Phyllotaxis
the spirals are counted, not admired
Where the angle comes from

Ten sequences, two of them the ladder's

Every parastichy pair the settling table produces is two consecutive terms of a sequence in which each term is the sum of the two before it. Ten such sequences account for all fifteen destinations, and the two the collection is built on are neither the largest nor the smallest.
12 min read 19 figures Counted, not admiredThe angle is an output

Worth reading first: How long a stem takes to settle · Counting the spirals · The angle is an output.

Counting every settled run in the settling table returns seventeen distinct parastichy pairs across fifteen destinations. Only five of the pairs are ones this collection has a name for.

The other twelve are not arbitrary. Every pair the counter returns — all seventeen — is two consecutive terms of a sequence in which each term is the sum of the two before it, which is the ladder’s own rule started from different terms.

The 10 sequences the settling table's destinations belong to. Every pair the counter returns is two consecutive terms of a sequence in which each term is the sum of the two before it, which is the ladder's own rule. two of these are the sequences this collection is built on. The rest are not, and they are not rare: the 1, 4, 5, 9, 14 sequence and the 2, 5, 7, 12, 19 sequence each supply several destinations, at every falloff exponent the table is grown at. The right-hand column is the divergences that land on each.
Fig. 1 The ten sequences the table’s destinations belong to, with the pairs seen on each and the divergences they were seen at.

Which is almost no constraint

Worth getting out of the way first. Any two whole numbers are consecutive terms of exactly one such sequence: walk backwards, subtracting the smaller from the larger, until the result stops being positive and smaller than what follows it. So “on an additive sequence” is true of every pair the counter could possibly return.

What carries information is which sequence, and how many destinations each one supplies. A table whose seventeen pairs sat on seventeen different sequences would be a table with no structure in it at all.

Every family but two is the sum of two others. Four heads and the contact families each actually has. An arc arrives at every family that is the sum of two smaller ones; the two with no arc arriving are the generators, and on every head they are the two smallest. whorled, 144°: 2, 3, 5 — golden, 137.508°: 8, 13, 21, 34, 55, 89 — Lucas, 99.502°: 11, 18, 29, 47, 76 — rational, 137.5°: 8, 13, 21, 34, 55, 89 — 137.0°: 8, 13, 21, 29, 50, 71, 92, 113. So once a pair is counted the rest is arithmetic, and a third counted family is a prediction rather than a second measurement.
Fig. 2 The additive rule the ladder is built on, which every sequence in this essay obeys and which by itself distinguishes nothing.

Ten sequences, not seventeen

They sort into ten. Four of the ten supply more than one destination and six supply one each.

The two largest are 1, 4, 5, 9, 14 — which supplies 77.7°, 79.2° and 80.5°, with the pairs 4/5, 4/9, 5/9 and 9/14 — and 2, 5, 7, 12, 19, which supplies 148.1° and 151.0° with the pairs 2/5, 5/7, 7/12 and 12/19. Both appear at every one of the four falloff exponents the table is grown at.

Every destination in the settling table, counted. One row per destination, placed across at the divergence the stems settled on and labelled with the parastichy pair a counter returns at the top of the run. 117 settled runs land on 15 destinations when two values within half a degree are called one. five of them sit on the golden or the Lucas sequence and ten do not, and the ones that do not are reached at every falloff exponent including the shallowest. Nothing here refuses to count: the settling test does not admit arrangements this collection would decline to call lattices.
Fig. 3 Every destination with its pair. The sequences are what these pairs sort into when they are grouped rather than listed.

The ladder’s two are third and fourth

The golden 1, 2, 3, 5, 8, 13 supplies three destinations — 136.8°, 137.8° and 139.1° — with the pairs 3/5, 5/8 and 8/13. The Lucas 1, 3, 4, 7, 11 supplies two, at 99.5° and 101.6°, with 3/4, 4/7 and 7/11.

So of the four sequences the table reaches more than once, the collection’s two are not the top pair. Measured by runs rather than destinations the picture is the same: 151.0° alone is reached by twenty-three runs, which is more than either ladder sequence’s total.

The golden sequence’s three destinations take twenty-seven runs between them and the Lucas sequence’s two take twenty-five, so the collection’s two account for fifty-two of the hundred and seventeen. That is a large minority and it is a minority, and the sentence the thread has been carrying — that the rule settles onto the ladder — was written when nobody had counted the other half.

