Phyllotaxis
the spirals are counted, not admired
Stems and cones

Two rungs, one angle

Five pairs of rises settle on the same divergence while a counter returns different pairs at them, and four of the five agree to 0.0000° — the same value of a quantity read on a grid of 1,536 azimuths. The rises differ by factors of 1.48 to 5.09.
12 min read 13 figures State the claim, state the testCounted, not admired

Worth reading first: The angle the ladder returns to · Where a handover sits · Counting the spirals.

Three quantities move together when the rise is swept along a rung: the pair a counter returns, the divergence the rule settles on, and which of the two contact steps is the shorter. A result that changes along a rung is a result that cannot be attributed to any one of them, and that is exactly the position a sweep of one rung was left in.

The way out is a design that holds two and moves one. One exists: a band holds the pair and the divergence and moves the ordering. This essay builds the second — hold the divergence, move the pair — and the whole of it is a search.

Which rungs of the golden branch share a divergence. One row per pair of rungs. A pair whose divergence ranges overlap has a rise on each rung where the rule settles on the same angle; a pair whose ranges do not overlap has none, whatever the search. On this branch four of six pairs match, and three of those match to 0.0000° — the same value of a quantity read on a grid of 1,536 azimuths. The rises differ by factors of 1.48 to 5.09, so the design holds one angle while changing everything the rise controls.
Fig. 1 Every pair of rungs on the golden branch, whether their divergence ranges overlap, and where the search puts the match.

What the design needs

Two rises. At the first, a stem settles on some divergence and a counter returns one pair; at the second, it settles on the same divergence and the counter returns a different pair. Then any experiment run at both is an experiment with the counted pair varied and the angle held.

That is only possible where the divergence curve returns to a value it has already taken, which it does on the golden branch and barely does on the Lucas. So the search is bounded before it begins: at most one match per pair of rungs whose divergence ranges overlap, and none at all for a pair whose ranges do not.

It is worth saying what the design is not. It is not a claim that two stems with the same divergence are the same stem — they plainly are not, and the whole point is that a counter says so. It is not a claim that the divergence is unimportant. And it is not a pairing chosen to make some later result come out: the target angle is the midpoint of whatever interval two rungs happen to share, computed before anything is cut, so the pairs are fixed by the ladder rather than by what is convenient at the far end.

The one thing a matched pair is is an experiment with a controlled variable. Anything measured at both rises of a pair is measured with the angle held; if the measurement differs, the angle did not cause the difference. That is a weaker statement than an account of what did, and it is the statement this collection has been unable to make about the divergence for four rounds.

The settled divergence down the golden branch. Every rise from 0.07 down to 0.00482, 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 three 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 3.195°, 3.352°, 0.961°, 0.422°.
Fig. 2 The curve the search is run over, with the horizontal lines that meet it twice being exactly the angles a match can sit at.

The overlap test comes first

Each rung covers an interval of divergences — the values at its two ends, since the curve is monotone inside a rung. Two rungs can produce a match only if their intervals intersect, and that is two comparisons rather than a sweep.

Of the six pairs of golden rungs, four overlap. The two that do not are 2/3 with 5/8 and 2/3 with 8/13: the 2/3 rung covers 137.930° to 140.703° and the 5/8 covers 136.641° to 137.902°, which miss each other by 0.028°. That is a near miss and it is still a miss — 0.028° is an eighth of the azimuth grid’s own step, so no resolution available here would close it.

Of the six pairs of Lucas rungs, one overlaps. Five do not, and they do not miss narrowly: the 1/3 rung ends at 102.617° and the 4/7 rung starts at 101.777°, which is nearly a degree of empty space.

Which rungs of the Lucas branch share a divergence. One row per pair of rungs. A pair whose divergence ranges overlap has a rise on each rung where the rule settles on the same angle; a pair whose ranges do not overlap has none, whatever the search. On this branch one of six pairs match, and one of those match to 0.0000° — the same value of a quantity read on a grid of 1,536 azimuths. The rises differ by factors of 1.68 to 1.68, so the design holds one angle while changing everything the rise controls.
Fig. 3 The same test on the Lucas branch, where five of the six pairs have nothing to search.

The search inside a rung

Where two intervals do overlap, the target is the midpoint of the intersection and each rung is searched for the rise nearest it. The ladder sweep’s own rows give a starting point at one per cent in the rise; the search then steps at two parts in a thousand, twelve steps either side, and keeps the rise whose settled divergence is closest to the target.

