Displacing a disc's organs makes no new near misses

A disc reads almost no Fibonacci heads by chance drew a Fibonacci census’s null the way a surveyor would meet it: a Vogel disc placed at every divergence angle on a fine grid, counted in a band from 0.55 to 0.95 of its radius. The null held a tenth to a fortieth of the Fibonacci heads a cylinder’s had, the census needed three kept specimens, and the heads a capped recount could move — a near-golden pair that a consecutive Fibonacci pair is read as by a closing error within twice the counter’s spread — were a few tenths of a per cent of the null, the same few pairs the cylinder had.

Every disc in that census was exact, each organ precisely at its Vogel position. No head is. A head that was grown has its organs placed against their neighbours, a fraction of a spacing from where the formula would put them, and a head that was photographed carries the photograph’s errors on top. The question that essay left was whether those fractions of a spacing matter to the census. A head displaced before it is counted had already shown that a displaced head’s counts change where an exact one’s would not. A band straddling a family the lattice holds only marginally might then read that family, and if a near miss is going to come from anywhere, it is from there.

So the null is drawn again, from the same discs with every organ moved.

The same discs, moved #

The heads are the same: at every angle 180·k/3600 degrees, a Vogel disc of 75, 150, 300 or 500 organs. Each organ is then displaced by a gaussian of a twentieth, a tenth or a quarter of a spacing — the unit every displaced head here has used — on a seed set by the angle, and the whole grid is drawn twice, with two independent seeds, so that 7,198 heads stand behind every size and displacement. Each head is counted in the surveyor’s band and in that band’s innermost, middle and outermost tenths, exactly as the exact heads were, so that the two can be set side by side angle by angle.

The census arithmetic is unchanged: closing errors spread over 7.2° and correlated at 0.9 between a head’s two counts, one recount of an announced reading, and an alternative of ninety per cent of grown plants at 34/55. Only the null is new.

One head at 137.6 degrees #

At 137.6 degrees a 300-organ head’s band reads 21 and 34 — a Fibonacci pair, near the golden angle — and it still does with every organ moved by a twentieth of a spacing and by a tenth. Moved by a quarter in one of the two draws, it reads 34 and 68: the two counts share a factor of 34, a reading the census treats as announced and recounts. A quarter of a spacing has scrambled the band’s inner rows enough that the counter’s shortest families are no longer the lattice’s.

That is the kind of change the question was about, and it went the other way: from a Fibonacci reading to one no capped aim could move. One head says little. The null says the rest.

How many readings change #

Displacement changes readings, and more than a twentieth of a spacing might suggest. At a twentieth it changes the band’s pair at 6.0 per cent of the angles on a 75-organ head, 4.7 on 150, 3.0 on 300 and 2.1 on 500. At a tenth, 8.2, 5.7, 3.5 and 2.7 per cent. At a quarter, 15.8, 10.2, 5.9 and 4.4 per cent — one angle in six on the smallest head, one in twenty-three on the largest. A large head is steadier because its band holds more organs, and the count is a statement about many of them at once.

Near the golden angle, within two and a half degrees either side, the same displacements change fewer readings or about as many, never noticeably more: at a quarter of a spacing, 9.5 per cent of those readings on 75 organs, 3.0 on 150, 5.0 on 300 and 4.0 on 500. Across every size and displacement the near-golden share is never more than a percentage point above the share everywhere else. The neighbourhood where a near miss would have to be made is not where displacement does its work.

Where the changes fall #

The changes are not spread evenly over the half turn, and where they gather says what they are. On 150 organs displaced by a quarter of a spacing, 58 per cent of the readable angles under fourteen degrees change their reading — the stacked arms again, which a quarter of a spacing scatters into whatever the counter can trace. Between fourteen and sixty degrees 5.4 per cent change; between sixty and a hundred, 7.1; a hundred to a hundred and thirty, 8.5; a hundred and thirty to a hundred and forty-five, the window around the golden angle, 5.0; then 6.6 per cent, and 9.8 in the last ten degrees before a half turn, where successive organs fall on nearly opposite sides and the head is two arms rather than one.

On 500 organs the pattern is the same and quieter: 17.8 per cent under fourteen degrees, 2.6 to 4.2 per cent across the middle of the range, 3.2 in the window around the golden angle and 6.3 near a half turn. The angles where a head is closest to a degenerate arrangement — a single arm, or two — are where a quarter of a spacing does most, and the window a near miss would come from is among the quietest stretches of the half turn on both sizes.