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. 4 The golden ladder itself, which is one of ten sequences here and the one this subject is named for.

The six with one destination each

1, 5, 6, 11, 17 at 65.2° with 6/11. 3, 7, 10, 17, 27 at 106.1° with 10/17. 2, 7, 9, 16, 25 at 157.7° with 7/9 and 9/16. 4, 9, 13, 22 at 79.2° with 4/9. 1, 8, 9, 17, 26 at 42.3° with 8/9. 1, 7, 8, 15, 23 at 47.9° with 8/15.

The last two are the ones only a steep falloff reaches. The other four are reached at three or four exponents each, including the shallowest.

Every destination in the settling table, counted. One row per destination, placed across at the divergence the stems settled on and labelled with the parastichy pair a counter returns at the top of the run. 117 settled runs land on 15 destinations when two values within half a degree are called one. five of them sit on the golden or the Lucas sequence and ten do not, and the ones that do not are reached at every falloff exponent including the shallowest. Nothing here refuses to count: the settling test does not admit arrangements this collection would decline to call lattices.
Fig. 5 The ten destinations off the ladder. Six of them are the only representative of their own sequence in the whole table.

Two sequences at one destination

79.2° is reached by nineteen runs and the counter returns 4/5, 4/9 and 5/9 among them. 4/5 and 5/9 sit on 1, 4, 5, 9, 14; 4/9 sits on 4, 9, 13, 22, because 4 and 9 are not adjacent in the first sequence.

That is a counter reading a slightly different patch and skipping a term, and it is worth taking seriously rather than tidying away: a pair that skips a term of its own sequence is what a count taken at the wrong radius produces, and the counting radius is known to move a count.

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. 6 How a count changes with where in the pattern it is taken, which is the effect a skipped term would be an instance of.

Which means the ten is an upper bound

If 4, 9, 13 is a reading artefact rather than a sequence, the table has nine. Nothing here settles that, and it is checkable on runs already grown: count the same runs over a hundred organs and over four hundred, and see whether 4/9 survives.

The reverse is also possible. Two of the ten sequences are represented by a single destination reached by a handful of runs, and a denser sweep of starting angles would very likely add more. So ten is a number this sample produced and not a property of the rule, and any use of it should carry the sample with it — the same discipline a count taken at one radius needs.

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. 7 Counts against angle at one rise. Whether a pair is stable under a change of window is a question this picture is the wrong shape to answer.

What the sequences have in common

Each has a limit ratio — the value its consecutive terms converge to — and a limit angle, which is a full turn divided by that ratio squared, folded into half a turn. Those limit angles are what the destinations approximate.

The golden sequence’s limit angle is 137.5° and its destinations are 136.8°, 137.8° and 139.1°. The 2, 5, 7, 12, 19 sequence’s is near 151°, and its destinations are 148.1° and 151.0°. So a sequence is a family of nearby destinations rather than a single one, which is exactly what the golden ladder already is: the ladder is one angle across a range of rises, giving a different pair at each rung, and the three golden destinations here are three readings of that.

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. 8 The angles a rule of this kind converges towards, which is where each sequence’s limit sits.

And what makes them different

The limit ratios are different irrationals. The golden sequence converges on the golden ratio, the Lucas one converges on the same ratio from different terms, and the others converge on their own values.

That is the part a reader coming from the standard account should notice. The usual story privileges the golden ratio because of a property of its continued fraction — it is the hardest number to approximate — and several of these sequences converge on ratios with no such property, and the rule reaches them anyway.

How nearly each angle is a simple fraction of a turn. A dip means a rational approximation and a lattice with visible rows. The golden angle's floor is 0.005; the rational angle reaches exactly zero.
Fig. 9 How well each angle is approximated by a fraction, which is the property the standard account rests on.

Which is not a refutation of that account

It is a measurement it has to accommodate. The standard account says an arrangement whose divergence is poorly approximated by simple fractions packs better, and it predicts that such arrangements should be preferred. It does not say others are unreachable.

What the table shows is that a placement rule minimising a sum over neighbours reaches ten of them from nine starting angles, and lands on the golden one from about a quarter of its settled runs. Whether the golden one is preferred in any sense that matters is a question this table can be asked and has not been.

There is also a measurement on this site that already complicates the standard story from a different direction: the most irrational angle is not the most disordered one by the site’s own disorder statistic. Two independent readings, neither of which refutes the account and both of which say it is doing less work than it is usually given.

