Phyllotaxis
the spirals are counted, not admired
The pattern itself

A counter that sees no positions

This site has counted spirals two ways, and both were handed coordinates. A third counter is handed a list of angles and nothing else. It returns one number instead of two, it refuses more often, and where it refuses it would have been wrong every time.
17 min read 10 figures Counted, not admiredState the claim, state the testThe round trip

Worth reading first: Counting the spirals · Counting up the stem · The sequence has a memory.

There are now three instruments on this site that report a parastichy number, and the useful thing about having three is not that they agree. It is that they are given different things.

The first is parastichy, which counts a disc. It receives an array of points and a radius band. The second is cylCount, which counts a stem. It receives an array of points and a height band. The third arrived with the previous essay and it receives a list of angles.

What follows is what that difference buys, what it costs, and the one property a blind instrument must have that the other two never needed.

What each one is not told

The discipline is the same in all three cases and it is the reason any of the numbers mean anything.

The two spiral families a counter finds between 0.43 and 0.67 of the radius21 spirals one way and 34 the other, found from the point positions alone — the counter is never told the divergence angle.21 and 34 spiralscounted, not assumed
Fig. 1 The first counter’s output, drawn. What it was given is a set of coordinates and a radius band; what it was not given is the divergence angle, the model, or the fact that Fibonacci numbers exist.

None of the three knows the divergence angle. None knows the rise, or the growth rate, or which numbers a reader is hoping for. Each is a function from data to a number, and the data does not contain the answer in any form that could be read off without doing the work.

That is easy to state and it is the thing that is most often quietly violated. A counter that took the divergence angle and returned the pair the ladder predicts would agree with all of these on every case in this collection, and would be a restatement rather than a measurement. The site’s foundation phase turned on exactly this: the recovery step refused a count, and it could only refuse because it had never been told what the count was supposed to be.

The third counter takes the discipline further than the other two, because a list of angles does not contain a picture. There is no arrangement in it to look at. The information is there — the previous essay is about why — but nothing in the data resembles the answer.

What each one returns

Here they differ, and the third comes off worst.

Which offsets give short hops, at a rise of 0.05The two lowest points are at 2 and 3, and those are the parastichy numbers. Offset 1 is high because a hop of one node is at least the rise, which is what makes a stem easier to count than a disc.00.50011.50102030index offsetmedian hop between node i and node i+m23260 nodes, 34 offsets triedshortest at 2 and 3
Fig. 2 What the position counters actually measure: for every offset in placement order, how far apart two nodes that many places apart really are. The offsets at which that is a local minimum are the parastichy numbers, and there are usually four or five of them.

The disc counter returns a pair — the two shortest offsets — along with the whole curve it picked them from, so a reader can see how close the third was. The cylinder counter returns a pair and the same curve. Both, on a good band, are unambiguous.

The angle counter returns one number. It is one member of the counted pair, it is the nearer family away from a transition, and there is no second number available: the sequence’s spectrum contains the other families, as the previous essay shows, but nothing in it distinguishes a second family from a harmonic of the first.

The order of the angles carries the countThree stems, each held at a fixed rise so the pattern sits on one rung of the ladder. At a rise of 0.032 the positions count 3 and 5 spirals and the angles peak at 3; At a rise of 0.013 the positions count 5 and 8 spirals and the angles peak at 5; At a rise of 0.005 the positions count 8 and 13 spirals and the angles peak at 8. Each panel marks the peak and its multiples; the pale strip is what an uncorrelated sequence of this length gives.rise 0.032counted 3/5angles say 336912150.5rise 0.013counted 5/8angles say 55101520250.5rise 0.005counted 8/13angles say 8816240.5151015202530lag, in internodescorrelation between a divergence and the one that many internodes later3 runs per rise · 320 internodes eachthe counter is never shown a position
Fig. 3 The third counter’s raw output. The tallest peak away from lag one is the number reported; the peaks at its multiples are what make it a period rather than a coincidence, and they are also what makes a second family unreadable.

So the third instrument is strictly weaker in what it produces. Its interest is entirely in what it consumes.

The round trip, run three ways

The site’s habit is to close a circuit: build a pattern from stated parameters, throw the parameters away, recover them from the output, and report the error.

