A drift along the spirals changes fewer counts

A census of Fibonacci heads needs a null — what a band counts 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. Displacing a disc’s organs then moved every organ by an independent gaussian of a twentieth, a tenth and a quarter of a spacing, and found that displacement changes readings, up to one angle in six on small heads, but not near the golden angle any more often than elsewhere, almost never into a Fibonacci pair, and that the heads a capped recount could move barely change.

Every organ there was moved on its own. That essay ended on why that is not what a grown head does. An organ placed against what is already there is placed relative to its neighbours, so their errors are carried into its position: an error laid down near the centre travels outward along the spirals that pass through it, and a whole family can drift together. A band then sees a family shifted rather than scattered, and a shifted family is exactly what might tip a near-golden band from one pair to its neighbour. The question was whether it does.

A drift laid down with the head #

The discs are the same — at every angle 180·k/3600 degrees, Vogel heads of 75, 150, 300 and 500 organs — and so are the seeds, angle by angle and draw by draw, so a drifted head and an independently displaced one at the same angle start from the same random numbers.

The drift is laid down in the order the head is: organ by organ from the centre out, each organ taking ρ times the mean error of its two nearest earlier organs and adding a fresh gaussian of its own. Those two nearest earlier organs are the ones it sits against, one on each of the two spiral families that cross there, so an error is passed along both. ρ is set at 0.6 and at 0.9. The finished field is then scaled so that its size per coordinate is exactly a tenth or a quarter of a spacing — the same size as the independent displacement it is compared with — so that the only difference between the two is how the errors are arranged, not how large they are.

Each head is counted in the surveyor’s band from 0.55 to 0.95 of its radius and in that band’s three tenths, exactly as before, and as a head displaced before it is counted counted its own. Thirty-two drifted grids, each of 3,599 heads, stand beside the independent ones.

One head, both ways #

The two panels carry the same total displacement, a quarter of a spacing per coordinate, and they look nothing alike. On the left the lines point every way, neighbour against neighbour. On the right they run together in patches: a stretch of the band moved one way, the next stretch another, with the organs inside each patch moved nearly alike. That is what inheriting nine tenths of the neighbours’ error does over a few steps, and it is the arrangement the question was worried about.

The band reads 21 and 34 on both, the lattice’s own pair at this angle. A single head says little, and at this angle neither displacement moved the reading.

How many readings change #

The drift changes fewer readings, not more. At a quarter of a spacing the independent displacement changes 15.8 per cent of the readings on 75 organs, 10.2 on 150, 5.9 on 300 and 4.4 on 500. The drift at ρ = 0.6 changes 14.4, 9.0, 5.6 and 3.6; at ρ = 0.9, 12.1, 7.6, 4.9 and 3.2. That is a sixth to a quarter fewer at the stronger correlation, on every size.

At a tenth of a spacing the three are closer together, as they must be when every one of them is small: 8.2, 5.7, 3.5 and 2.7 per cent independently, and 7.4, 5.1, 3.3 and 2.3 at ρ = 0.9. The weaker drift sits between, never more than a third of a point above the independent displacement anywhere.

So the obvious worry runs backwards. Arranging the same amount of error along the spirals, rather than scattering it, makes the count steadier.

Where along the half turn #

The fewer changes are not spread evenly, and where the drift saves most says what it is saving. On 150 organs at a quarter of a spacing, the independent displacement changes 57.6 per cent of the readings under fourteen degrees, where successive organs nearly stack into arms; 5.4 to 8.5 per cent across the middle of the half turn; 5.0 in the window from 130 to 145 degrees; and 9.8 in the last ten degrees, where successive organs fall on nearly opposite sides and the head is two arms. The drift at ρ = 0.9 changes 54.9 per cent under fourteen degrees — the stacked arms are scrambled whichever way the error is laid down — and then 3.9 to 5.5 across the middle, 2.5 in the window and 3.5 in the last ten degrees.

So the drift’s saving is largest exactly where a head is closest to two arms. There a counter’s chains run between organs laid down one after the other, and the drift has made consecutive organs’ errors alike; an independent displacement scatters the two arms against each other and the drift moves them together. On 500 organs the pattern repeats, with the window around the golden angle the one place the drift does not help: 3.3 per cent there against the independent displacement’s 3.2.

