Phyllotaxis
the spirals are counted, not admired
Stems and cones

The hops cross once

Walking a whole rung at the grid its rises are named on costs a few hundred stems and no cuts at all, and it answers a question nobody had asked: whether a rung has one handover or several. It has one, and the ladder had recorded it in the wrong place.
15 min read 21 figures Counted, not admiredState the claim, state the testTwo routes, one number

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

A lattice’s two contact families are the two lags whose hop across the surface is shortest. A handover is a rise at which they change places. Every rung of the ladder was searched for one, and six of the ladder’s rungs have one.

That search swept the ladder at a ratio — a constant step in the logarithm of the rise, which is the right instrument for finding rungs across two decades and is coarser than the grid the rises are named on. It reports one handover per rung and nobody had asked whether that is because there is one or because the sweep found the first.

The two shortest hops across the Lucas 7/11 rung, and where they cross. The lengths of the two shortest hops at every one of the 378 rises of the rung, fine on the left, computed from the settled divergence rather than measured off a cut. They cross exactly once, between 0.00804 and 0.00803, which is this rung's handover. The rise at which a cut first keeps the longer family is 164 steps of the grid further down and is marked separately; nothing happens to either length there.
Fig. 1 The two shortest hops at every rise of one rung, walked at the grid the ladder names its rises on.

What it costs to ask

Almost nothing, and that is the whole reason this exists. A hop length is closed-form geometry: the settled divergence gives the angle between consecutive organs, and the hop of a lag follows from the angle and the rise without growing anything.

One stem a rise to find the divergence, and then forty hop lengths evaluated and sorted. The Lucas 7/11 rung holds 378 rises at the hundred-thousandth grid, and walking all of them takes about three minutes.

That is two orders of magnitude cheaper than any question that needs a cut, and it is the reason a rung can be walked where a band has to be sampled.

Which offsets give short hops, at a rise of 0.008The 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.00.50011.5051015index offsetmedian hop between node i and node i+m2324 nodes, 18 offsets triedshortest at 2 and 3
Fig. 2 Every lag’s hop length at one rise, of which only the top of the ordering is ever read.

The answer

Exactly one crossing, between h = 0.00804 and h = 0.00803. Above it the 7-hop is shorter; below it the 11-hop is, at every one of the remaining rises down to the rung’s fine end at 0.00570.

So this rung has one handover. That is the answer the coarser sweep gave and it is now a measurement rather than the first thing found.

It is also the answer that makes a band meaningful. A band is grown outwards from a handover, and a rung with two would have two bands and no reason to prefer either.

The two shortest hops across the Lucas 7/11 rung, and where they cross. How much longer the second-shortest hop is than the shortest, at every one of the 378 rises of the rung, coarse on the left. The ratio falls to 1.0003 at the crossing between 0.00804 and 0.00803 and rises away from it on both sides, which is what a crossing of two lengths looks like from above. It never reaches one, because the two lengths are equal only between two rises of the grid.
Fig. 3 How much longer the second hop is than the shortest, across the rung, dipping to a minimum at the crossing.

The ladder records it somewhere else

At 0.00800, which is neither of the two rises the fine walk brackets the crossing between.

That is not an error. The ladder sweep steps by a ratio, and near the fine end of the Lucas branch that step is about half a per cent of the rise, so it visits 0.00800 and does not visit 0.00804 or 0.00803. It reports the rise it visited that is nearest the crossing.

The difference is 0.44 per cent of the rise, and nothing built on the handover moves: the band grown around it is 124 rises wide and the four-thousandths of a per cent of a rung in question is inside it many times over.

Where a rung's handover sits, over six rungs. The rise at which the two contact steps change places, as a fraction of the way down each rung from its coarse end, over every rung on both branches that has one. All six fall in the coarse half and none in the middle: the positions measure 6, 12, 15, 33, 35, 40 per cent. That is a fact about the arithmetic of the lattice rather than about any experiment run on it, because the crossing is where the two step lengths are equal and both are functions of the rise and the divergence alone.
Fig. 4 Where each rung’s handover sits inside it, as recorded by the ladder sweep that steps by a ratio.

Which is worth writing down anyway

Because the handover has been used as a rise in comparisons, and a rise used in a comparison should be a rise somebody has located.

The comparison it is used in is the one this thread rests on: whether the rise where the survivor changes is the rise where the hops cross. Until now the second of those was known to a sweep step and the first to a bracket, and neither was known to the grid.