Six stems built, forgotten and recoveredEach row is a lattice built from a divergence and a rise, counted by machinery shown only the coordinates, and reconstructed from the counts and the two hop lengths. The worst error in the recovered angle is 3.0e-13°.137.51°, rise 0.098.5e-14°counted 2/3137.51°, rise 0.032.0e-13°counted 3/5137.51°, rise 0.0122.8e-14°counted 5/899.50°, rise 0.083.0e-13°counted 1/3151.14°, rise 0.071.1e-13°counted 2/399.50°, rise 0.021.1e-13°counted 4/7error in the recovered divergence anglecounts and hop lengths onlyworst 3.0e-13°
Fig. 4 The existing circuit. A lattice is built at a stated divergence and rise, the parameters are discarded, and two counts and two hop lengths recover them — on a cylinder, to thirteen digits.

Run on the same stems, the three counters agree. At a rise of 0.032 the positions say 3 and 5 and the angles say 3; at 0.013 the positions say 5 and 8 and the angles say 5; at 0.005 the positions say 8 and 13 and the angles say 8.

That agreement is worth something specific and it is easy to overstate. It does not show that either instrument is correct — two instruments implementing the same mistake would agree too. What it shows is that they do not share a mistake, because they do not share a code path, a data type, or a definition. One measures distances between points. The other measures products of angle differences. They have the integer in common and nothing else.

The same stem, not unrolled33 of the 64 nodes face the reader and 31 are behind the stem, drawn open. The count is 2 and 3 either way; the unrolling changes nothing but the visibility.near facefar face64 nodes at 137.51°2 and 3, both faces
Fig. 5 The object all three are describing, drawn as it actually is. The lattice is the same whether it is read as a set of positions or as a sequence of turns; the two readings share the arrangement and no arithmetic.

What a counter got wrong once, and how it was caught

The site’s first counter was wrong, and the way the error surfaced is the reason this essay treats refusal as a feature rather than as a nuisance.

parastichy computes, for every offset, the median distance between nodes that many places apart, and returns the offsets at which that distance is smallest. The first version returned the two smallest offsets among the local minima rather than the two shortest hops. On a head whose real neighbours are 34 and 55 it returned 21 and 34.

Nothing looked wrong. Twenty-one spirals were drawn and there were twenty-one of them; the figure was correct, the assertions inside it passed, and a reader checking the picture against the caption would have found them in agreement. The pattern genuinely has a 21 family. It is simply not one of the two nearest.

What caught it was the recovery step refusing. Given the pair 21 and 34 and the band they were counted in, the recovery asks which divergence angles make those two the shortest offsets — and there is no such angle. Not a large error: no solution at all. The instrument downstream could not be satisfied, and its complaint was specific enough to locate the fault.

That is the shape every check on this site is built to have, and it is why the third counter’s threshold is not an afterthought. An instrument that can only return answers can only be wrong. An instrument that can refuse has a second output, and the second output is where errors show up first.

The same story has repeated twice since. A cylinder counter applied a local-minimum rule that is right on a disc and wrong on a stem, drew five families where there were five, and was caught by a recovery that refused. A figure asserted a tolerance tighter than its own library’s and failed on a run the library passed. In all three cases the picture was fine.

The Lucas control, which is the one that matters

Agreement on Fibonacci numbers is weak evidence, because Fibonacci numbers are what a careless instrument would produce.

What the positions say, and what the angles sayEach row is one stem at one rise. The left column is the parastichy pair counted from the coordinates; the right is the single number read out of the divergence angles alone, over 5 runs. On the Lucas ladder — 3/4, 4/7, 7/11 — the readout returns the smaller number too, so it is reading the lattice rather than Fibonacci. The last row is the one that matters: at a rise of 0.05 the positions give an unarguable 2/3 and the angles give 4, 23, 12, 2, 9 — all five wrong, and all five refused.counted from the pointsread from the anglesgolden, rise 0.0323 / 535/5 clear · peak 0.72golden, rise 0.0135 / 855/5 clear · peak 0.59golden, rise 0.0058 / 1385/5 clear · peak 0.78Lucas, rise 0.0323 / 435/5 clear · peak 0.52Lucas, rise 0.024 / 745/5 clear · peak 0.45Lucas, rise 0.0087 / 1175/5 clear · peak 0.59golden, rise 0.052 / 34, 23, 12, 2, 9refused — peak 0.13 under 0.345 runs per rise · the readout sees a list of angles and nothing elsethe refusal is the gate working
Fig. 6 The three-way comparison in full. The middle block is a pattern seeded at the Lucas angle, where the ladder runs 3/4, 4/7, 7/11; the angle counter returns 3, 4 and 7. Eleven is not a Fibonacci number and none of these instruments has ever heard of it.