The gaps close faster than the dips narrow. For each Fibonacci fraction, the distance to the nearest other rational with a denominator of 60 or less, and the half-width of its own dip at the smallest head that resolves it. The gaps fall from 0.462° at 5/13 to 0.1925° at 21/55; the dips stay between 0.0103° and 0.0155°. The dips never touch — the closest they come is a factor of 15 — so what stops the measurement is not the dips overlapping but the background between them ceasing to be flat. The clear offsets available fall from 70 to 54.
Fig. 10 How the simple fractions crowd the circle, which is the geometry the standard account is built on.

The question the table can be asked

Whether the basins differ in size. Each destination’s basin is the set of starting angles that reach it, and the table samples nine of them per cell. Counting how many of the nine reach each destination is a measurement of basin width at that resolution.

At nine angles the answer is noisy — a binomial with a standard error of about 0.17 — and it is the same instrument the settling shares already broke on. Twenty angles halves the error at 640 runs against 288, and it is the obvious next sweep.

The share of starting angles that reach a lattice, by exponent. Each row pools one exponent's whole column — every rise, every starting angle — into one share, with a bar an error either side of it. Pooling is what makes the comparison possible: a single cell is 9 runs and carries an error of about 0.17, which is three times the whole spread between the four exponents. Pooled, the four sit at 30/72, 31/72, 29/72, 27/72 — a spread of 0.056 against an error of 0.058. Nothing in the falloff moves the basin. A five-angle version of the same sweep gave a clean monotone trend, which is what a null looks like when it is sampled too thinly.
Fig. 11 The share of starting angles reaching a lattice, which is the reading a basin-width question would refine.

What this changes about the site’s premise

Nothing about the premise and something about a habit. The premise is that spirals are counted rather than admired, and counting is exactly what produced this: the sequences were invisible while the table reported angles.

The habit is quoting the golden ladder as though it were what the rule does. It is what the rule does from the starting angles this collection has been seeding stems with, which is a different sentence and the one the table supports.

A round trip on four heads of 900 primordia: the divergence angle recovered from each. The counter is shown the points and nothing else. The worst recovery across the four is 0.012°.
Fig. 12 The angle recovered from a settled run, which is the reading that made the ladder look like the rule’s output rather than one of its outputs.

The sequences are not equally spread across the exponents

Four of the ten appear at all four falloff exponents: the golden, the Lucas, 1, 4, 5, 9, 14 and 2, 5, 7, 12, 19. Three appear at three of them. Two appear at exponents four and five only, and one appears at exponents two, four and five and not at three, which is a gap that is almost certainly the sample rather than the rule.

That last one is worth naming because it is a small warning about reading a column as absent. 157.7° is reached at exponents two, four and five, and its absence from the exponent-three column is one cell of nine runs not finding a destination that neighbouring columns find. Nothing follows from it, and if it had been the only column it was absent from somebody could have made something of it.

How many starting angles reach a lattice, by rise and by falloff. Every cell is 9 stems grown from 9 starting angles spanning 40° to 180°, at one rise and one exponent in the placement rule's falloff, for 1200 organs each. The number is how many of them settle onto a lattice. Reading down a column, the basin closes as the rise falls, which is the wall. Reading across a row, nothing much happens: the exponent-three column is the published settling table digit for digit, and the other three are the same column inside their own sampling error. The account that a steeper falloff should wall somewhere else does not survive the table it predicted.
Fig. 13 The table’s four columns, from which any claim about a sequence being absent at one exponent has to be read.

Where these sequences have names

Outside this collection they do. Sequences of this kind, and the arrangements they describe, have been catalogued in the botanical literature for a long time under several naming conventions, and 2, 5, 7, 12 and 1, 4, 5, 9 are among the ones that recur.

This essay does not use those names, for the reason the collection does not use borrowed names anywhere: a name imported without its definition is a claim that two things are the same, and checking that is a piece of work nobody here has done. The same caution applies to the pairs: 5/7 in a description of a plant and 5/7 out of this counter are two readings that would have to be shown to be one, and that is a survey rather than a footnote.

Every open question here needs under 28 specimens. The sample size at which each comparison reaches 80 per cent power at a 5 per cent false-positive rate, from the exact binomial rather than a normal approximation. The census question — do plants show consecutive Fibonacci pairs far more often than the geometry does — needs 4: 14.7% is the share of divergence angles giving a consecutive Fibonacci pair at a fine rise; 90% is what a grown history gives.
Fig. 14 What it would take to connect a computed arrangement to a described one, which is a survey rather than a citation.