Why a steadier count #

A parastichy count does not read where an organ is. It reads which organs are each other’s neighbours, and in what direction: the families a counter traces are chains of nearest neighbours. Moving a whole patch of the band by the same amount leaves every neighbour where it was relative to the organ beside it, and the chains with it. Only the part of the displacement that differs from one organ to the next can change a chain.

Measured directly — the root-mean-square difference between an organ’s displacement and each of its four nearest neighbours’, inside the band, over thirty heads spread across the half turn — an independent displacement of σ moves an organ against a neighbour by $\sqrt{2}\,\sigma$, as two independent errors must. A drift of the same size moves it by 0.79 to 0.81 of that at ρ = 0.6, and by 0.54 to 0.58 at ρ = 0.9, the larger heads at the lower end. Nearly half of what the stronger drift does to an organ, it does to the organ’s neighbours too.

The count follows the relative size #

That gives a prediction to check rather than a story to tell. If a count reads only the relative part, a drift should change as many readings as an independent displacement whose relative movement is the same — an independent displacement of the drift’s relative size divided by $\sqrt{2}$, read off the independent displacements already measured at a twentieth, a tenth and a quarter of a spacing.

It holds to within two percentage points everywhere. On 150 organs at a quarter of a spacing and ρ = 0.9 the drift moves organs against their neighbours as an independent displacement of 0.14 of a spacing would, which changes 6.9 per cent of the readings; the drift changes 7.6. On 500 organs, 3.0 against 3.2. On 75 organs, where the band holds fewest organs, the gap is largest: 10.3 against 12.1.

The points sit on or above the diagonal, all but one. The drift does a little more than its relative size alone accounts for — up to 1.8 points on the smallest heads, a few tenths on the largest — and that remainder is the coherent part: a patch moved together still moves against the patch beside it, and a band narrow enough to be crossed by a few patches feels their edges. That is the effect the question imagined, and it is real, and it is smaller than the relative steadiness that comes with it.

Near the golden angle #

The neighbourhood a near miss would have to come from is no more disturbed by a drift than the rest of the half turn. Within two and a half degrees of the golden angle, at a quarter of a spacing and ρ = 0.9, the drift changes 7.5 per cent of the readings on 75 organs against 12.1 overall, 3.0 on 150 against 7.6, 2.5 on 300 against 4.9 and 2.0 on 500 against 3.2. Across every size, displacement and correlation, the near-golden share is never more than a percentage point above the whole half turn’s, and usually below it.

The window around the golden angle is among the quietest stretches of the half turn under either kind of displacement. Its organs are as evenly spaced as any angle’s — the first three hundred organs found the golden head’s closest pair the widest of forty-six rivals — and a counter’s chains there have room to absorb a shift. A drift does not find a weakness there that an independent displacement missed.

What the changed readings become #

Most of the readings a drift changes on a large head are a pair the exact band already reads somewhere across its width, as they were independently: at a quarter of a spacing and ρ = 0.9, 62 per cent on 500 organs and 49 on 300, against 57 and 50 independently. The band straddles a transition, and a drift tips it between the two pairs its edges read, as an independent displacement did; it does not reach for pairs the lattice does not hold.

Fibonacci readings made by a drift are as rare as those made independently. No drift turns more than two readings into a Fibonacci pair at any size and displacement. On 75 organs at a quarter of a spacing the weaker drift reads 13 and 21 at 137.15 degrees where the exact band reads 8 and 21, and at 138.00 where it reads 13 and 26; the independent displacement made the same first change at the same angle. On 300 organs at ρ = 0.6 the 21/55 head at 137.45 degrees reads 21 and 34 — the near miss the independent displacement also made, on the same head from the same seed. The stronger drift makes only one of the three, the 138.00-degree head on 75 organs.

A changed reading belongs to the draw #

Each null is drawn twice, from two independent seeds, and the two draws say how much of a change belongs to the angle and how much to the particular errors. On 150 organs at a quarter of a spacing, of the angles whose reading the stronger drift changes in either draw, 37 per cent change in both; independently displaced, 39 per cent. On 500 organs, 37 and 32 per cent. About two thirds of the changes, under either arrangement, are one draw’s errors at one angle, and the third that recur are mostly the stacked arms and the bands that straddle a transition.

