Phyllotaxis
the spirals are counted, not admired
Stems and cones

A band that holds the angle still

Around every handover the settled divergence has a shallow floor, so a run of rises either side of it share a divergence to a twentieth of a degree while their two contact steps change places. That is a matched pair with one quantity varying, and it is the design the ablation thread had no way to state.
12 min read 26 figures Counted, not admiredState the claim, state the test

Worth reading first: Where a handover sits · A head is a set of points · Counting the spirals.

Two quantities move inside a rung that no counter can read. The settled divergence slides. The two contact steps change places. Sweeping the rise moves both at once, so a result that changes along a rung cannot be attributed to either — and that is exactly what a sweep of one rung found, leaving the attribution open and saying so.

Separating them needs a place where they come apart: rises with the same counted pair, the same settled divergence, and opposite step orderings. This essay is about finding two such places and checking that they are real.

A divergence that does not move across the 5/8 band. Measured at every rise of a band on the golden branch, where a counter returns 5 and 8 spirals throughout. The settled divergence moves by 0.0469 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 1.0013 and the ordering changes hands at a rise of 0.0156, so above that rise the shorter step belongs to the 5 family and below it to the 8 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. 1 Eighteen rises whose settled divergence moves by five hundredths of a degree, with the step ordering changing hands in the middle of them.

The floor

The settled divergence does not slide monotonically down a rung. Across the golden 5/8 rung it climbs from 136.64° to 137.87° — but not smoothly. In the middle it flattens, and around the flattening it takes the same value at two different rises, one either side.

The divergence slides along the 5/8 rung. Measured at every rise on one rung, where a counter returns 5 and 8 spirals throughout. The settled divergence slides from 136.6406 to 137.8672 degrees. The ratio of the two contact steps falls to 1.0131 at a rise of 0.016 and the ordering changes hands at 0.015: above it the shorter step belongs to the 5-family and below it to the 8-family. Neither quantity is available to a counter, which is shown positions and reports a pair, and both of them move while that pair does not.
Fig. 2 The slide across the whole rung, with the flat middle the band is cut out of.

That flattening is where the handover is. Both facts have the same cause: the rule’s own divergence crosses the curve on which the two contact steps are equal, and near a crossing of two lines with similar slopes the difference between them is stationary.

So the band is not a lucky feature of one rung. It is a consequence of the arithmetic and it is available on every rung with a handover — which is six of the eight this ladder carries.

Two lines across the 5/8 rung, crossing once. The divergence the rule settles on, against the divergence at which the two contact steps would be exactly the same length. The second is arithmetic on the lattice and no stem is grown for it. Across this rung the balanced line moves 2.281 degrees and the rule's own line moves 1.262, so the shallower line crosses the steeper one, and it does so exactly once at a rise of 0.0154 — 16 per cent of the way down from the coarse end. That crossing is the handover: above it one family has the shorter step and below it the other does. So a rung has one handover, its position is fixed by the arithmetic rather than by any experiment, and a sweep of the rise carries a stem across it at a place nobody chose.
Fig. 3 The crossing the flat sits at, drawn as the two lines whose difference is stationary there.

The two bands

A band is defined by a bound rather than by hand: it is every rise whose settled divergence sits within a twentieth of a degree of the divergence at the crossing, grown outwards from the crossing until the bound or the rung’s edge stops it. That way the extent is measured and the only judgement in it is the bound.

On the golden branch the 5/8 band runs from 0.0176 down to 0.0142 — eighteen rises at a step of 0.0002, over which the settled divergence moves 0.047°, with the five-step giving way to the eight-step at 0.0156.

The two steps changing places inside the 5/8 band. Measured at every rise of a band on the golden branch, where a counter returns 5 and 8 spirals throughout. The settled divergence moves by 0.0469 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 1.0013 and the ordering changes hands at a rise of 0.0156, so above that rise the shorter step belongs to the 5 family and below it to the 8 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. 4 The one quantity that does move across the golden band: the ratio of the two contact steps, passing through one.