Seeded at 99.502° the pattern walks the Lucas ladder and the angle counter follows it: 3, then 4, then 7. Every one of those agrees with the position counter on the same stem, and the last two are numbers that no Fibonacci-flavoured bug could produce.

This is the same control the site’s disc counter was given in the foundation phase and it is worth applying to every counting instrument as a matter of course. An instrument that only ever returns 8, 13, 21 and 34 has not been shown to be counting.

A blind instrument must be able to refuse

Here is the property the other two counters never needed, and it is the reason this essay exists rather than being a paragraph in the last one.

The position counters can fail visibly. A band with too few points, a curve with no local minimum, a pair sharing a factor — each of those is a recognisable state and each throws. More to the point, their output is checkable by eye: draw the polylines joining every 34th point, and if there are thirty-four of them the counter was right.

The angle counter has no such recourse. It is an argmax over thirty lags, and an argmax always returns an index. Handed a sequence with no period in it at all it returns a number, promptly and with no sign of difficulty. There is no picture to check it against, because the whole point of the instrument is that it never saw one.

So it needs a threshold, and the threshold has to be set from outside the data it is judging. It is set from sequences with no structure whatever: white noise of the same length gives a largest-of-thirty-lags value of about 0.10, worst case 0.15, and the threshold sits at three sampling bands, which is 0.34. Anything under that is refused.

Where it refuses, it would have been wrong

A threshold that never fires is decoration. This one fires, and the case is instructive.

At a rise of 0.05 the pattern is a 2/3 lattice. The position counter reports 2 and 3 without hesitation, and the pattern is intact by every measure the site has — the scatter is four tenths of a degree, every run is coherent, nothing is wrong with it.

The angle counter reports 4, 23, 12, 2 and 9 across five runs.

Four of those five are not in the counted pair. The fifth, 2, is correct by accident. And the clearance test refuses all five, because the peaks are 0.10 to 0.22 and the line is at 0.34.

That is the whole argument for the gate in one row of a table: five wrong answers, five refusals. Without it, this essay would be reporting that a 2/3 lattice sometimes has a period of twenty-three.

Why it fails there, and what that says about the instrument

The failure is at the coarse end, and it is not arbitrary.

A period of two or three has few cycles inside a thirty-lag window, and the family carrying the correlation has few members to carry it. Going finer, the peak grows: 0.32–0.46 at the 2/3 rung, 0.72 at 3/5, 0.59 at 5/8, 0.78 at 8/13.

The spiral counts, band by band, in one headThe 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.0255075406080fraction of the head's radiusparastichy numbers found in a band there1000 primordia at the golden angle4 different pairs
Fig. 7 The other counters’ equivalent difficulty, for comparison. Every count is a statement about a band, and near the radius where the pair changes, three offsets have almost the same length and no instrument can prefer one.

So the three instruments have different blind spots, which is the useful property. The position counters are weakest near a transition, where two pairs are equally short. The angle counter is weakest at the coarse end, where there is barely a period to find. A pattern that defeats one is not usually the pattern that defeats another.

What each one costs to use on a plant

The three instruments are not interchangeable in the field, and the differences are larger than the differences in what they return.

The disc counter needs a head, photographed square on, with the primordia resolvable and the centre locatable. That is an afternoon with a camera and a sunflower, and it is the reason this is the count the literature is full of. Its weakness is that it answers about a band and is usually reported as though it answered about a head.

The cylinder counter needs a stem with its leaves or scales still attached and their positions measurable in two dimensions. Harder — the organs are on a curved surface and the far side is hidden — but it has the compensating property that the answer does not depend on where it is taken, which on a disc it always does.

The angle counter needs sixty consecutive internodes, a steady rung, and the angles measured to better than a quarter of a degree. The first two are cheap and checkable in advance. The third is not: it is a demand for photogrammetry rather than a protractor, and the essay two along is about exactly how the requirement arises and why it cannot be relaxed by averaging.

So the order of difficulty is not the order of sophistication. The blind instrument is the most demanding of the three, and it is demanding in precision rather than in quantity — sixty internodes is one good stem, and a quarter of a degree is an instrument most fieldwork does not carry.

A count of m and n pins the divergence to 221°/mnEach dot is one reported pair, and its height is the total width of the divergence angles that could have produced it at some rise. 2/3 leaves 38.8° open; 34/55 leaves 0.118°. The line is 221°/mn, taken from the three highest pairs and drawn back through the rest.-10111.5022.503product of the two counts, log₁₀angles left open, log₁₀ °7 pairs · edges found by bisectionwidth × mn = 221°
Fig. 8 What a count buys once you have one. A reported pair pins the divergence angle to an interval of about 221°/mn — so a count of 8 and 13 is worth a great deal more than a count of 2 and 3, whichever instrument produced it.