What a sequence is, as an object

Worth being precise, because two things are being called one. A sequence here is a list of whole numbers with a rule. A destination is a measured divergence. A pair is what a counter returns from a patch.

The chain from one to the next is not automatic. A pair determines a sequence uniquely, by the walk backwards. A destination does not determine a pair — three of the fifteen return more than one — and a sequence certainly does not determine a destination, since four of the ten supply several. So the grouping in this essay is a map from measurements to a structure, and it is many-to-one in both directions in places.

Every destination in the settling table, counted. One row per destination, placed across at the divergence the stems settled on and labelled with the parastichy pair a counter returns at the top of the run. 117 settled runs land on 15 destinations when two values within half a degree are called one. five of them sit on the golden or the Lucas sequence and ten do not, and the ones that do not are reached at every falloff exponent including the shallowest. Nothing here refuses to count: the settling test does not admit arrangements this collection would decline to call lattices.
Fig. 15 The measurements themselves: a divergence and the pairs found at it. The sequences are a structure laid over these rather than something read off them.

What the collection’s own ladder is, restated

A rung of the golden ladder is a range of rises over which a stem grown at the golden angle shows one counted pair. The ladder is one angle and a sequence of pairs down it.

A sequence here is the same thing seen from the other end: a family of angles, each carrying its own ladder of rungs. So there are ten ladders in the table, of which the collection has built machinery for two, and every measurement in this collection about rungs, bands, handovers and cuts has been made on one of the two.

That is not a criticism of those measurements. It is a statement about their scope, and it is the sort of statement that is easy to make once and hard to remember to attach to results.

Every transition as the rise falls. The pair climbs 1/2 → 2/3 → 3/5 → 5/8 → 8/13 → 13/21. Consecutive transitions are 0.382, 0.382, 0.383, 0.382 of the previous rise — 1/φ² is 0.3820.
Fig. 16 One ladder, drawn from a coarse rise to a fine one. There are nine more of these implied by the table and none of them has been swept.

The honest limits

Nine starting angles, eight rises, four exponents, and a counter reading two hundred organs. Ten sequences is what that sample reaches; a denser sample would reach more, and a different counting window might merge or split two of them.

And the sample of starting angles is not a sample of anything in particular. They span the useful half-circle at even intervals, which is a reasonable thing to do and is not a model of how a stem chooses where to begin. A sweep that sampled starting angles the way a meristem might would give a different distribution over these ten and this table says nothing about what it would be.

Everywhere a cut of one to five organs can send a 5/8 stem. Every settled divergence reached by any arrangement of up to five organs removed from a stem at the 5/8 rung, on one axis. There are six of them and no more. two are slips of the lattice the stem was cut from: each keeps the lag-5 family intact and sits a whole number of turns of it from the next, which is the ladder marked below the axis with rungs 72.0 degrees apart. The other three keep no lag at all and sit near a fraction of a turn, marked above: 175.0 degrees near 1 of 2 turns, 190.0 degrees near 1 of 2 turns, 235.0 degrees near 2 of 3 turns. A stem that is cut either slides along the ladder it was on or leaves it for a lattice with files in it.
Fig. 17 The destinations reachable across a dense range of starting angles, which is what this table samples nine points of.

What a basin over ten sequences would look like

Worth sketching, because the picture the collection has been carrying is a basin structure over one ladder with strays. The measured picture is different in a specific way: several sequences with several destinations each, and a run’s starting angle deciding which sequence it lands on before it decides which rung.

That predicts something checkable and unchecked. Starting angles near a sequence’s own limit should land on that sequence, and starting angles between two limits should be the ones that fail to settle at all. Nine angles cannot show it and the table records which angle produced which destination, so a denser sweep would settle it in one picture.

Everywhere a cut of one to five organs can send a 5/8 stem. Every settled divergence reached by any arrangement of up to five organs removed from a stem at the 5/8 rung, on one axis. There are six of them and no more. three are slips of the lattice the stem was cut from: each keeps the lag-5 family intact and sits a whole number of turns of it from the next, which is the ladder marked below the axis with rungs 72.0 degrees apart. The other three keep no lag at all and sit near a fraction of a turn, marked above: 175.0 degrees near 1 of 2 turns, 190.0 degrees near 1 of 2 turns, 235.0 degrees near 2 of 3 turns. A stem that is cut either slides along the ladder it was on or leaves it for a lattice with files in it.
Fig. 18 Destinations against starting angle across a dense range, which is the picture a basin structure over ten sequences would be read from.