On the Lucas branch the 4/7 band runs from 0.0237 down to 0.0201 — nineteen rises, over which the divergence moves 0.020°, with the four-step giving way to the seven-step at 0.0225.

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. 5 The Lucas band, on a different branch and a different pair, with the same structure.
The two steps changing places inside 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. 6 And its ordering, which changes hands nearer the coarse end because that rung’s crossing does.

Two hundredths of a degree is a fifth of the run-to-run scatter of a settled divergence and a twelfth of the azimuth grid step. Against the 1.262° the same quantity travels across the whole rung, it is a hold rather than a slide.

What “the same divergence” is being claimed

A settled divergence here is the mean of a run’s last sixty divergences, and two runs reporting the same mean is not the same thing as two runs having the same angle. It is worth separating the two, because the whole design rests on the distinction.

What is claimed is that the lattice is the same to within a twentieth of a degree — that if the arrangement of points at the coarse end of the band and the arrangement at the fine end were each described by a divergence and a rise, the divergences would agree far more closely than either differs from anything else on the rung. That is a claim about a summary of each run.

What is not claimed is that the two runs are the same run. They differ in the rise by a fifth, so their organs sit at different heights, their fronts contain slightly different numbers of organs, and their scatters differ a little. A band is a matched comparison in one quantity, not a pair of identical stems.

Two stems at 0.75° of scatter, one angle at a time. The divergence of each node, with the slow climb of the ladder removed. Both stems scatter by about 52.26° and a botanist would report them identically. The jostled one wanders in runs — its lag-one correlation is 0.70 — and the one with placement noise alternates about the mean at 0.21, because a displacement of one node enters two consecutive divergences with opposite signs.
Fig. 7 Two stems that share a summary statistic, which is the kind of agreement a band asserts and the kind it does not.

The distinction matters most for a null result. If cutting an organ out at both ends of a band gives the same answer, the honest statement is that the answer did not move when the ordering reversed and the rise moved by a fifth and the front changed by an organ. Those are three held-or-varied quantities, and only the first is the one the design was built for. Reading such a null as “the ordering is irrelevant” is stronger than the evidence; reading it as “the ordering is not what was doing the work in the sweep that raised this” is exactly right.

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. 8 The general form: which differences a design can attribute and which it can only bound.

The check the band needed

A quantity that comes out constant is the first thing a discretised measurement should be suspected of. Every azimuth here lands on a grid of 1,536 steps, a quarter of a degree apart; a settled divergence is a mean of sixty such angles, so it is not itself confined to the grid, but a stem that locks onto exactly the same sequence of grid values at two rises will report exactly the same mean at both. A flat that is really a grid artefact would look identical to a flat that is real.

The spiral counts four different divergence angles produce. Fibonacci counts come from one angle. The Lucas angle — which the same dynamical model reaches on a different branch — gives 47 and 76, and neither number is a Fibonacci number.
Fig. 9 The grid every angle here lands on, which is fine enough for a count and coarse enough to manufacture a flat.

So both bands were grown again on a grid four times finer — 6,144 steps, 0.059° apart — and the answer is that the flat is not exactly flat and the band survives anyway.

The flat band, re-measured on a finer grid. A quantity that comes out constant is the first thing an azimuth grid should be suspected of, so the whole band is grown again on a grid of 6144 steps against the 1536 the site uses. The finer grid does resolve structure the coarse one flattened: a shallow minimum 0.0820 degrees deep, with its floor at a rise of 0.0158. What it does not do is separate the ends, which still agree to 0.0000 degrees while carrying opposite step orderings. The matched pair the band is for survives the check that would have broken it.
Fig. 10 The golden band on both grids. The finer one resolves a shallow minimum the coarse one flattened; the ends still agree.

