Phyllotaxis
the spirals are counted, not admired
The claims, measured

A streak is a shear the count can see

Let each organ of a census disc inherit its error from its nearest earlier neighbour alone, rather than from the two it sits against, and the error travels along one family of spirals: streaks instead of patches. Measured against its four nearest neighbours a streak moves an organ exactly as much as a patch does — just over half what an independent displacement of the same size moves it — and yet it changes more of the band's readings at every size and in every draw, and on heads of three and five hundred organs as many as the independent displacement itself, or more. Its movement is nearly all across its own streak and hardly any along it, so it shears one family's chains against the other's, and the relative size that predicted a patch's effect underestimates a streak's by up to three points. Near the golden angle it changes no more readings than anywhere else, and the census still needs three kept specimens.
15 min read 7 figures State the claim, state the testCounted, not admired

Worth reading first: How many plants would it take.

A Fibonacci census needs a null, the pairs a band reads on heads that are not Fibonacci by design, and a disc reads no Fibonacci by chance drew it from Vogel discs at every divergence angle, counted in a surveyor’s band from 0.55 to 0.95 of the radius. Two essays since have moved those discs’ organs off the lattice to see what a real head’s imperfection does to the null. Displacing a disc’s organs moved each organ on its own, and a drift along the spirals laid the error down with the head, each organ inheriting part of the mean error of its two nearest earlier organs. The drift changed fewer readings than the independent displacement, not more, because a count reads an organ against its neighbours and a drift moves neighbours together.

That drift inherits from two organs, one on each of the two families of spirals crossing at the new organ, so its errors spread along both and make round patches. A placement rule need not weigh its neighbours equally. A new organ usually sits closer to one of the two it touches, and if it takes its error from that one alone, the error travels along one family only. The patches become streaks — long runs of organs moving together along one spiral, sliding past the next spiral’s run. The earlier essay ended by asking whether streaks change more readings than patches, whether near the golden angle, and whether the relative size of the movement still predicts what a count does.

The streak, laid down with the head

The streak is the drift with one change. Organs are taken in the order the head lays them down; each inherits a share ρ\rho of the error of its single nearest earlier organ and adds a gaussian of its own; and the whole field is then scaled so that its size per coordinate is a tenth or a quarter of a spacing — exactly the size of the independent displacement and of the patchy drift, on the same seed at each angle. The shares are 0.6 and 0.9, the same two the patches were read at. Heads of 75, 150, 300 and 500 organs are read at every one of 3,599 angles across the half turn, in two draws, counted in the same band and its three tenths.

The same head drifted in patches and in streaks, the displacements drawn four times their sizeA Vogel head of 300 organs at 137.6°, every organ in the band from 0.55 to 0.95 of its radius drawn with a line from where the lattice puts it, four times the length of its displacement, both fields a quarter of a spacing in size on the same seed. On the left each organ inherits 0.9 of the mean error of its two nearest earlier organs, which carries an error along both families crossing there and makes round patches. On the right it inherits 0.9 of its single nearest earlier organ's error, which carries it along one family and makes streaks. Both move an organ against its four nearest neighbours by about the same amount — 0.55 and 0.55 of an independent displacement — but the streak moves it against the neighbours on its own streak by 0.34 and against the rest by 0.70. The band reads 21/34 in patches and 21/34 in streaks.patches, ρ 0.9band reads 21/34streaks, ρ 0.9, the same sizeband reads 21/34300 organs · 137.6° · lines ×4generated from a stated rule, not drawn to look right
Fig. 1 One 300-organ head drifted in patches and in streaks of the same size, on the same seed, every band organ’s displacement drawn four times its length; the dial sets the share each organ inherits.

The picture shows the difference at once. The patches are blobs: neighbouring organs move the same way in every direction from a spot. The streaks are lines: runs of organs moving together along one direction round the head, and the run beside them moving another way. On this head at 137.6° both read 21/34; the difference is in how often, across the half turn, they read something else.

The same relative size

The earlier essay’s explanation of the patch’s mildness was a single number: how far an organ moves against its four nearest neighbours, as a share of what an independent displacement of the same size moves it. An independent displacement moves an organ against a neighbour by 2\sqrt{2} times its size, since both move; a drift moves neighbours together and so moves an organ against them less.

