps. The four the correction was fitted over run from 19.5 to 39.1 degrees; the new one sits at 12.78 degrees, a third smaller than any of them. A fifth point beyond the near end of a fitted range is a test of the fit, where a fifth point between two old ones would mostly have been a restatement.
Fig. 19 The hops the correction is fitted over, with the new one beyond the near end of the old range.

What a search has to report that a sweep does not

A sweep visits every point of a stated range and its report is its output. A search goes looking, and its report has to include where it looked, because a negative from a search is only as strong as its coverage.

So the twenty rises are named in full and the six that could not be counted as lattices are in the table with the reason attached. A search that dropped its unreadable points would be reporting its own reach, which is the failure this collection has caught in three separate instruments already.

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 3 per cent error on each ring position leaves, so no ruler separates it from a flat disc.
Fig. 20 What a measurement can and cannot report about itself, which is why an unreadable point stays in the table.

The banked search, and why it lives on its own

The twenty rises cost about eight minutes of stems, and the analysis that reads them costs nothing. Those two are kept in separate files on purpose: a cached sweep is keyed on the code that produced it, so a file holding both would discard eight minutes every time a sentence in the analysis changed.

That is not a hypothetical. It happened once during this work — an edit to a comment above the reading function invalidated the whole search — and the split is what stops it happening again.

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. 21 The reading the analysis produces, which is cheap once the search behind it is banked.

What would make this two cases instead of one

The Lucas 7/11 rung supplies one lattice that keeps 11. A second case would have to come from somewhere the ladder does not obviously offer.

The rung’s own fine end is the first place to look: below 0.006 there are rises down to 0.0057 that were not cut, and if the whole lower quarter of the rung keeps 11 then there are several lattices rather than one, though they would be several rises of one rung and so not independent in the way two rungs would be.

The honest statement is that this ladder has one rung whose larger number can survive, and that a second would need a subject with a longer ladder in it.

The ratio is a U across every rung, and its floor is the number that was reported. The ratio of the second comb to the main comb, on five stems at each of 15 rises spanning two rungs, against the ladder's own coordinate for where each rise sits inside its rung. Both rungs give the same shape: a floor of 0.71 and 0.79 about two thirds of the way up, climbing towards the transition at either end. The dashed line is a transported disturbance with no rule in it at 1.29, which does not vary with the rise at all — a kinematic lattice's angle sequence has no rise in it. Where the rule's curve crosses that line the two accounts are indistinguishable.
Fig. 22 One rung read across its whole span, of which the search cut four rises.

What a reader should carry

That a census’s answers can be bounded by its own sampling in a way nobody notices, because a list of four numbers attached to ten lattices reads like a fact about lattices.

And that the repair here cost eight minutes and one extra row. The lattice was on a rung the collection already sweeps, one rise below one already in the table, with nothing about it out of the ordinary except that it had not been cut.

The 20 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 20 rows is fitted over five 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. 23 The extended table, whose fifth cluster is a lattice the collection already had.

What the picture at the top shows

Ten rows, one per Lucas rise searched, coarse at the top. Each names the counted pair, the settled divergence, whether the rise is on a rung, and the lags its wrecking cuts leave standing.

The top three rows keep 7. The fourth — the rise at 0.006 — keeps 7 and 11, and is the only row in either branch’s block that keeps a lag outside the census’s four while still on its branch. The six below it are pale, at divergences thirty to sixty-six degrees off.

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. 24 The Lucas branch’s fine end, with the one on-branch row that answers the search.

The one line

A Lucas lattice at a rise of 0.006, on the 7/11 rung, settled divergence 99.344 degrees: three of its six wrecking cuts keep a lag of 11, which is the larger counted number and the shortest hop, and no other lattice on either branch does.

It is one rise below a lattice the census already holds, and the whole search that found it was eight minutes.

Where this goes next

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. The lag that never survives A correction to the exchange's size rests on four hop clusters, and a fifth would be the first real test of it. The prediction was written for a golden lattice at a lag of eleven. No golden lattice on this ladder reaches one, and the reason is a fact about the rule rather than about the search. The hops the correction is fitted over, with the new one at the near end. Each cluster of rows placed by the lag it kept and the angle of the hop that lag keeps. The four the correction was fitted over run from 19.5 to 39.1 degrees; the new one sits at 12.78 degrees, a third smaller than any of them. A fifth point beyond the near end of a fitted range is a test of the fit, where a fifth point between two old ones would mostly have been a restatement. A fifth cluster A correction to the exchange's size was fitted over four hop clusters and its own file said so. A search turned up a fifth, at a hop smaller than any of the four, and the rule is right on it — which is what a prediction being confirmed looks like when the confirmation is worth having. 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. One offset, two answers Which contact family a wrecked stem keeps is decided by where the cut landed, at twenty-five of thirty offsets, by the simplest rule anybody would write down. It is refuted by two runs: the same counted pair, the same offset, two different rises, and two different surviving families.

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.

AblationCensus designClaim testingContact familyHop lengthLattice offsetParastichy pairReplicationRiseRungSamplingSearch