What the third counter is for

Not for counting spirals. Given a photograph, the spirals should be counted.

It is for the case of a sequence and no photograph — which is not a contrivance, because a botanist walking up a stem with a protractor produces exactly that, and because the angles are the quantity every published claim about divergence is stated in. It is also for the case where the count and the angles are both available and disagree, which is a state no single instrument can detect.

A round trip on four heads of 900 primordia: the divergence angle recovered from eachThe counter is shown the points and nothing else. The worst recovery across the four is 0.012°.the first of the four — 300 of its 899 pointsused to buildcountsrecovered137.508°55 · 89137.520°99.502°47 · 7699.500°151.100°31 · 81151.105°77.960°37 · 6077.960°worst error 0.012°counts in, angle outthe recovery never sees the angle
Fig. 9 The other direction of the same circuit: given a counted pair, what divergence angles are consistent with it. Two instruments that take different data and return the same integer make this recovery worth more than either alone.

And it is for the argument the next essays make, which is that the order of the angles carries several things nobody had looked for. The count is the first, and it is the one that could be checked against an instrument the site already trusted.

A head of 300 primordia at a divergence of 137.51°Nothing is placed by hand: the nth point sits at n·137.51° and radius √n. The closest any two points come is 1.60 of the mean spacing.divergence 137.508°closest pair 1.60 × mean spacing
Fig. 10 Where the site started: a pattern, and the question of how many spirals run through it. Four phases of counting later, the answer turns out to be readable from a list of numbers with no pattern in them at all.

What a fourth counter would have to use

It is worth asking what is left, because the answer is short and it says something about how much of a pattern a count uses.

A stem hands over three kinds of information: where the organs are, what order they were made in, and how big they are. The disc and cylinder counters use the first two — positions, indexed by placement order, which is why they can speak about an offset at all. The angle counter uses the second and a projection of the first.

Nothing on this site uses the third. Organ size varies systematically up a stem, it is measurable with far less precision than an angle needs, and it is not obviously independent of anything: a primordium’s size and the space available to it are the same quantity seen twice. Whether a size sequence carries a parastichy number the way an angle sequence does is a question with a definite answer and this collection does not know it.

What is not left is a fourth reading of the positions. The disc and cylinder counters differ in their metric and not in their data, and a third metric on the same coordinates would be a third way of saying the same thing — which is worth having for robustness and is not worth calling an independent instrument.

The honest summary

Three instruments, three kinds of data, one integer. The third is weaker in every respect except independence, and independence is what a round trip is for.

Its distinguishing feature is not accuracy. It is that it has a threshold with something behind it — a measured noise floor, a case where it fires, and a demonstration that the case where it fires is the case where the answer would have been wrong. That is a smaller claim than the other two counters make and it is supported by a kind of evidence they were never asked for.

Where this goes next

rise 0.032counted 3/5angles say 336912150.5rise 0.013counted 5/8angles say 55101520250.5rise 0.005counted 8/13angles say 8816240.5151015202530lag, in internodescorrelation between a divergence and the one that many internodes later3 runs per rise · 320 internodes eachthe counter is never shown a position The order carries the count Take the divergence angles off a stem, throw away every coordinate, and autocorrelate what is left. The result is periodic at the smaller parastichy number — peaks at it and at every multiple of it. A list of angles, with no picture and no position in it, carries the spiral count. 21 and 34 spiralscounted, not assumed Counting the spirals Almost every claim about phyllotaxis is a claim about how many spirals run through a pattern, and the count is almost never done. It can be done from the points alone, by machinery that is never told what angle built them — and then a count of 34 is evidence rather than a restatement. stem, lower third2 and 3stem, middle third2 and 3stem, upper third2 and 3disc, r = 0.18–0.3221 and 34disc, r = 0.45–0.6234 and 55disc, r = 0.82–0.9955 and 89stem at rise 0.050, disc of 1600 points, both at 137.51°counted from coordinates onlyone answer against three Counting up the stem The same counting machinery, pointed at a stem instead of a seed head, returns one answer three times where the head returned three answers. That contrast is a measurement rather than a preference, and it is the one the whole cylindrical argument rests on.

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.

AutocorrelationCylinderDiscretisationDivergence angleEnsembleEquilibriumLattice offsetMeasurementNoiseParastichyRiseRound tripSamplingSummary statistic