By that number a streak and a patch are the same. At ρ=0.9\rho = 0.9 the patch moves an organ against its four nearest neighbours by 0.54 to 0.58 of the independent displacement, depending on the head size; the streak by 0.53 to 0.59. At ρ=0.6\rho = 0.6 both sit at 0.78 to 0.81. Over every head size and both correlations the two never differ by more than five hundredths. If the relative size is what a count reads, a streak should change exactly as many readings as a patch.

It changes more

How many band readings an independent displacement, a patchy drift and a streaky drift of the same size change. The share of angles, over two draws, at which moving the organs changes the pair the band reads, at each head size, at a quarter of a spacing (solid) and a tenth (dashed). independent, 0.1: 8.2%, 5.7%, 3.5%, 2.7%; patches, ρ 0.9, 0.1: 7.4%, 5.1%, 3.3%, 2.3%; streaks, ρ 0.9, 0.1: 7.8%, 5.8%, 3.7%, 2.7%; independent, 0.25: 15.8%, 10.2%, 5.9%, 4.4%; patches, ρ 0.9, 0.25: 12.1%, 7.6%, 4.9%, 3.2%; streaks, ρ 0.9, 0.25: 13.3%, 8.9%, 6.1%, 4.8%, on 75, 150, 300, 500 organs. Streaks change more readings than patches everywhere, and on the two larger heads more than the independent displacement.
Fig. 2 The share of angles at which moving the organs changes the pair the band reads, at each head size, for the independent displacement, the patchy drift and the streaky drift at ρ = 0.9, at a quarter and a tenth of a spacing.

It does not. At every head size, both sizes of displacement and both correlations the streak changes more readings than the patch. At ρ=0.9\rho = 0.9 and a quarter of a spacing the patch changes 12.1, 7.6, 4.9 and 3.2 per cent of readings on heads of 75, 150, 300 and 500 organs; the streak 13.3, 8.9, 6.1 and 4.8. On the two larger heads that is as much as the independent displacement of the same size changes, 5.9 and 4.4 per cent, or more — so a correlated error, which on the earlier evidence made a count steadier, here makes it no steadier than no correlation at all.

The same holds at a tenth of a spacing in smaller numbers: the streak changes 3.7 and 2.7 per cent on the larger heads against the independent displacement’s 3.5 and 2.7, and the patch’s 3.3 and 2.3. And at ρ=0.6\rho = 0.6, where both drifts are weaker, the streak still changes more than the patch everywhere, by margins from three hundredths of a point to half a point.

Two draws, read apart

Every share above pools two draws of the error field. Read apart, the streak changes more readings than the patch in each draw of every cell, not just on average: at a quarter of a spacing on the 500-organ head, 4.76 and 4.85 per cent against 3.52 and 2.92; on the 300-organ head, 5.90 and 6.21 against 4.72 and 5.00; on the 75-organ head, 13.79 and 12.82 against 11.92 and 12.25. The gap between streak and patch is larger than either one’s scatter between its own draws everywhere.

The comparison with the independent displacement is less clean. The streak changes more in both draws on the 300-organ head at a tenth of a spacing, 3.66 and 3.82 against 3.57 and 3.34, and on the 500-organ head at a quarter, 4.76 and 4.85 against 4.42 and 4.37. On the other two large-head cells the draws overlap: 5.90 and 6.21 against 6.06 and 5.73, and 2.90 and 2.50 against 2.69 and 2.61. So the firm statement is that a streak is no gentler on a count than independent error on a large head, and sometimes harsher; the firm statement against the patch is that it is always harsher.

Why the small heads differ

The streak does not overtake the independent displacement everywhere. On the 75- and 150-organ heads it changes fewer readings than the independent displacement, 13.3 against 15.8 per cent and 8.9 against 10.2 at a quarter of a spacing, and more only on the 300- and 500-organ heads. The crossover has a measurable cause in the band’s geometry.

An error inherited at ρ=0.9\rho = 0.9 keeps 0.9j0.9^{j} of itself after jj links, a third after ten: a streak has a length of about ten organs before the inheritance fades. Followed link by link inside the surveyor’s band, over thirty heads at each size, a streak runs unbroken for 4.9 organs on average on the 75-organ head, 7.8 on the 150, 10.6 on the 300 and 11.6 on the 500 — because the band holds 44, 89, 179 and 300 organs, and a streak that starts inside it leaves it, or reaches a rim, sooner on a small head. So only on the two larger heads does a streak run its full length inside the band where the count is taken, and only there does it move a whole run of organs as one body against the next.