At the finer resolution the golden band shows a minimum 0.082° deep, with its floor near the handover, and the Lucas band one 0.054° deep. That is real structure the coarse grid was hiding, and it is exactly what the arithmetic predicts a crossing of two nearly parallel lines should look like.

But the two ends of each band — which is where the claim lives, because that is where the two step lengths are far enough apart for the ordering to mean anything — still agree. On the golden band they agree to 0.000°, and on the Lucas band to 0.020°.

The flat band, re-measured on a finer grid. A quantity that comes out constant is the first thing an azimuth grid should be suspected of, so the whole band is grown again on a grid of 6144 steps against the 1536 the site uses. The finer grid does resolve structure the coarse one flattened: a shallow minimum 0.0537 degrees deep, with its floor at a rise of 0.0201. What it does not do is separate the ends, which still agree to 0.0195 degrees while carrying opposite step orderings. The matched pair the band is for survives the check that would have broken it.
Fig. 11 The same check on the Lucas band, whose ends agree to two hundredths of a degree.

So the matched pair is a matched pair at four times the resolution that would have broken it, which is the only reason to believe it at the original one.

Why the ends and not the middle

At the handover itself the two step lengths are equal, so there is no ordering to speak of. This collection’s own bound for calling two steps ordered is one per cent, and within about a per cent of the handover the gap is under it.

Which offsets give short hops, at a rise of 0.018. The two lowest points are at 5 and 8, and those are the parastichy numbers. Offset 1 is high because a hop of one node is at least the rise, which is what makes a stem easier to count than a disc.
Fig. 12 The ranking at the coarse end of the golden band, where the two steps differ by four per cent.
Which offsets give short hops, at a rise of 0.013. The two lowest points are at 5 and 8, and those are the parastichy numbers. Offset 1 is high because a hop of one node is at least the rise, which is what makes a stem easier to count than a disc.
Fig. 13 And below it, where they differ by seven and the other family is first.

That is why the design puts its weight on the ends. The coarse end of the golden band has its two steps four per cent apart with the five first; the fine end has them three per cent apart with the eight first. Both are properly ordered, they are ordered oppositely, and their settled divergences agree to a thousandth of a degree.

The rises in between are not wasted. They are the evidence that the two ends are connected by a continuum rather than being two arbitrary lattices that happen to share an angle, and they are where the handover is located.

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. 14 The band with the whole ablation across it, which is what the design was built to make possible.

What is held and what is not

Worth listing, because a matched design is only as good as its list.

The counted pair is held. Every rise on each band returns the same pair from a counter shown nothing but positions, and that is asserted rather than assumed.

The settled divergence is held, to 0.047° and 0.020° on the two bands.

The branch is held. Each band is one branch, seeded from one angle.

The front depth is nearly held. The number of offsets that wreck at all changes by one or two across a band, which is the front deepening as the rise falls, and it is the reason the survivor grids have a ragged edge rather than a straight one.

How many offsets wreck, along the 5/8 rung. The count of offsets that never repair, at each rise on one rung. It runs from 1 at the coarse end to 5 at the fine end, while a counter returns 5 and 8 spirals at every one of them. The front — the run of recent organs at which a removal is felt at all — deepens as the rise falls, so there are simply more places a cut can land and fail to heal. That is the mechanism under the grid: the offsets that appear at the fine end are the ones beyond the smaller contact number, and those are the ones that keep the larger family.
Fig. 15 The front deepening across the rung the golden band sits in, which is the quantity a band cannot fully hold.

The rise itself is not held, and cannot be: it is the knob the band is swept with. The golden band spans a factor of 1.24 in the rise and the Lucas one a factor of 1.18. Anything that depends smoothly and strongly on the rise is not controlled here, and the honest reading of any null result from a band is that the ordering does not act and that the rise does not act over a range of a fifth.

Every transition as the rise falls. The pair climbs 1/2 → 2/3 → 3/5 → 5/8 → 8/13. Consecutive transitions are 0.381, 0.382, 0.382 of the previous rise — 1/φ² is 0.3820.
Fig. 16 The ladder the bands are two small pieces of, for the scale of what a fifth in the rise is.