That is worth knowing before reading anything into a single changed head, as the 137.45-degree near miss shows: it is made by one seed and not the other, under both kinds of displacement. A census reads a population of heads, and how often a head is Fibonacci is a share, not a verdict on any one of them.

The stacked arms, both ways #

The exact bands refuse a reading at a few small angles, where the organs stack into arms a counter cannot trace, and displacement lets some of them be read: the independent displacement unlocks 73 such heads over the two draws on 150 organs and 133 on 500, every one under six degrees. The stronger drift unlocks 71 and 89 — fewer on the large head, because moving consecutive organs together keeps an arm an arm — and it also does something the independent displacement almost never does: it makes seven heads on each size unreadable that the exact band read, all under seven degrees, by bending an arm until no chain of nearest neighbours runs cleanly along it. None of these heads is within a hundred and twenty degrees of the golden angle, so none of them is a head a capped aim could move; the capped aim’s reach below counts them all and is not moved by them.

The heads a capped aim can reach #

The question that matters for the census is whether the heads a capped recount could move grow, since those are the near misses that would count against a Fibonacci claim. They do not. On 150 organs the share within a capped aim’s reach is 0.281 per cent exact, 0.333 independently displaced, 0.278 under the weaker drift and 0.292 under the stronger. On 300, 0.168 exact and 0.153 under both drifts; on 500, 0.142 exact and 0.125 or 0.126 drifted. Every drifted share is within six hundredths of a percentage point of the exact one, and the census needs three kept specimens under every null.

So the census as it would be run on grown heads, with a recount aimed where the counter expects — whose errors, if they look like either of these drifts, are arranged along the spirals — has the null the exact disc gave it, to within a few heads in seven thousand.

The null’s Fibonacci share #

The null’s own Fibonacci share — the fraction of non-designed heads that read a consecutive Fibonacci pair anyway — moves by a head or two. On 150 organs it is 0.168 per cent exact, 0.195 at a quarter of a spacing independently, and 0.181 and 0.167 under the two drifts. On 500 organs, 0.057 exact and 0.056 under both drifts. The stronger drift leaves the null slightly nearer the exact disc’s than an independent displacement does, which is the steadiness above seen from the census’s side.

What a stand-in for growth can say #

The drift is not a grown head. It is an error laid down with the head and passed along both families at each organ, at two strengths, chosen because it has the property the question was about — a family moving together — and nothing else of growth. A placement rule’s errors could be correlated over longer stretches, or along one family more than the other, or tied to where the rule hesitates between two positions. None of that is drawn here.

What the measurement does establish is the direction. For any arrangement of errors, the count reads the part that moves an organ against its neighbours, and correlation along the spirals shrinks that part. A grown head whose errors are more coherent than independent noise of the same size is a steadier head for a counter, not a less steady one, unless its coherence comes in patches about the size of the band.

A drift that changes more would undo it #

A head size and displacement at which the stronger drift changes more readings than the independent displacement of the same size. A drift changing near-golden readings more often than the rest by more than a point. A drift turning three or more readings into a Fibonacci pair. A capped aim’s reach moving by a tenth of a point. Each is checked against every null drawn above.

Still open: errors that run along one family only #

Each organ here inherits from both of its two nearest earlier organs, so its error travels along both families at once, and the patches it makes are round. A placement rule weighs its neighbours unequally — a new organ sits closer to one of the two it touches — and its errors could travel along one family and not the other, making long thin streaks rather than patches. A streak along one family moves that family’s chains together and shears them against the other family’s, which is a different thing to a counter.

The measurement is the same null with each organ inheriting from its nearest earlier neighbour alone: whether errors streaked along one family change more readings than patches do, whether they do so near the golden angle, and whether the relative size still predicts the count once the relative movement is all across one family’s chains.

Measured, a streak is a shear the count can see: inheriting from the nearest earlier organ alone, the error moves an organ against its four nearest neighbours exactly as much as the two-parent drift does, yet changes more readings in every draw of every cell — on large heads as many as an independent displacement, or more — because its movement is nearly all across its own streak. The relative size underestimates it by up to three points; near the golden angle it does no more, and the census still needs three kept specimens.