What carries forward

Ten sequences, four of them reached more than once, two of them the collection’s own and neither of those the largest. A pair that skips a term, which is either an eleventh sequence or a counting artefact and is cheap to settle. And a basin-width sweep at twenty starting angles, which would turn a list of destinations into a distribution over them.

The 10 sequences the settling table's destinations belong to. Every pair the counter returns is two consecutive terms of a sequence in which each term is the sum of the two before it, which is the ladder's own rule. two of these are the sequences this collection is built on. The rest are not, and they are not rare: the 1, 4, 5, 9, 14 sequence and the 2, 5, 7, 12, 19 sequence each supply several destinations, at every falloff exponent the table is grown at. The right-hand column is the divergences that land on each.
Fig. 19 The ten again, which is the table every one of those follow-ups would be adding rows or confidence to.

The one line

Every parastichy pair the settling table produces sits on an additive sequence, which constrains nothing; they sort into ten such sequences, which constrains a good deal.

The golden and the Lucas are two of the ten, they supply five of the fifteen destinations and fifty-two of the hundred and seventeen settled runs, and the single most-visited destination in the table is on neither of them.

Where this goes next

Every destination in the settling table, counted. One row per destination, placed across at the divergence the stems settled on and labelled with the parastichy pair a counter returns at the top of the run. 117 settled runs land on 15 destinations when two values within half a degree are called one. five of them sit on the golden or the Lucas sequence and ten do not, and the ones that do not are reached at every falloff exponent including the shallowest. Nothing here refuses to count: the settling test does not admit arrangements this collection would decline to call lattices. A counter on the settling table The settling table has reported an angle for every run that reaches a lattice, and nobody has ever counted one. A hundred and seventeen settled runs, regrown and counted: every one of them has a parastichy pair, and sixty-five of them land on sequences the ladder does not carry. Every transition as the rise falls. The pair climbs 1/2 → 2/3 → 3/5 → 5/8 → 8/13 → 13/21. Consecutive transitions are 0.382, 0.382, 0.383, 0.382 of the previous rise — 1/φ² is 0.3820. The Fibonacci ladder Lower the rise on a cylinder and the parastichy pair climbs — 1 and 2, then 2 and 3, then 3 and 5 — each rung the sum of the two before it. Nothing in the arithmetic mentions Fibonacci, the transitions sit at computable rises, and consecutive ones stand in the ratio 1/φ². What the 3 steep-only destinations count as. Each block is a divergence that only the two steeper falloff exponents settle a stem on, with the pair a counter returns and the sequence that pair belongs to. All three count. None is a rung of either ladder. Two of them are the only members of their sequence anywhere in the table; the third, at 148.1 degrees, is the coarsest rung of a sequence that supplies another destination at every exponent — so it is not a new kind of arrangement but a coarser one of a kind the table already reached. What a steep rule counts as Exponents four and five settle stems on 42.3°, 47.9° and 148.1°, and the two shallower ones reach none of them. All three count: 8/9, 8/15 and 2/5. None is a rung of either ladder, and one of them is the coarsest rung of a sequence the table already had.

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 list that was a rounding — both name attractor, basin, census, claim testing, divergence angle, honest limits, measurement, parastichy pair, selection effect
  • The angles name the branch — both name census, continued fraction, divergence angle, fibonacci, ladder, lucas numbers, measurement, parastichy pair
  • Destinations only a steep rule reaches — both name attractor, basin, claim testing, divergence angle, honest limits, measurement, selection effect
  • A file has to close — both name continued fraction, divergence angle, measurement, noble number, parastichy pair, rational approximation
  • A period the grid invented — both name attractor, claim testing, divergence angle, honest limits, measurement, parastichy pair
  • Fibonacci is a branch, not a law — both name attractor, continued fraction, divergence angle, fibonacci, lucas numbers, selection effect

Named objects

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

AttractorBasinCensusClaim testingContinued fractionDivergence angleFibonacciGeometric ladderHonest limitsLadderLucas numbersMeasurementNoble numberParastichy pairRational approximationSelection effect