Two guards are on that loop and both matter. A candidate rise outside its rung’s measured bounds is rejected, and so is one whose counter returns a different pair — because a search that wandered off the end of a rung would come back with a rise on the next rung, and the whole design is that the two rises are on different rungs by construction rather than by luck.

The refinement step is a ratio rather than an absolute amount, for the same reason the ladder itself is swept at one. The four golden rungs span the rise from 0.0700 down to 0.00482, a factor of fifteen, and a step of two parts in a thousand is two parts in a thousand at every one of them. An absolute step fine enough for the 8/13 rung would take thousands of stems to cross the 3/5 rung, and one coarse enough to cross the 3/5 rung would step over the 8/13 rung’s whole band.

Each candidate rise costs one grown stem, and a stem at these rises is a few hundred organs placed one at a time against a sum over their neighbours. The twenty-five candidates a refinement takes are therefore the expensive part of the search and the overlap test is free, which is the second reason to do the arithmetic first.

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.
Fig. 4 The rung bounds the search is confined to, measured by the counter rather than asserted.

The pairs

Five matches, four on the golden branch and one on the Lucas.

The 2/3 and 3/5 rungs match at 139.297°, at rises of 0.06151 and 0.03065. The 3/5 and 5/8 rungs match at 137.266°, at 0.02251 and 0.01074. The 3/5 and 8/13 rungs match at 137.859° and 137.844°, at 0.02547 and 0.00500. The 5/8 and 8/13 rungs match at 137.844°, at 0.00739 and 0.00500. And on the Lucas branch the 4/7 and 7/11 rungs match at 99.273°, at 0.01046 and 0.00624.

Two of those share a rise. The 8/13 stem at 0.00500 appears in both the 3/5-and-8/13 pair and the 5/8-and-8/13 pair, because 137.844° lies inside the interval the 3/5 rung covers as well as inside the 5/8’s. That is not a duplication to be tidied away: it means one stem is the far end of two different comparisons, against a 3/5 stem grown at five times its rise and against a 5/8 stem grown at one and a half times it, with the angle held in both.

And one of the five sits at an angle nowhere near the golden one. 139.297° is 1.789° from 137.5078°, which is more than seven grid steps and further from the limit than any rise on the two fine rungs ever gets. A matched pair does not have to sit near the branch’s limit; it has to sit where the curve crosses itself, and on the coarse half of this ladder that is a long way out.

The angles the five sit at are worth listing together for a different reason. Two of them — 137.844° and 137.859° — are within a grid step of each other while belonging to different pairs of rungs, so the ladder crosses that neighbourhood three times: on the 3/5 rung, on the 5/8, and on the 8/13. Three rises, three counted pairs, one angle to within 0.0156°. Nothing in the design needs a triple, and having one means the two pairs that use it are not independent of each other.

The same angle, two rungs, two answers. Each block is one matched pair: two rises whose stems settle on the same divergence and whose counters return different pairs. Under each is the family every wrecked cut leaves standing. On three of the four pairs both rises wreck at some offset, and on every one of those the two stems keep different families — so the divergence, which is held, is not what decides the survivor. The two stems keep exactly the counted numbers their two pairs share, including the pair that shares none and keeps none.
Fig. 5 The five pairs as blocks, each with the two rises and the angle they share, before anything is done to them.

How close is close

Four of the five agree to 0.0000°. That is not a rounding of something small: it is the same value of a quantity read on a grid of 1,536 candidate azimuths, which is to say the two stems settled into arrangements whose measured divergence is one number. The fifth agrees to 0.0156°, which is a fifteenth of one grid step.

It is worth being explicit about what that does and does not claim. The divergence is a mean over the last stretch of a run, so it is not itself quantised at 0.234°; two stems can agree to better than a grid step. What the grid does set is the floor below which “the same angle” stops meaning anything, and 0.0156° is comfortably under it. The bound the design is held to is a twentieth of a degree — the same figure a band is held to, kept rather than re-chosen so that the two designs are held to one standard.

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. 6 The other design at the same tolerance: a band holding its angle to a twentieth of a degree while the ordering reverses.

What is not held

Everything else. The two rises of a matched pair differ by factors of 1.48, 1.68, 2.01, 2.10 and 5.09, and the rise is the parameter that sets the whole geometry: how far apart the organs are up the cylinder, how many of them are within reach of the placement rule, how long the two contact steps are, which pair a counter returns.