Now both are known to one step of the grid on this rung, and the comparison is between two located numbers rather than between two approximations.

Two rises on the Lucas 7/11 rung, each located to one step of the grid. The whole of the Lucas 7/11 rung, coarse on the left, with the two rises this round located. The two contact steps change places between 0.00804 and 0.00803, found by walking all 378 rises of the rung and evaluating hop lengths, with no cut stems at all. The family a cut leaves standing changes between 0.0064 and 0.00639, found by cutting 89 stems. They are 164 steps of the grid apart, which is 45 per cent of the rung's whole span, so the geometry crossing is not what moves the survivor.
Fig. 5 The two located rises on this rung, and the recorded handover beside the crossing it stands for.

How near the two hops come

At the crossing the second-shortest hop is longer than the shortest by a factor of 1.0003, which is three parts in ten thousand. That is what a crossing looks like when it is sampled at a grid: the two lengths never come out equal, they come out as near as the grid allows.

Away from the crossing the ratio rises to 1.104 at the rung’s ends. So the ordering is decisive over most of the rung and marginal over a handful of rises around the crossing.

That has a consequence for anything read near a handover. A quantity that depends on which hop is shorter is reading a distinction of three parts in ten thousand there, and a quantity that depends on how much shorter is reading nearly nothing.

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. 6 How near the two contact steps come across each band, which is what “separated” means.

The separation rule

The counter that returns a contact pair declines to order the two steps when they are within a stated ratio of each other, and the rule is not a convenience: two steps that differ by less than the grid can resolve are two steps whose ordering is a property of the grid.

On this rung the ordering is refused at the crossing and nowhere else, which is what a single crossing on a smooth pair of curves should produce.

A rung where the refusal covered a stretch would be a rung whose handover is not a rise but an interval, and none of the six is like that.

The families left standing, on both sides of every handover. One row per band, with every offset cut at rises spread across it and always at both ends and at the handover itself. The last column is every family left standing anywhere on that band. On three of the four bands that wreck at all it is a single family, unchanged across a rise at which the two contact steps swap places — so the step ordering is not what decides which family survives, and the result now rests on four counted pairs rather than on two. The shortest-hop reading scores 44 of 114 across the whole set, which is what a reading looks like when the quantity it is stated over is not in the mechanism.
Fig. 7 The ordering across every band on the ladder, with the stretch around each crossing where it is refused.

Why one crossing is not obvious

The two hop lengths are not monotone in the rise. Each is the distance across a cylinder between organs a fixed number apart, and as the rise falls the whole lattice compresses in one direction while the divergence drifts, so both curves bend.

Two bending curves can cross any number of times. What the walk shows is that on this rung they cross once, and it shows it by evaluating both at every rise rather than by an argument.

The same walk on the other five rungs is the obvious extension and has not been run. It is three minutes a rung, and the six bands built around those handovers are all built on the assumption this walk has now tested on one of them.

The two shortest hops across the Lucas 7/11 rung, and where they cross. The lengths of the two shortest hops at every one of the 378 rises of the rung, fine on the left, computed from the settled divergence rather than measured off a cut. They cross exactly once, between 0.00804 and 0.00803, which is this rung's handover. The rise at which a cut first keeps the longer family is 164 steps of the grid further down and is marked separately; nothing happens to either length there.
Fig. 8 The two curves whose crossings are being counted, neither of them monotone in the rise.

What a second crossing would have meant

That the band built around the recorded handover is one of two, chosen by which the ladder sweep found first — and that every claim about the handover on that rung is a claim about an arbitrary one of them.

It would also have meant something about the counted pair. A rung is a stretch over which one pair is returned, and a pair whose two members exchange the shortest hop twice inside one rung is a stranger object than one that exchanges once.

Neither is the case here. The result is a negative and it is the negative that lets everything else stand.

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. 9 The Lucas branch’s rungs, each of which was searched for a handover by a sweep coarser than this walk.

The rung’s own extent

From 0.00947 to 0.00570, which is a factor of 1.66 in the rise and 0.508 in the logarithm. The crossing at 0.008035 sits 33 per cent of the way down it from the coarse end.

That is the quantity a handover’s position inside its rung is measured by, and across the six rungs with handovers it runs from 5 per cent to 40. So this rung’s handover is neither central nor at an end.

Where it sits is not decided by anything anybody chose: a rung ends where the counted pair changes and the handover falls where the two hops cross, and both are read off the same geometry.