That last one is the design’s real limitation and it is not fixable by a narrower band: narrowing it holds the rise better and shrinks the ordering difference at the ends, until at the handover itself there is no ordering left to compare. The two things trade off directly, and the bound chosen here — a twentieth of a degree — is where both are still large enough to mean something.

What it is for

The design exists because an earlier sweep found the family a wrecked stem keeps changing along a rung, and could not say which of the two movers was responsible. A band holds one and varies the other, which turns an ambiguous sweep into a test.

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 sweep that raised the question: one rung, one offset, two different answers.

Run on both bands, at every offset that wrecks, the answer is that the survivor does not move. That is a negative result and it is the strongest kind available here: not a low score on a census but a quantity reversed under a controlled comparison with the outcome unchanged.

Both bands, on two branches and two pairs. One row per band. Each runs from its coarse end on the left to its fine end on the right, with the rise at which the two contact steps change places marked, and the family that survives every wrecked cut written at the end. The 5/8 band on the golden branch keeps the 5 at all 24 of them and the 4/7 band on the Lucas branch keeps the 4 at all 31. Two branches, two counted pairs, one result: the quantity the band varies is not the quantity that decides the answer.
Fig. 18 Both bands and both answers, on two branches and two counted pairs.

The cost of it

Worth recording, because a design that is cheap gets used and one that is not does not.

The geometry of both bands — thirty-seven rises, each grown once to read its settled divergence and its hop ranking — is under a minute of work. The check on the finer grid is four times that, because the placement scan is linear in the number of azimuth samples, so a few minutes. Neither is a consideration.

The ablation is the expensive half: one cut stem per offset per rise, each grown to settle and compared against a control that shares its history. Fifty-five wrecked cuts came out of about a hundred and fifty grown ones, since most offsets repair, and the whole of it is a few minutes on one core.

The next organ moves for the last 8, and for no others. One row per organ removed, counted back from the tip of a stem at a rise of 0.013 whose counted pair is 5 and 8. Removing any of the last 8 moves the next organ by 4.9° to 164.1°; removing an older one moves it by at most 0.70°, which is under the azimuth grid. The boundary is at 8, and 8 is the larger parastichy number — so the experiment counts the spirals without measuring an angle.
Fig. 19 What one row of that costs: a cut, a control, and a comparison organ by organ.

Against the census this replaces the design is smaller in every direction — one rung instead of a ladder, one branch per band instead of two, and no attempt to span the counted pairs. That is the point. A comparison that varies one thing does not need to cover a subject; it needs to cover the one thing, twice.

What else a band is good for

Anything stated over the step ordering. Two candidates are already in this collection.

The first is the front. Its width is the larger of the two counted numbers, which is a statement about the pair rather than the ordering, so a band should leave it alone — and that is a prediction rather than a hope, because the front depth across a band changes by one or two offsets rather than by the several a reversal would imply.

Two answers 138° apart, and one organ holding the second one up. The repulsion the rule minimises, around the circumference of a stem at a rise of 0.008, at the height the next organ will sit at. It has two low points 138.3° apart: the slot the next organ takes, and the slot the organ after it will take. The runner-up is 13.6% higher. The organ 13 places back carries 14.6% of the energy at the winning slot and twelve places back carries 16.3% at the runner-up — and that is more than the gap, so taking that organ away makes the runner-up win and the next organ appears a whole divergence away. Neither guard is a contact of the organ being placed; twelve is one place inside the larger number of the pair this stem is climbing towards.
Fig. 20 The front the prediction is about, whose width is read from the pair.

The second is the rigid hop. Which lag a wrecked stem holds steady is the block it repeats, and the block is measured against each stem’s own control, so nothing about a comparison between two rises enters it. A band should leave that alone too.

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. 21 The measurement the second prediction is about, which is internal to one pair of runs.

