Phyllotaxis
the spirals are counted, not admired
Stems and cones

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.
12 min read 15 figures Counted, not admiredThe angle is an output

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

A stem grown at a fixed rise settles on one divergence angle, and the counter is never shown that angle: it reads the arrangement’s spirals from the points and returns a pair. Sweep the rise from coarse to fine and the pair changes at computable places, which is the ladder this collection has been walking for several rounds.

What the ladder has always been drawn as is a staircase of pairs. This essay draws the other quantity — the angle itself — down the whole of both branches, and it does not do what a staircase does.

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. 1 Every rise on the golden branch against the divergence its stem settles on, each rung in its own stroke, with the limit angle marked.

What a slide would look like

The convenient picture is a slide. The golden angle is 137.5078°, the ladder is the sequence of Fibonacci pairs, and a stem at a coarse rise sits some way off that angle and creeps towards it as the rise falls. On that picture the divergence is a monotone function of the rise: every value is taken once, the approach is from one side, and asking which rise settles on 137.3° has exactly one answer.

Nothing in the rule promises that. The rule places each organ where a sum over its neighbours is least, and what a stem settles on is an output of that placement rather than a parameter of it. A monotone approach would be a fact about the output, and facts about outputs are measured.

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. 2 The ladder as it is usually drawn: the counted pair against the rise, a staircase whose steps are the transitions.

The measurement

The ladder sweep already has the number in it. Each branch is swept from a rise of 0.0700 down to 0.0040 at a constant ratio of one per cent — a geometric step, because the ladder’s own transitions are geometric and a constant absolute step would resolve the coarse rungs finely and the fine ones hardly at all. At each rise a stem is grown, a counter is run over its points, and the settled divergence is read from the last stretch of the run.

That gives 288 rises a branch and, on the golden one, four rungs: 2/3 from 0.0700 to 0.04702, 3/5 from 0.04655 to 0.01809, 5/8 from 0.01791 to 0.00689 and 8/13 from 0.00682 to 0.00482. The divergences at the ends of those four are 137.930° and 140.703°, then 140.703° and 136.641°, then 136.641° and 137.902°, then 137.902° and 137.785°.

Read that list again as a path rather than as four intervals. It goes up, then down, then up, then down.

Two things about that list are worth pinning before anything is built on it. The first is that the rung bounds are measurements and not choices: a rung is a maximal run of consecutive rises returning one pair, so its ends are the rises at which the counter’s answer changes, found by the counter and not by a person deciding where a rung should be. The second is that the divergence at a rung’s end and the divergence at the next rung’s start are not the same number — 0.04702 gives 140.703° and 0.04655 gives 140.703° as well, but 0.01791 gives 136.641° where 0.01809 gives 136.641°. The two agree at the transitions to the resolution the grid allows, which is what says the curve is continuous through them.

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. 3 The four rungs of the golden branch on one axis, with their bounds measured rather than asserted.

Three turns in four rungs

The divergence climbs across the 2/3 rung, falls across the 3/5, climbs across the 5/8 and falls across the 8/13. Three changes of direction in four rungs, and the turns happen at the transitions rather than inside the rungs — each rung is traversed in one direction and the direction reverses when the counted pair does.

That is a zig-zag about the limit angle, and it is not a shape anybody drew here before. Every previous figure of this quantity has been a figure of one rung — the slide across the 5/8 that a sweep of a single rung measured, or the flat middle a band is cut out of. Inside a rung there is nothing to see but a slide. The turn is between rungs and needs the whole ladder in one picture.

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. 4 The same curve without the annotation, so that the turns can be counted rather than read off a label.

The excursions, which do not simply shrink

The obvious reading of a zig-zag about a limit is a spiral closing in: each excursion smaller than the last, the pattern converging on the golden angle as the rise falls. Measured, the widest excursion from 137.5078° on each rung, coarse rung first, is 3.195°, 3.352°, 0.961° and 0.422°.

The second is larger than the first. The 3/5 rung swings further from the limit than the 2/3 rung does, and only after that does the collapse begin — and when it begins it is abrupt, a factor of three and a half between the second rung and the third and another factor of two between the third and the fourth.

A phrase like “converging on the golden angle” would have hidden that. It is true of the last three rungs and false of the first two, and the one number in the set that does not fit is the one worth printing.