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. 10 A rung’s geometry, from which both its extent and the position of its handover are read.

What the walk does not answer

Whether the crossing means anything to a cut. That is the question the same rung was cut for, and the two answers sit far apart: the family a cut keeps first includes 11 between 0.00640 and 0.00639, which is a hundred and sixty-four grid steps below the crossing.

So the geometry and the ablation disagree about where anything happens on this rung, which is the point of having both.

The walk’s contribution is to make that a disagreement between two located numbers rather than between a sweep step and a bracket.

The Lucas 7/11 rung between 0.007 and 0.006, cut at every rise. One row per offset, one column per rise of the search, coarse on the left. A pale cell is a rise at which that offset's cut recovers; a dark cell is a cut that wrecks and keeps a lag of 7; a warm cell is one that keeps 11. The nine intermediate rises of the ten-thousandth grid were cut first and the nine of the hundred-thousandth grid inside the one bracket they opened. The change sits between 0.0064 and 0.00639, one step of the grid the ladder names its rises on, and above it no cut keeps 11 anywhere.
Fig. 11 The cuts on the same rung, whose transition sits far below the crossing this walk locates.

Sampling a smooth quantity, for once

Everything else in this thread is discrete: a surviving family is an integer, a cut wrecks or does not, an offset is in the wrecking set or is not. Those quantities change at single rises and can only be found by looking at every rise.

Hop lengths are not like that. They are smooth functions of the rise, they can be interpolated, and the crossing could have been found by bisection in a dozen evaluations rather than 378.

The walk was run anyway, because bisection assumes one crossing and one crossing was the thing being tested. Having the whole curve is also what makes the ratio at the ends available, which bisection would not have produced.

The two shortest hops across the Lucas 7/11 rung, and where they cross. How much longer the second-shortest hop is than the shortest, at every one of the 378 rises of the rung, coarse on the left. The ratio falls to 1.0003 at the crossing between 0.00804 and 0.00803 and rises away from it on both sides, which is what a crossing of two lengths looks like from above. It never reaches one, because the two lengths are equal only between two rises of the grid.
Fig. 12 The smooth quantity this walk reads, which could have been bisected if the number of crossings were known.

What the ratio at the ends is worth

It bounds how much of the rung is near the crossing. At 1.104 at the ends and 1.0003 at the crossing, the ratio is above 1.05 over about two thirds of the rung.

So a claim of the form this rung’s cuts keep the shorter hop is a claim about a well-ordered pair over most of the rung and about a marginal one near the middle. The census cuts this rung at 0.008, which is one grid step from the crossing and therefore in the marginal region.

That is worth knowing about the census row rather than about the rung: the lattice the census holds here is the one whose two contact steps are most nearly tied on the whole ladder.

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. 13 The census, one of whose rows sits one grid step from a handover where the two hops are three parts in ten thousand apart.

The refusal built into the walk

A rise the counter cannot read is recorded as a refusal on its own row rather than allowed to end the walk. A walk that stopped at its first unreadable rise would report where the reading fails and not where the crossing is.

On this rung nothing refuses. All 378 rises return a pair and an ordering, which is what a rung means — the pair is constant across it by construction.

The provision is there because the same machinery is used on stretches below the ladder’s finest rung, where refusals are common and are the informative part.

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 Where the same reading is taken below the ladder, and where refusals are the informative part.

Five rungs unwalked

The golden 3/5, 5/8 and 8/13 and the Lucas 3/4 and 4/7 each have a recorded handover and none has been walked at the grid.

Three minutes each. What it would buy is the same two things it bought here: a crossing count per rung, and a located rise to compare anything against.

The crossing count is the interesting half. One rung with two crossings would change what a band is on that rung, and there is no argument available that says there is not one.

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. 15 Every rung with a handover, of which one has now been walked at the grid its rises are named on.

Why a rung is swept at a ratio in the first place

The ladder runs from a rise of 0.07 to 0.003, which is a factor of twenty-three. A fixed step in the rise is a step of about one per cent at the coarse end and a step of about thirty per cent at the fine end, so a sweep by fixed step either spends most of its budget where nothing changes or steps clean over whole rungs at the bottom.

Sweeping by ratio makes every step the same size in the quantity that matters, which is the argument the ladder sweep states for itself and the argument a band’s own step follows.