At the weaker inheritance the length is two or three organs and the picture is mixed: at ρ=0.6\rho = 0.6 the streak changes slightly more than the independent displacement on the 300-organ head at both sizes of displacement, the same on the 500-organ head at a tenth of a spacing, and fewer at a quarter. A shear needs runs long enough to slide, and at 0.6 there are hardly any.

Along the streak and across it

How far a streak moves an organ against the neighbours on its own streak and against the rest, beside a patchy drift. For each head size and correlation, at a quarter of a spacing, the root-mean-square movement of a band organ against its four nearest neighbours, as a share of what an independent displacement of the same size does, averaged over thirty heads. 75 organs, ρ 0.6: patches 0.81; streaks 0.79 overall, 0.65 along the streak and 0.90 across it; 75 organs, ρ 0.9: patches 0.58; streaks 0.56 overall, 0.38 along the streak and 0.69 across it; 150 organs, ρ 0.6: patches 0.81; streaks 0.80 overall, 0.65 along the streak and 0.92 across it; 150 organs, ρ 0.9: patches 0.57; streaks 0.59 overall, 0.36 along the streak and 0.74 across it; 300 organs, ρ 0.6: patches 0.80; streaks 0.79 overall, 0.65 along the streak and 0.91 across it; 300 organs, ρ 0.9: patches 0.55; streaks 0.55 overall, 0.34 along the streak and 0.70 across it; 500 organs, ρ 0.6: patches 0.80; streaks 0.78 overall, 0.64 along the streak and 0.90 across it; 500 organs, ρ 0.9: patches 0.54; streaks 0.53 overall, 0.33 along the streak and 0.67 across it. Overall the two drifts agree; the streak's movement is nearly all across its own streak.
Fig. 3 For each head size and correlation, how far a streak moves a band organ against the neighbours on its own streak and against the rest, beside how far a patch moves it against all four, as shares of an independent displacement.

The single number hides how the movement is distributed. Split an organ’s four nearest neighbours into the one or two it shares a streak with — its parent, or an organ whose parent it is — and the rest. At ρ=0.9\rho = 0.9 a streak moves an organ against its streak-mates by only 0.33 to 0.38 of an independent displacement, and against the others by 0.67 to 0.74. At ρ=0.6\rho = 0.6 the split is 0.64 to 0.65 along and 0.90 to 0.92 across. A patch, which inherits from both sides, has no such split; a streak puts nearly all its relative movement across itself. Since a hexagonal cell’s six neighbours are two from each of three families, that means two of an organ’s six contacts barely move and the other four carry the whole displacement.

That is a shear. A run of organs along one spiral moves as a body, and the run beside it moves as another body, so the two spirals slide past each other while each stays nearly intact. The overall relative size is the same as a patch’s because the sliding is large where the streaks meet and nothing along them; the average over four neighbours cannot tell a little everywhere from a lot in one direction.

Which way the streaks run

Which family each organ's nearest earlier neighbour lies on, and so which way the streaks run. For each head size, over thirty heads spread across the half turn, the share of band organs whose nearest earlier organ is one step back along the smaller of the two families the band reads, along the larger, or along neither. 75 organs: 79% along the smaller, 21% along the larger; 150 organs: 89% along the smaller, 10% along the larger; 300 organs: 67% along the smaller, 33% along the larger; 500 organs: 90% along the smaller, 10% along the larger. Most streaks run along one family, and on most heads it is the smaller.
Fig. 4 For each head size, the share of band organs whose nearest earlier organ lies one step back along the smaller of the two families crossing in the band, and along the larger.

An organ’s nearest earlier neighbour is a fixed distance back in placement order — a parastichy number — so its streak runs along one family. Over thirty heads spread across the half turn, 79, 89, 67 and 90 per cent of band organs on heads of 75, 150, 300 and 500 organs have their parent along the smaller of the two families crossing there, and nearly all the rest along the larger. The streaks run mostly along one family, and it changes with the head: the smaller family’s step is the shorter over most of a band, and where the larger takes over, the streaks turn with it.