A tenth of a spacing, the size every photograph here carries #

A tenth of a spacing is not an arbitrary middle value. It is the displacement every golden head in the photographed-head essays was given before it was photographed. At that size the census’s null barely notices: readings change at 8.2 per cent of the angles on 75 organs and 2.7 on 500, the near-golden window changes at ten, two, seven and three readings in two hundred, and not one changed reading is a Fibonacci pair on any head of 150 organs or more.

So the census as it would actually be run — on heads displaced about that much, counted in the surveyor’s band — has the null the exact disc gave it, to within a few heads in seven thousand. The quarter of a spacing is there to find where that stops being true, and it does not stop.

What the changed readings become #

The question imagined a displaced band reading a family the lattice holds only marginally, and on large heads that is exactly what happens. A band from 0.55 to 0.95 of the radius crosses a range of rises, and half of a large head’s bands straddle a transition with their two edges reading different pairs. Displaced, such a band tips from one of those pairs to the other. At a quarter of a spacing, 57 per cent of the changed readings on 500 organs are a pair the exact band already reads in its innermost, middle or outermost tenth, and 50 per cent on 300; at a tenth, 60 and 39 per cent. On 150 organs it is under a fifth, and on 75 nothing can be said, since tenths that narrow cannot be counted on a head that small.

The rest are other pairs, and nearly none of the changed readings is a Fibonacci pair: none, one or two in each size and displacement, against 146 to 1,085 changes each. About a quarter share a factor, like the 34/68 above — counts the census never keeps without a recount — and the remainder are ordinary spiral pairs far from golden. Displacement mostly trades one non-Fibonacci reading for another.

The rare reading that becomes Fibonacci #

Two of the changes near the golden angle are worth naming because they are the shape the question feared. On a 300-organ head at 137.45 degrees the exact band reads 21 and 55 and, displaced by a quarter of a spacing in one draw, 21 and 34: a reading off the Fibonacci sequence has become one on it. On 150 organs at 137.95 degrees a tenth of a spacing moves 13 and 47 to 13 and 34, two Fibonacci numbers that are not neighbours in the sequence, which the census does not count as a Fibonacci pair. The first is a real near miss made by displacement, and it is nearly the whole of its kind: of the 7,198 heads behind each size and displacement, at most two change into a Fibonacci pair.

The null’s Fibonacci share moves by about that much. On 150 organs it is 0.168 per cent exact and 0.195 at a quarter of a spacing; on 300, 0.084 and 0.111; on 500, 0.057 and 0.070; on 75, 0.43 and 0.52 — in each case one to six heads of seven thousand, and at a twentieth and a tenth of a spacing by no more.

They matter to the census less than they seem to, for a reason the census arithmetic already contains. A displaced head that reads 21/34 is a head the null counts as Fibonacci, and the null’s Fibonacci share is what the grown plants have to beat. At a share this small, one or two heads in seven thousand move it by a few hundredths of a per cent.

The heads a capped aim can reach #

The capped aim, which the recount essay found to be the honest way to aim a recount, moves a reading only when a consecutive Fibonacci pair, misread by a closing error within twice the counter’s spread, rounds to it. On the exact discs its reach was 20/33 and 22/35 on 150 and 300 organs and 33/53 and 35/57 on 500: 0.28, 0.17 and 0.14 per cent of the null. Displaced by a quarter of a spacing, the reach is 0.33, 0.18 and 0.13 per cent. At a twentieth and a tenth it is within three hundredths of a point of the exact value either way.

Displacement adds two pairs to the reach, and both are rare. On 150 and 300 organs a quarter of a spacing makes 57/92 appear, a near-golden pair that 55/89 rounds to under a closing error of 3.4 per cent; it is two heads of 7,188 on 150 organs and one on 300. On 300 organs a tenth of a spacing makes one head read 33/53. Every change in the reach, at every size and displacement, is under six hundredths of a percentage point.

The census needs three specimens still #

The census follows. Unaimed, the share of the null’s kept specimens reading Fibonacci by chance is 0.73, 0.24, 0.12 and 0.06 per cent on exact discs of 75, 150, 300 and 500 organs, and 0.86, 0.30, 0.16 and 0.08 displaced by a quarter of a spacing. Capped, 0.73, 0.28, 0.14 and 0.10 exact, and 0.86, 0.35, 0.19 and 0.12 displaced. Aimed without a cap, the counter the recount essay found doing the damage, 5.4, 3.5, 2.3 and 1.8 exact and 5.2, 3.4, 2.3 and 1.6 displaced: the uncapped aim’s damage is set by the whorled heads it reads as Fibonacci, and displacement does not add whorls.

At every size and every displacement, unaimed and capped, the census needs three kept specimens — where counted on a cylinder with every count right it needed four. Displacement has raised the chance of a Fibonacci reading by a few hundredths of a per cent on a null so small that three specimens out of three were already well beyond it.