The cost is that a ratio step lands where it lands. It never visits the five-decimal grid except by coincidence, so a rise it reports is a rise it sampled and not a rise anybody named. That is the whole of the discrepancy here.

Every transition as the rise falls. The pair climbs 1/2 → 2/3 → 3/5 → 5/8 → 8/13. Consecutive transitions are 0.382, 0.382, 0.382 of the previous rise — 1/φ² is 0.3820.
Fig. 16 The ladder, swept by ratio across a factor of twenty-three in the rise.

What the five-decimal grid is

Every rise on this site is rounded to five decimal places before anything is grown from it. That is not a display convention; it is the set of rises that exist.

The reason is reproducibility. A stem grown at 0.0064000001 and one grown at 0.0064 are two different runs, and a table that quotes the first is a table nobody can reproduce from what it prints. Rounding at the point of use makes every rise in every file a rise anybody can type.

It also sets the finest resolution any claim here can have. A transition located to one step of that grid is located as finely as this collection can locate anything, and asking for more would mean changing what a rise is.

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. 17 The grid rises are named on, which is the finest resolution any claim on this site can have.

What the walk says about the census’s own rise

The census cuts this rung at 0.008, which is four steps of the grid from the crossing. Of the ten lattices the census holds, that is the one nearest a handover by a wide margin.

Whether that matters is not something this walk can say, and it is worth stating as an open question rather than as a caveat. The claims the census makes are about which family a cut keeps, and the family it keeps is not the shorter hop — so a row where the two hops are nearly tied is a row where the rejected account has least to lose.

If anything it is the useful row. An account that says the cut keeps the shorter hop makes its weakest prediction where the two are tied, and the census’s nearest-to-tied row still refuses it.

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. 18 The census’s rows against which hop is shortest at each, including the one nearest a handover.

Two routes to one number, again

The crossing can be computed two ways and both are in the walk. The ordering flips between two rises: at 0.00804 the shortest lag is 7 and at 0.00803 it is 11. The ratio between the two shortest lengths dips to its minimum at the same pair of rises.

Those are the same fact read off two different quantities, and they agree — which is what the site asks of every number it reports. The ratio’s minimum is the more useful of the two because it is continuous: it says not only where the crossing is but how sharp it is.

Two routes to one number is the discipline this whole collection runs on, and it is cheapest exactly here, where both routes are arithmetic on a settled divergence.

The two shortest hops across the Lucas 7/11 rung, and where they cross. How much longer the second-shortest hop is than the shortest, at every one of the 378 rises of the rung, coarse on the left. The ratio falls to 1.0003 at the crossing between 0.00804 and 0.00803 and rises away from it on both sides, which is what a crossing of two lengths looks like from above. It never reaches one, because the two lengths are equal only between two rises of the grid.
Fig. 19 The ratio between the two shortest hops, whose minimum locates the same crossing the ordering does.

The stem behind each rise

One per rise, grown until its divergence stops changing, and the divergence is what every hop length is computed from. So the walk is not free of stems — it is free of cut stems, which is the expensive kind.

A cut needs two runs: the control and the cut run, grown to the same length with the same history below the hole. A hop length needs one run and then closed-form arithmetic, which is why 378 rises of geometry costs three minutes and twenty rises of cutting costs seven.

That ratio is what makes the design of this round possible at all. The geometry can be walked at the grid and the cuts have to be sampled, so the two quantities being compared are known to very different precisions unless the cutting is aimed at a bracket somebody has already narrowed.

How many organs settling takes, by falloff exponent. One mark per stem that settles, placed at the organ from which every later divergence stays within a degree and a half of the run's own final value. At the exponent every other measurement here uses, the slowest is 290 organs — the number four rounds of this collection have carried as though it belonged to the rule. It belongs to the exponent: at 4 the slowest is 808 and at 5 it is 674. A table of shares cannot see this, because a stem that settles at organ 800 and one that settles at organ 8 are the same entry in it. And it is still not a shortage of time: of 72 pairs of runs grown to 1200 organs and then to 3200, 0 settle at the longer length after failing at the shorter.
Fig. 20 The run behind each rise, grown until its divergence stops changing before any hop is computed.

What the walk is a template for

Any question about a lattice that does not need a cut. The expensive thing in this thread is growing a second stem with an organ removed and comparing it to a control; everything read off an intact stem is one run and closed-form arithmetic afterwards.