So a streaked head has one family of spirals whose chains each move as a body, and a crossing family whose chains are cut across by those moving bodies. What a count is worth and counting the spirals both read a pair as two families of chains, each linking organ to neighbour; a shear leaves one family’s links intact and stretches every link of the other.

The relative size stops predicting

The readings a streak changes against those its relative size predicts, beside a patchy drift's. For every head size, size of displacement and correlation, the share of readings each drift changes against the share an independent displacement would change if it moved organs against their neighbours as much as the drift does. The patches sit within two percentage points of the diagonal, at most 1.8 above it; the streaks sit above it by up to 3.2. The relative size predicts what a patch does and underestimates what a streak does.
Fig. 5 For every size, displacement and correlation, the readings each drift changes against those an independent displacement of the same relative size would change, streaks and patches.

The earlier essay’s account of the patch was quantitative: take the independent displacement whose relative size matches the drift’s, and it changes the same share of readings to within two percentage points. That still holds of the patches here, at most 1.8 points above the diagonal. It fails for the streaks. At ρ=0.9\rho = 0.9 and a quarter of a spacing the streak changes 3.2 points more than its relative size predicts on the 75-organ head, 1.8 on the 150, 2.0 on the 300 and 1.8 on the 500 — consistently above, never below.

That is the useful negative result. A count reads an organ against its neighbours, as the patch essay found, but not against all of them equally: a count reads chains, so it reads the links along each family, and a displacement that stretches one family’s links while leaving the other’s alone does more to a count than one that stretches all links a little. The relative size averages over directions, and the count does not.

Not near the golden angle

Near the golden angle a streak changes readings no more often than anywhere else. For every head size, size of displacement and correlation, the share of readings a streak changes within two and a half degrees of the golden angle against the share it changes over the whole half turn. 75 organs, 0.1, ρ 0.6: 4.5% near, 8.1% overall, 1 into a Fibonacci pair; 75 organs, 0.1, ρ 0.9: 2.5% near, 7.8% overall, 1 into a Fibonacci pair; 75 organs, 0.25, ρ 0.6: 12.0% near, 14.6% overall, 4 into a Fibonacci pair; 75 organs, 0.25, ρ 0.9: 11.0% near, 13.3% overall, 1 into a Fibonacci pair; 150 organs, 0.1, ρ 0.6: 2.0% near, 5.6% overall, 0 into a Fibonacci pair; 150 organs, 0.1, ρ 0.9: 2.5% near, 5.8% overall, 1 into a Fibonacci pair; 150 organs, 0.25, ρ 0.6: 3.5% near, 9.2% overall, 0 into a Fibonacci pair; 150 organs, 0.25, ρ 0.9: 3.5% near, 8.9% overall, 0 into a Fibonacci pair; 300 organs, 0.1, ρ 0.6: 2.5% near, 3.6% overall, 0 into a Fibonacci pair; 300 organs, 0.1, ρ 0.9: 2.0% near, 3.7% overall, 0 into a Fibonacci pair; 300 organs, 0.25, ρ 0.6: 4.0% near, 6.0% overall, 1 into a Fibonacci pair; 300 organs, 0.25, ρ 0.9: 5.0% near, 6.1% overall, 0 into a Fibonacci pair; 500 organs, 0.1, ρ 0.6: 1.5% near, 2.6% overall, 0 into a Fibonacci pair; 500 organs, 0.1, ρ 0.9: 1.5% near, 2.7% overall, 0 into a Fibonacci pair; 500 organs, 0.25, ρ 0.6: 3.0% near, 3.9% overall, 0 into a Fibonacci pair; 500 organs, 0.25, ρ 0.9: 3.0% near, 4.8% overall, 0 into a Fibonacci pair. Every point sits on or below the diagonal.
Fig. 6 For every size, displacement and correlation, the share of readings a streak changes within two and a half degrees of the golden angle against the share it changes over the whole half turn.

The census turns on what happens near the golden angle, where a band is closest to tipping between a Fibonacci pair and its neighbour. Under every streak the readings within two and a half degrees of it change no more often than readings elsewhere: on the 300-organ head at a quarter of a spacing and ρ=0.9\rho = 0.9, 5.0 per cent near against 6.1 overall; on the 500-organ head, 3.0 against 4.8. Every point sits on or below the line where the two are equal.