There is a second reason not to summarise the four numbers as a trend. An excursion is a maximum over a rung, and the rungs are not the same width: the 2/3 rung carries 41 rises, the 3/5 carries 96, the 5/8 carries 97 and the 8/13 carries 36. A wider rung has more chances to reach a large value, so a sequence of maxima taken over runs of unequal length is not a sequence of comparable quantities. Here that argues in the direction of the finding rather than against it — the two rungs whose excursions collapse are one wide and one narrow — but a comparison that is only valid in one direction should say which.

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. 5 The slide across a single rung, which is all any previous figure of this quantity has shown.

The Lucas branch does something else

The same sweep on the Lucas branch gives four rungs — 1/3, 3/4, 4/7 and 7/11 — and a different shape. The divergences run 104.766° to 102.617°, then 102.422° to 101.777°, then 101.777° to 99.141°, then 99.141° to 99.402°. Down, down, down, and then a single small climb on the finest rung.

One turn in four rungs, against three. The excursions from the Lucas limit of 99.5016° are 5.264°, 2.920°, 2.295° and 0.427°, and those do shrink monotonically. So the Lucas branch is the picture everyone expects and the golden branch is not.

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. 6 The Lucas branch: three rungs traversed in one direction, one turn at the end, and excursions that do shrink.

Why the difference matters more than it looks

A curve that turns takes some of its values twice. A curve that slides takes each value once. That is the whole of it, and it is the reason this measurement is worth a figure rather than a footnote.

On the golden branch, 137.266° is the settled divergence somewhere on the 3/5 rung and again somewhere on the 5/8. 137.844° is reached on the 5/8 and again on the 8/13. 139.297° is reached on the 2/3 and again on the 3/5. Those are pairs of rises with different counted pairs and the same settled angle, and they exist because the curve came back.

On the Lucas branch only one such pair exists, between the 4/7 and the 7/11 rungs, and it exists because of the one turn at the fine end. Every other pair of Lucas rungs has divergence ranges that do not overlap, so no search however fine will produce a match between them.

The asymmetry is worth stating as a negative rather than as a shortfall. It is not that the Lucas branch was searched less carefully; it is that the search has nothing to find there, and a report saying “one match on the Lucas branch” is reporting a property of that branch’s curve. A file that had asserted a floor of two matches per branch would have turned that property into an error, and the measurement would have been thrown away as a bug in the search.

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. 7 Every pair of Lucas rungs, and whether their divergence ranges overlap at all. Five of the six do not.

What that buys, stated now and used later

Sweeping the rise along a rung moves three things at once: 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 cannot be attributed to any one of them, which is exactly the position a sweep of one rung was left in.

A band holds the first two and moves the third. What the zig-zag makes available is the design that holds the second and moves the first: two rises, one angle, two different counted pairs. That is the last of the three quantities and nothing here could separate it before.

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. 8 The design that already existed: a band holding the pair and the angle while the step ordering reverses.

The turn is not an artefact of the sweep

Two cheap ways for a turn to be manufactured were checked before anything was built on it.

The first is the sweep’s resolution. At one per cent in the rise the narrowest rung here carries thirty-six rises and the widest a hundred, so a turn is not two points joined by a line; it is a direction held across dozens of consecutive rises and reversed at a transition.

The second is the azimuth grid. The divergence is read from a stem placed on a grid of 1,536 candidate azimuths, which quantises it at 0.234°. The excursions being compared are 0.422° to 3.352°, so the smallest of them is under two grid steps and would be worth doubting on its own. It is not on its own: the sign of the excursion is what the turn is about, and the sign is stable across a whole rung.

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. 9 The interior of a rung, measured on the same sweep, as a check that the resolution is doing work rather than joining dots.

What a counter would see

Nothing. This is a fact about the angle and a counter is never shown the angle; it reads the arrangement’s spirals from the points and returns a pair. Two rises that share a divergence return different pairs, so a counter separates them perfectly and says nothing about the quantity they share.

That cuts both ways, and the second way is the useful one. A real specimen is counted, not protractored — the divergence is the harder measurement and the count is the easy one — so an observation that a plant sits at 137.3° is an observation nobody makes, while an observation that it carries 5 and 8 is made routinely. A curve that takes the same angle at two counted pairs is a curve on which the count carries information the angle does not.

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. 10 The two branches read from their angles rather than from their counts, which is the reading a specimen cannot supply.

Where the turns sit

The direction reverses at a transition, which is where the counted pair changes. That is worth stating separately because it is the one place where the two quantities — the pair and the angle — are locked together.