Two draws of the same displacement #

Every number above pools two independent draws, and the draws are worth reading apart, because they say how large a difference has to be before it is a difference. At a quarter of a spacing on 150 organs the capped aim’s reach is 0.334 per cent in one draw and 0.333 in the other, and the null’s unaimed Fibonacci share among kept specimens 0.285 and 0.308. On 300 organs the reach is 0.195 and 0.167 and the share 0.194 and 0.131; on 500 organs 0.111 and 0.139, and 0.100 and 0.063.

So on the two larger heads the two draws of one displacement differ by as much as displacement moves the pooled numbers from the exact ones — the 500-organ census share goes from 0.063 exact to 0.081 pooled while its two draws sit at 0.063 and 0.100. A tenth of a spacing is quieter still: 0.278 and 0.278 on 150 organs, 0.167 and 0.167 on 300. What displacement does to the census is the size of the difference between two heads with the same displacement, which is to say it is the scatter of a few heads in seven thousand, and both draws need three kept specimens at every size.

The angles displacement unlocks #

The exact discs refused to be counted at a handful of angles — 74 of 7,198 heads on 150 organs, 142 on 500 — and every refusal sat under fourteen degrees, where successive organs nearly stack into a few tightly wound arms and a band holds no two families a counter can trace. Displacement unlocks them. At a quarter of a spacing the refusals fall to one on 150 and 300 organs and ten on 500, and the band reads the rest.

What it reads there is noise: a quarter of a spacing scatters a tight arm into a cloud the counter traces as small, arbitrary pairs — 3 and 12, 4 and 6, 2 and 65 — of which, at every size and displacement, at most two are Fibonacci and about two in five share a factor. Every angle displacement unlocks lies under thirteen degrees. These heads enter the null, which slightly enlarges its denominator, and nowhere near the golden angle.

What it says about the marginal family #

The intuition behind the question was sound as far as it went. A band that straddles a transition holds two families near its edges, and displacement can tip its reading from one to the other — which on large heads is the commonest thing displacement does. What it missed is that tipping is not making. A band tipped between two families it already held reads one of them, and the pairs it held were the exact head’s own — so a tip near the golden angle reads a pair the exact null already counted at a neighbouring angle, not a new one. Why near-golden bands tip no more often than others is measured here and not traced; the counts say only that they do not, at any size or displacement read.

So a displaced disc’s null is the exact disc’s null with its noisiest angles shuffled. The near misses that a capped aim could move were made by the lattice’s own geometry at a handful of angles, and they stay at those angles, at the same few tenths of a per cent.

A gaussian is not a grown head #

Every displacement here is a gaussian drawn independently for each organ. A grown head’s errors are not independent: an organ placed against its neighbours inherits their errors, so a grown head’s displacement runs along its spirals rather than scattering, and a correlated error could tip a whole band’s family at once in a way independent errors do not. Nothing here grows a head; a placement rule would. And the band is the surveyor’s band of the earlier essay, 0.55 to 0.95 of the radius, a stated choice.

Nor does it settle what the census’s null should be. A divergence angle drawn evenly from a half turn is a model of a plant that is not Fibonacci, and a displaced disc is that model with noise, not a better model.

Readings that would undo it #

A size and displacement at which readings near the golden angle change more than a percentage point more often than elsewhere. More than two changes into a Fibonacci pair at any size and displacement. A change in the capped aim’s reach of six hundredths of a point or more, or a census needing more than three kept specimens. A refusal, or an unlocked angle, above fourteen degrees. Each is checked against the displaced discs whenever they are read.

Noise shuffles the noisy angles #

Displacing every organ of a Vogel disc by up to a quarter of a spacing changes the surveyor’s reading at one angle in six on a small head and one in twenty-three on a large one, mostly by tipping a straddling band between families it already held. Near the golden angle it changes no more readings than elsewhere, and turns at most two into a Fibonacci pair. The heads a capped aim can reach stay at a few tenths of a per cent, within six hundredths of a point, and the census still needs three kept specimens.

Still open: a disc whose errors run along its spirals #

Every displacement here is independent from organ to organ. A grown head’s error is not: an organ placed against what is already there carries their errors forward, so a whole spiral can drift together, and a band then sees a family shifted rather than scattered. The next measurement draws the null from discs displaced by an error correlated along each parastichy, at the same total size as the gaussians here, and asks whether a correlated drift tips near-golden bands between Fibonacci and non-Fibonacci families where an independent one does not — the place the census’s near misses would come from if displacement makes any at all.