That distinction is worth keeping in view when a question is being designed. Where do the two contact steps cross is geometry and costs three minutes for a whole rung. Where does the family a cut keeps change is ablation and costs seven minutes for twenty rises. The two differ by two orders of magnitude in coverage per minute.

So a round that wants both quantities at the same resolution has to spend its cutting where the geometry has already narrowed the bracket. That is exactly the design this round used, and the walk is the half that made it affordable.

The five rungs this leaves

Each is three minutes and each could hold a second crossing. There is no argument available that says it does not — the two hop lengths are non-monotone in the rise, and two bending curves can meet any number of times.

What a second crossing would mean is worth stating in advance, because it decides how interesting the check is. It would mean that rung has two handovers, that the band grown around the recorded one is a band around whichever the ladder sweep found first, and that every claim about the handover on that rung is a claim about an arbitrary one of two.

Fifteen minutes for all five. It is the cheapest unspent check in this thread.

What is claimed

That the two contact steps of the Lucas 7/11 lattice change places exactly once across the whole rung, between 0.00804 and 0.00803, over 378 rises walked at the hundred-thousandth grid with no cut stems.

That the ladder sweep records the handover at 0.00800, which is 0.44 per cent of the rise away and is the nearest rise it visits, and that nothing built on the handover moves as a result.

And that the two hops come within three parts in ten thousand of each other there and within ten per cent at the rung’s ends, so the ordering is marginal near the crossing and decisive over most of the rung.

The two shortest hops across the Lucas 7/11 rung, and where they cross. The lengths of the two shortest hops at every one of the 378 rises of the rung, fine on the left, computed from the settled divergence rather than measured off a cut. They cross exactly once, between 0.00804 and 0.00803, which is this rung's handover. The rise at which a cut first keeps the longer family is 164 steps of the grid further down and is marked separately; nothing happens to either length there.
Fig. 21 The whole walk: 378 rises, one crossing, and no cut stems anywhere in it.

Where this goes next

The Lucas 7/11 rung between 0.007 and 0.006, cut at every rise. One row per offset, one column per rise of the search, coarse on the left. A pale cell is a rise at which that offset's cut recovers; a dark cell is a cut that wrecks and keeps a lag of 7; a warm cell is one that keeps 11. The nine intermediate rises of the ten-thousandth grid were cut first and the nine of the hundred-thousandth grid inside the one bracket they opened. The change sits between 0.0064 and 0.00639, one step of the grid the ladder names its rises on, and above it no cut keeps 11 anywhere. One step of the grid, again A search of the fine end stepped from one rise to another a hundred grid steps away, and the family a cut keeps changed somewhere between. Cutting the rises in between puts the change inside one step, with the lattice identical on both sides. The golden 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 12 and 35 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; seven of them sit past the mark. Where a handover sits Inside every rung there is a rise at which the two contact steps change places, so that the shorter hop belongs to the other family below it. Six of the eight rungs on this ladder have one, each has exactly one, and every one of them sits in the coarse half. Two rises on the Lucas 7/11 rung, each located to one step of the grid. The whole of the Lucas 7/11 rung, coarse on the left, with the two rises this round located. The two contact steps change places between 0.00804 and 0.00803, found by walking all 378 rises of the rung and evaluating hop lengths, with no cut stems at all. The family a cut leaves standing changes between 0.0064 and 0.00639, found by cutting 89 stems. They are 164 steps of the grid apart, which is 45 per cent of the rung's whole span, so the geometry crossing is not what moves the survivor. Two rises far apart The whole handover thread rests on the claim that where the two contact steps change places is not where the survivor does. On one rung both rises are now located to a single step of the grid, and they sit forty-five per cent of the rung apart.

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.

  • Nine rises were not enough — both name claim testing, contact family, discretisation, handover, honest limits, instrument setting, negative result, resolution
  • A band with nothing inside it — both name claim testing, contact family, handover, honest limits, negative result, resolution, rung
  • Every rise of a band — both name claim testing, handover, honest limits, measurement, negative result, resolution, rung
  • The alternation is not a period — both name claim testing, handover, honest limits, measurement, negative result, resolution, rung
  • The offsets that never change — both name claim testing, handover, honest limits, measurement, negative result, resolution, rung
  • The third band, cut whole — both name claim testing, contact family, handover, honest limits, negative result, resolution, rung

Named objects

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

Claim testingContact familyDiscretisationGeometric ladderHandoverHonest limitsHop lengthInstrument settingMeasurementNegative resultResolutionRung