And streaks turn almost nothing into a Fibonacci pair. Across all sixteen streaked nulls, the readings that become Fibonacci number nought to four, the four on the smallest head at the weaker correlation. A shear makes a band read a different pair more often, but not a Fibonacci one.

Where the changed readings go

A changed reading can be a pair the exact head’s band already reads at one of its edges — the band straddling a transition and the displacement tipping it — or a pair the band never reads. On heads of 300 and 500 organs about half the streak’s changes are of the first kind, 41 to 57 per cent; on the 75-organ head none are, because its band is narrow in organs and has no edge readings to tip to. The streak does not invent new pairs any more readily than the patch does: the patches’ share of edge readings is 16 to 18 per cent on the 150-organ head, 43 to 49 on the 300 and 59 to 62 on the 500, against the streaks’ 16 to 22, 41 to 50 and 53 to 57. It just tips more bands, which is what a head displaced before it is counted found displacement does to a band that straddles a transition.

The census and the capped aim

The heads a capped aim can reach in the census null, exact, in patches and in streaks. The share of the null's readable heads a capped recount could move, at each head size, for the exact disc and for a patchy drift and streaks at ρ = 0.6 and 0.9, all a quarter of a spacing. 150 organs: 0.281%, 0.292%, 0.292%, 0.279%; 300 organs: 0.168%, 0.153%, 0.153%, 0.139%; 500 organs: 0.142%, 0.126%, 0.112%, 0.126%. Every streaked share is within four hundredths of a percentage point of the exact one, and the census needs 3 kept specimens under every one.
Fig. 7 The share of the census null’s heads a capped recount could move, at each head size, exact, under a patchy drift, and under streaks at ρ = 0.6 and 0.9, a quarter of a spacing.

The census’s two figures come through unmoved. The heads a capped recount could move — near-golden pairs some Fibonacci pair reads as within reach — are 0.281, 0.168 and 0.142 per cent of the null on heads of 150, 300 and 500 organs exactly, and under streaks they move by at most three hundredths of a point: to 0.139 and 0.126 per cent on the larger heads. And the census still needs three kept specimens under every streak, as it did exact, displaced and drifted. How many plants would it take set that number, and three different errors have now failed to move it.

What a streak stands for, and does not

A streak is the second stand-in for a grown head’s errors, and it is no more grown than the patch was. It assumes an organ takes its error from one neighbour, wholly at a fixed share; a real placement rule would weigh its two neighbours by how close they are, which puts it between the patch and the streak, and its errors would not be gaussians added to a lattice but the rule’s own response to what was there. The organ that was taken away is a reminder of how far that response can carry. What the two stand-ins together say is that the shape of a correlated error matters to a count in a way its size does not: two errors of identical size and identical relative size differ by half again in the readings they change.

The rule this replaces

The patch essay ended with a rule a survey could use: a correlated error changes a band reading about as often as an independent error of its relative size. A streak breaks that rule upward, by up to three points on a share of four to thirteen. The replacement is less tidy — the count follows the relative movement across a family’s chains, not the average over all neighbours — and it needs the direction of the errors, which a survey of real heads does not usually have.

Readings that would overturn it

A streak moving an organ against its own streak-mates as much as against the rest. A streak changing fewer readings than the patch of the same size and correlation anywhere. A quarter-spacing streak at ρ=0.9\rho = 0.9 changing within a point and a half of what its relative size predicts. Near-golden readings changing more often than the rest, or the census needing a different number of kept specimens. Each is checked on every null above.

Still open: a count weighted by direction

The relative size failed because it averages over directions and the count does not. The natural repair is a relative movement measured the way the count reads it: for each organ, its movement against its neighbour along each of the two families the band reads, taken separately, and a prediction built from the larger of the two, or from both. The measurement is that prediction tested on all three errors at once — independent, patch and streak — asking whether one directional number places every point on the diagonal, and if it does, whether a survey could estimate it from a photograph by comparing how straight each family’s spirals run.

Measured, a count weighted by direction closes a third of the gap: the larger family’s relative movement closes between a quarter and under half of what the all-direction size misses, and a wobble read off a photograph does as well. But the two families are not lopsided on the larger heads, so direction was not all of it, and a strong streak is still a fifth short.