Inside a rung the two are independent: the pair is constant and the angle slides. At a transition both change at once, and the angle’s change is a change of direction rather than a jump. The curve is continuous through the transition on every one of the six transitions swept here; what is discontinuous is its slope.

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 A transition resolved finely: the angle passes through it without a step, and turns.

Two routes to the same pair

A matched pair can be found two ways and both were run. The first is the sweep: take every rise on both rungs, look for two whose settled divergences agree. The second is arithmetic on the sweep’s own output: take the interval of divergences each rung covers, intersect the two intervals, and if the intersection is non-empty search inside each rung for the rise nearest its midpoint.

The second is what the file does, because it separates the question is there a match from the question where is it. A pair of rungs whose intervals do not overlap cannot produce a match at any resolution, and saying so costs two comparisons rather than a sweep. Of the six pairs of golden rungs, four overlap and four match; of the six Lucas pairs, one overlaps and one matches. So the overlap test and the search agree everywhere, which is the check that the search is finding what the arithmetic says is there and not something else.

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. 12 Every pair of golden rungs, with the two that cannot match marked as such rather than left blank.

What is not answered

Why the golden branch turns three times and the Lucas branch once is not answered here. Both branches are the same rule at the same rises; the difference is which lattice the stem was started into, and it produces two qualitatively different curves.

Nor is the size of the excursions accounted for. Three of them shrink by factors of three and a half and two; the first two do not shrink at all. A curve converging on a limit at a geometric rate would give a constant ratio, and this one gives 0.94, 3.49 and 2.28 — so whatever sets the excursion is not one parameter with one exponent in it.

And the number of rungs is a property of where the sweep stopped rather than of the ladder. It runs from 0.0700 to 0.0040 because below about 0.004 a stem does not settle onto a lattice at any run length and above 0.0700 the coarse rung has no interior worth sampling. Four rungs a branch is what that range holds. Whether the alternation continues below the wall is not a question this sweep can be extended to answer, because there is nothing to measure down there: the runs that reach a lattice at all settle on 106.1° and 151.3°, which is neither branch.

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. 13 The coarse end of the ladder, where the first two excursions sit and where the shrinking has not started.

What is answered

That the divergence curve returns to its own values, on one branch and barely on the other, and that the return is a property of the rule rather than of the sweep.

Everything the next few essays do rests on that one sentence. A matched pair is two rises the curve reaches the same angle at; without the turn there is no pair, and the third of the three quantities that move along a rung stays where it has been for four rounds — moving with the others, and untestable.

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. 14 The pairs the golden branch’s turns produce, with the rises they sit at and how closely the two angles agree.

The one line

Down the golden branch the settled divergence climbs, falls, climbs and falls again, turning three times in four rungs and swinging 3.195°, 3.352°, 0.961° and 0.422° from the limit angle. Down the Lucas branch it turns once and shrinks steadily. The golden branch therefore offers four rises where two different counted pairs settle on the same angle, and the Lucas branch offers one.

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. 15 The census those matched rises will be cut at, as it stands before any of them are used.

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 angles against the positions, rise by rise. five 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 0.01 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. At 0.008 the counter says 5/8 and the angles agree on 2 of 5, refusing 3. The two instruments share no code path: one is given a list of angles, the other a list of coordinates. The angles name the branch Seed the same rule at the Lucas angle and the readout returns 4 and 7, then 7 and 11 — the pairs the position counter finds, and not Fibonacci numbers. So a list of divergence angles carries not only how many spirals there are but which family of ladders the plant is on. 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. 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.

What links here

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

Reads more easily once this is understood

Essays that name this one as worth reading first.

Shares its objects with

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

  • Four crossings nobody visited — both name discretisation, divergence angle, geometric ladder, ladder, matched design, measurement, parastichy pair, resolution, rise, rung
  • A band that moves nothing — both name discretisation, divergence angle, matched design, measurement, parastichy pair, resolution, rise, rung
  • The response with a hole in it — both name counting blind, discretisation, divergence angle, ladder, measurement, parastichy pair, rise, rung
  • The rung was not the instrument — both name counting blind, discretisation, divergence angle, ladder, measurement, parastichy pair, rise, rung
  • A front with no middle — both name discretisation, divergence angle, ladder, measurement, parastichy pair, rise, rung
  • Seven rises and two seeds — both name ladder, lucas numbers, matched design, measurement, parastichy pair, rise, rung

Named objects

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

Counting blindConvergenceDiscretisationDivergence angleGeometric ladderGolden angleLadderLucas numbersMatched designMeasurementParastichy pairResolutionRiseRung