That is the design working rather than a defect in it. A matched pair is not two similar stems; it is two very different stems that happen to have settled on one angle. The 3/5-and-8/13 pair is the extreme — 0.02547 against 0.00500, a factor of five — and it is the most useful of the five for exactly that reason.

Some numbers to put on how different. At a rise of 0.02547 the placement rule’s neighbourhood holds a few dozen organs and the front — the stretch of the stem a removal is felt across — runs to five. At 0.00500 the neighbourhood holds several times as many and the front runs to thirteen. The two contact steps are different lengths, sit at different angles round the cylinder, and belong to different families. Every geometric quantity this collection measures differs between the two ends of that pair except one.

This is also why the design cannot be run the other way round. Holding the pair and moving the divergence is what a rung already does, and it moves the ordering as well; holding the divergence and moving the pair moves the rise, and the rise moves everything. There is no design here in which exactly one thing changes. What there are are two designs each holding a different two, and between them they pin all three.

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. 7 How far apart the matched rises are on the ladder as a whole, which is what “the same angle at two rungs” costs in rise.

A counter separates them perfectly

Point a counter at the two stems of any matched pair and it returns 3/5 at one and 5/8 at the other, or 3/5 and 8/13, or 5/8 and 8/13, or 4/7 and 7/11. It never hesitates, because it is never shown the angle and the two arrangements are genuinely different arrangements.

So a matched pair is a case where the two most obvious measurements of a stem disagree about whether the two stems are alike. Counted, they are as different as two stems on this ladder get. Protractored, they are indistinguishable. Anything that follows the counts at a matched pair is following the pair; anything that follows the angle is following nothing, because the angle does not vary.

The angles against the positions, rise by rise. three rises, five seeded stems each. A filled mark is a run whose angle readout returned the pair the position counter finds in the same stem; an open mark is a refusal. At 0.032 the counter says 3/5 and the angles agree on 0 of 5, refusing 5. At 0.013 the counter says 5/8 and the angles agree on 5 of 5. At 0.005 the counter says 8/13 and the angles agree on 5 of 5. The two instruments share no code path: one is given a list of angles, the other a list of coordinates.
Fig. 8 The counter’s answer at three rises, which is the reading that separates a matched pair without reference to any angle.

The pair that produces no experiment

The 2/3-and-3/5 match is a match and not an experiment. Cut the 2/3 stem at every offset and two cuts wreck it, both keeping nothing rigid at all; cut the 3/5 stem at 0.03065 and none wreck. So the coarsest match has no surviving family at either rise, and comparing them would be comparing two absences.

That is the coarse end of the ladder behaving as it has been measured to behave — a front of five organs is its own two edges with no middle, and a single removal has nothing to sever. It is kept in the table rather than dropped, because a design that quietly drops its one unusable cell is a design reporting four for four.

Which rises are a lattice, from 0.04 to 0.13. How much the divergence wanders over the last sixty organs, at each rise up the coarse ladder. 13 of the 19 settle, scattering between 0.0000 and 0.3356 degrees. six do not: from 0.09 to 0.115 the divergence sticks on exactly 135.0000 degrees, which is three eighths of a turn, and wobbles about it by 0.79 to 1.60 degrees. A counter shown either kind returns the same pair, so the counts cannot tell them apart. The line is the threshold a rise has to pass before a cut is made on it, and it sits in the gap rather than among the measurements.
Fig. 9 The coarse end, where a single removal cannot wreck a stem and a matched pair therefore produces nothing to compare.

Why the Lucas branch supplies one

Not because it was searched less carefully. The same sweep, the same overlap test, the same refinement. The Lucas divergence curve descends across three successive rungs — 104.766° to 102.617°, 102.422° to 101.777°, 101.777° to 99.141° — and only turns at the finest rung, climbing back to 99.402°. A curve that descends monotonically takes each value once, so the only overlapping pair it can produce is the one straddling its single turn.

Reporting that as a shortfall would be reporting a property of the branch as a failure of the search. It is worth saying plainly because the temptation runs the other way: a table with four rows on one branch and one on the other looks like an uneven effort, and it is an even effort on an uneven object.

six limit divergences, all of them 137.5078 over a whole number. The golden angle is the k = 1 member of a family. Real bijugate plants — teasel, Cephalaria — are reported near 68.75°, which is exactly half of it, and the pairs they are counted at are the Fibonacci pairs doubled.
Fig. 10 The limit angles the two branches converge on, which is what the two curves are converging towards from different sides.