Where this goes next

The same head displaced organ by organ and by a drift along its spirals, the displacements drawn four times their size. A Vogel head of 300 organs at 137.6°, every organ in the band from 0.55 to 0.95 of its radius drawn with a line from where the lattice puts it, four times the length of its displacement. On the left each organ is moved by an independent gaussian of a quarter of a spacing; on the right by a drift of the same size, each organ inheriting 0.9 of the mean error of its two nearest earlier organs. The drift's lines run in the same direction over whole stretches of the band, so neighbours move together: at this correlation an organ moves against its four nearest neighbours by 0.55 of what the independent displacement moves it. The band reads 21/34 independently displaced and 21/34 drifted. A drift along the spirals changes fewer counts A grown head's errors are not independent: an organ placed against the organs already there inherits their errors, so whole stretches of a spiral move together. Laid down that way over the census's Vogel discs, at the same size as the independent displacement measured before, the drift changes fewer of the band's readings, not more — a sixth to a quarter fewer at a quarter of a spacing — never more often near the golden angle, and at most two into a Fibonacci pair. A count reads an organ against its neighbours, and a drift moves neighbours together: at a strong correlation an organ moves against its four nearest by just over half what an independent displacement of the same size moves it, and the readings changed are about what an independent displacement of that relative size changes, plus a little for the coherent part. A head of three hundred organs at 137.6°, exact and displaced, counted in a surveyor's band. A Vogel head of 300 organs at a divergence of 137.6°, the organs between 0.55 and 0.95 of its radius filled, exact and with every organ displaced by a gaussian of 0.25 of a spacing. The band reads exact: 21/34, 0.05: 21/34, 0.1: 21/34, 0.25: 34/68. A quarter of a spacing moves the reading at this angle from the Fibonacci pair 21/34 to 34/68, a pair whose two counts share a factor, not to a near miss; across the null such changes are rare near the golden angle and almost never make a Fibonacci reading. Displacing a disc's organs makes no new near misses A Fibonacci census's null, drawn from Vogel discs counted in a surveyor's band, holds a few tenths of a per cent of heads that a capped recount could move. Every disc was exact, and a real head's organs sit a fraction of a spacing off the lattice, where a band can read a family the lattice holds only marginally. Displaced by a twentieth, a tenth and a quarter of a spacing, the discs' readings do change — at up to one angle in six — and on large heads mostly to a pair the exact band already reads at one of its edges. But near the golden angle they change no more often than anywhere else, almost never into a Fibonacci pair, and the heads within a capped aim's reach move by under six hundredths of a percentage point. The census still needs three kept specimens. The share of the geometry's heads that read as consecutive Fibonacci numbers, on a disc and on the matched cylinder. At every divergence angle on the cylinder census's grid, a Vogel head counted in the band from 0.55 to 0.95 of its radius, against a cylinder at the rise the band's middle stands at. 40 organs: disc 1.23% of 3084 readable angles, cylinder at 0.004 9.97%; 75 organs: disc 0.44% of 3445 readable angles, cylinder at 0.002 6.75%; 150 organs: disc 0.17% of 3562 readable angles, cylinder at 0.001 4.70%; 300 organs: disc 0.08% of 3563 readable angles, cylinder at 0.0005 3.28%; 500 organs: disc 0.06% of 3528 readable angles, cylinder at 0.0003 2.50%. Whorled shares: disc 46.4%, 47.4%, 48.0%, 48.8%, 48.2%; cylinder 39.4%, 41.7%, 43.7%, 45.0%, 45.6%. The disc's null holds a tenth to a fortieth of the cylinder's Fibonacci heads at every size. A disc reads almost no Fibonacci heads by chance A Fibonacci census draws its null from the geometry, and every null so far was a cylinder's: one rise, one dominant pair per divergence angle. Read the same angles as discs, counted in the band a surveyor uses, and the null's Fibonacci share falls by a factor of ten to forty — because nearly all of the cylinder's Fibonacci heads were the pair 1/2 at the smallest angles, a single tight arm that no band reads as 1 and 2. The census then needs three kept specimens. Half of a large head's bands do straddle a transition, but a band that straddles reads one edge's pair or its own middle pair, and the handful of true mixtures are nowhere near a Fibonacci pair.

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BiasCensusClaim testingDisplacementFibonacciHonest limitsMeasurement errorNull modelParastichy pairSample size