Neither is a hard test, and that is the point of writing them down: a design that changes nothing it should not change is a design worth reusing, and the way to find out is to state in advance which quantities are supposed to be inert.

What the finer grid does to the rises already published. The two rises this site has argued from and the one it published as having no answer, each read at both azimuth grids, five stems apiece. A filled mark agrees with the position counter, a half mark contradicts it, an open mark is a refusal. At 0.013 and 0.005 the readings are identical at both grids, so nothing already written depends on the sample count. At 0.008 they are not: 384 azimuths gives 5/8, 5/8 and 1152 gives 8/13, 8/13, against a counter that says 5/8. That rise was chosen in the earlier work because the three shortest lattice offsets there are within a fifth of each other, and a stem with no answer answering differently on a different grid is the object behaving as it was said to.
Fig. 22 The habit this belongs to: naming what an instrument fixes before reading what it reports.

Why this was not available before

Because the flat is only visible once the rung has been swept finely enough, and the rungs were swept at one rise each for three rounds.

A census taking one rise per rung cannot see a flat: a flat is a property of a run of rises, and one rise is not a run. The sweep that first went along a rung finely enough went at a step of 0.001, which puts eleven rises on the golden 5/8 rung and about five inside the flat — enough to notice, and not enough to know whether the flat had structure or whether the grid had made it.

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. 23 The finer sweep, at the resolution where the flat first became visible.

What made the band a design rather than an observation is the arithmetic: once the crossing is computable, the flat has a predicted location and a predicted reason, and a band can be grown around a computed rise rather than searched for. Both bands here were located that way, and neither cost a search.

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. 24 The six computed crossings, of which two carry a band and four do not yet.

What is left

Four rungs carry a handover and have no band grown around them. The 8/13 golden rung and the 7/11 Lucas rung are narrow — thirty-six and fifty-two sampled rises — so their bands would be narrow too, and whether the divergence flattens enough inside them to give a usable band is a question of arithmetic rather than of taste. The 3/5 golden and 3/4 Lucas rungs are wide and their bands should be comfortable.

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. 25 The rungs with handovers, four of which have no band around them yet.

The more interesting extension is a band at a transition rather than at a handover. A transition is where the counted pair changes, so no band can hold the pair across it — but two rises either side share a great deal else, and the pair changing is precisely the quantity a great many readings here are stated over. A matched comparison across a transition is a different design from this one and it would test different claims.

The spiral counts, band by band, in one head. The same flower gives 13/21, 21/34, 34/55, 55/89 at different radii. A caption saying "34 and 55 spirals" is a statement about one annulus.
Fig. 26 Transitions as a disc shows them, where the same question is about an annulus rather than about a rise.

Where this goes next

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. 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. The ordering was not the actor Cut an organ out of every rise of a band where the counted pair and the settled divergence are held and the two contact steps change places, and the family left standing does not change. Fifty-five wrecked cuts on two branches, and the ordering reverses underneath every one of them. Which family survives, along the 5/8 rung. The family a wrecked stem keeps, at every offset that wrecks and every rise on one rung. A counter shown any of these stems returns 5 and 8 spirals, at all twelve rises, so nothing in the pair distinguishes the columns. The cells say otherwise: at an offset of 4 the stem keeps the 5-family at the coarse end of the rung and the other one at the fine end. The offsets that wreck at all grow from 1 to 5 as the rise falls, because the front deepens, and the extra offsets are the ones that keep the larger number. So the rule stated over the offset alone is a rule with the rise left out of it. One rung, two answers The offset accounts for twenty-five wrecked stems of thirty and is refuted by a single pair of runs that differ in nothing but the rise. Sweep one rung at a thousandth and the refutation stops being an anomaly: the same offset on the same lattice keeps one family at the coarse end and the other at the fine one.

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.

ArtefactCounting blindControlDiscretisationDivergence angleHandoverLatticeMatched designMeasurementNearest neighbourParastichy pairResolutionRiseRung