What could still be wrong with a matched pair

Two things, and both were checked.

The first is that the two stems might not have settled at all. A run whose divergence is still wandering has a mean and no settled value, and two wandering means can agree by accident. Every rise here is inside a rung the ladder sweep found, and a rung is a run of at least four consecutive rises returning one pair — which a wandering stem does not produce.

The second is that the match might be an artefact of reading the angle over a particular window. The divergence is a mean over the tail of each run, and two different tails could agree where two different runs do not. That is the same concern the finer grid was run for on a band, and the answer here is the same: the agreement survives, because what is being compared is a settled value rather than a transient.

The 3/5 rung at a tenth of the ladder's step. Every rise of one rung, sampled ten times as finely as the ladder that found the locked band. All 23 are counted at 3 and 5 spirals and all 23 settle: the largest wander is 0.221 degrees, against the 0.5 degree threshold and against the 0.79 to 1.60 degrees the band on the coarse rung wobbles by. The divergence slides smoothly from 139.0625 to 136.7344 degrees with no rise stuck on a rational and none stuck on anything else. Whatever the band is, it is not something a coarser sampling was hiding here.
Fig. 11 The angle read finely near a transition, which is where a settled value and a wandering one would come apart.

What this is for

One sentence. There are three quantities that move along a rung and a design that separates each of them from the others is worth having; the ordering has had one for a round, and this is the divergence’s.

What it will be used for is the cut. Two stems, one angle, different pairs: if the family a removal leaves standing follows the pair, the angle is not what decides it, and the last of the three is closed. That measurement is the next essay and it takes four of these five pairs.

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. 12 The census the matched rises will be cut into, as it stands before any matched pair is used.

The one line

Five pairs of rises settle on the same divergence at different counted pairs. Four agree to 0.0000° and the fifth to 0.0156°; the rises differ by factors of 1.48 to 5.09; four of the five wreck at some offset and are therefore usable, and the fifth sits at the coarse end where nothing wrecks.

Shared counted numbers against shared survivors. One row per matched pair, over both branches. The third column is the counted numbers the two rungs have in common and the fourth is the families both stems leave standing; on every row the two are the same set. The row whose rungs share no counted number is the one whose stems share no survivor, which is what makes this a claim about an intersection rather than a restatement that a survivor is usually a contact family. five rows, and the empty case is one of them.
Fig. 13 Every matched pair on both branches, listed with the counted pairs it holds apart.

Where this goes next

The settled divergence down the golden branch. Every rise from 0.07 down to 0.00482, 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 three 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 3.195°, 3.352°, 0.961°, 0.422°. The angle the ladder returns to Down the golden branch the settled divergence climbs across one rung and falls across the next, turning three times in four rungs. That is why the same angle is reached at two different rises — and why the Lucas branch, which turns once, almost never offers the same thing. 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. 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. The same angle, two rungs, two answers. Each block is one matched pair: two rises whose stems settle on the same divergence and whose counters return different pairs. Under each is the family every wrecked cut leaves standing. On three of the four pairs both rises wreck at some offset, and on every one of those the two stems keep different families — so the divergence, which is held, is not what decides the survivor. The two stems keep exactly the counted numbers their two pairs share, including the pair that shares none and keeps none. The angle is not the actor Cut an organ out of two stems that settled on the same divergence and return different counted pairs, and the family left standing is different at every one of the four pairs where both stems wreck. The angle is held to a hundredth of a degree underneath.

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.

  • A band that moves nothing — both name control, discretisation, divergence angle, matched design, measurement, negative result, parastichy pair, resolution, rise, rung
  • Four crossings nobody visited — both name control, discretisation, divergence angle, ladder, matched design, measurement, parastichy pair, resolution, rise, rung
  • The last of three quantities — both name control, divergence angle, identifiability, matched design, negative result, parastichy pair, rise, rung, underdetermination
  • The response with a hole in it — both name counting blind, discretisation, divergence angle, identifiability, ladder, measurement, parastichy pair, rise, rung
  • The rung was not the instrument — both name counting blind, discretisation, divergence angle, identifiability, ladder, measurement, parastichy pair, rise, rung
  • What a count cannot decide — both name counting blind, control, divergence angle, measurement, negative result, parastichy pair, rise, rung, underdetermination

Named objects

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

Counting blindControlDiscretisationDivergence angleIdentifiabilityLadderMatched designMeasurementNegative resultParastichy pairResolutionRiseRungUnderdetermination