A chosen mark errs by a third of a spiral

Two marks chosen by one eye let each family of a reported pair be closed at a mark of its own, the two closing errors drawn from a normal distribution with a spread of a few degrees and a correlation $\rho$ between them. Two findings came out of it. The pair 34/55 loses its ten-degree margin to marks that err on opposite sides, because 33/56 and 35/54 share no factor; and the second annulus, the check that reads the next pair out and asks that its counts add up, tells right readings from silent ones only while $\rho$ stays below about 0.98.

Both the spread and the correlation were assumptions. A careful counter does not draw a starting organ at random. They pick one that can be found again, usually because it sits near something fixed — the top of the photograph, a pin in the rim, the edge of a bract — and that choice is where the closing error comes from. That essay ended by asking for the mark modelled as a choice: start each family at the organ nearest one fixed place, and ask whether the correlation that produces is a property of the head rather than a habit of the counter.

A count that closes on a place #

The model is the simplest version of how a count by hand runs. The counter fixes a place on a circle around the head, inside the band of radii where the pair being counted are the two shortest families of spirals — the band how many organs a pair needs and counting the spirals work inside. For each family they start on the spiral through the organ nearest that place, run round the circle counting spirals as they cross it, and stop at the spiral whose crossing is nearest the place again.

If the organ sat exactly on the circle and exactly at the place, that count would be right every time. It sits within half an organ spacing of the place in some direction, so its spiral crosses the circle a little way off. Counting from that crossing round to the place, the counter is off by the gap between them measured in spiral spacings, rounded. When the gap is under half a spacing, the count is right; over half, it is one out. That gap is exactly the closing error the drawn-mark model sampled from a normal distribution — the share of a turn by which the count overshoots or falls short — and here it is computed rather than drawn, at six thousand places spread evenly through the band and evenly round the head.

Both families start on one organ #

The detail that decides the correlation is that both families start on the same organ. One place, one nearest organ: the 34-family’s spiral and the 55-family’s spiral both pass through it. So the two closing errors share whatever the organ’s angular offset from the place is — if the organ sits a little clockwise of the place, both spirals start a little clockwise — and that shared part pushes the correlation towards one.

But the organ also sits a little inside or outside the circle, and its two spirals carry that radial offset to the circle in opposite directions. The 34-family winds one way and the 55-family the other — the two families of any visible pair run in opposite senses, which is what makes them two of the head’s neighbour directions rather than one — so an organ outside the circle has one spiral reaching the circle clockwise of it and the other anticlockwise. That part of the error is shared with the opposite sign, and it pushes the correlation towards minus one. Which part wins is a question about the lattice’s shape at that radius, and it can be measured.

Three tenths of a spacing, in every family #

Measured in each family’s own spiral spacing, the closing error has the same spread everywhere: between 0.292 and 0.301 of a spacing, in all eight families of the four pairs. The largest single error seen in forty-eight thousand closings is 0.72 of a spacing. A chosen mark is never a whole spiral out, and it is never two.

So in degrees round the head the error falls as one over the count. On the 13-family it is 8.3°, on the 21-family 5.1°, on the 34-family 3.2° and on the 55-family 1.9°. The drawn-mark essays read every pair at a spread of 7.2°, which a chosen mark matches only on the lowest pair. On 34/55, where those essays did most of their arithmetic, a chosen mark is more than twice as accurate as was assumed — not because the counter is more careful at high counts, but because a spiral spacing at a high count is narrow, and the organ nearest a place is always within half an organ of it.

What the geometry alone reads #

With no error of the eye at all, a chosen mark reads 34/55 right 91.1 per cent of the time. Nearly every wrong reading has one count wrong and the other right: 33/55, 35/55, 34/54 and 34/56, each about two per cent. Both counts wrong together happens twice in six thousand places. And every one of the four single miscounts shares a factor — eleven, five, two and two — so it announces itself, as a count that can be wrong by one showed it must for this pair: 34/55 is one of the Fibonacci pairs that cannot be miscounted silently by one count.

One reading in six thousand is silently wrong, and it is 33/56: both counts wrong, on opposite sides. That is the reading the drawn-mark essay found costs 34/55 its margin, and here it is, from the geometry alone, at a rate of under two in ten thousand. The same holds for 21/34, whose single miscounts also all share factors: not one of six thousand readings is silently wrong.

Where the protection runs out #

The pair above has no such protection. Of 55/89’s single miscounts, 55/88 and 55/90 share factors, but 54/89 and 56/89 are coprime, because 89 is prime and any count beside 55 that is not a multiple of it shares nothing with it. So a chosen mark reads 55/89 silently wrong 4.7 per cent of the time from geometry alone, while getting it right 91.4 per cent. The 13/21 head is in between: 12/21 and 14/21 announce themselves, 13/20 and 13/22 do not, and 3.8 per cent of its readings are silent.

This is the arithmetic of what a count is worth and its successors, unchanged. What the chosen mark adds is the rate: about nine readings in a hundred are one count out, almost none are two counts out, and so whether a pair is protected depends entirely on whether its single miscounts share factors.

The correlation is set by where the place is #

Over a pair’s whole band the two families’ errors are anti-correlated: −0.27 on 13/21, −0.20 on 21/34, −0.16 on 34/55, −0.13 on 55/89. But that average hides the real dependence, which is on where in the band the place sits. At the band’s inner edge the correlation is positive, between 0.22 and 0.39; it crosses nought a fifth to two fifths of the way out; and at the outer edge it is about −0.5 on every pair. The four pairs lie on very nearly one curve.

That is the two parts of the error trading places. At the inner edge the shared angular offset wins and the errors move together; at the outer edge the radial offset, carried by the two spirals in opposite directions, wins and they move apart. Why the trade runs that way across the band — what changes in the spirals’ slope against the circle between one edge and the other — is not measured here; that it runs the same way on all four pairs, at nearly the same fraction of the band, is. So the correlation the drawn-mark model needed is not a property of the counter, nor of the pair. It is a property of the radius at which the head is counted, and it can be read off the lattice.

And so is the error rate #

The miscount rate follows the same curve. It is 6.5 to 11.8 per cent at the inner edge of a band, 15 to 17 per cent at the outer, and it falls to its least about a third of the way out: 1.1 per cent on 34/55 and 55/89, 2.6 on 21/34, 2.8 on 13/21. The least sits where the correlation crosses nought.

The same spread, a different shape #

What changes across the band is not the size of the error. Read slice by slice, its spread is 0.29 to 0.31 of a spacing everywhere — on 34/55, 0.292 at the inner edge, 0.289 a third of the way out, 0.312 at the outer. What changes is its shape. A third of the way out on 34/55 the largest error in three thousand places is 0.511 of a spacing, and a spread of 0.289 is exactly what an error spread evenly over one spacing has: the crossing of the starting organ’s spiral falls anywhere within half a spacing of the place and almost never beyond it, so the count almost never rounds off. At the inner edge the largest error is 0.62 and at the outer 0.70, with the same spread, which means the errors pile nearer nought and send a tail past half a spacing — and every closing in that tail is a miscount. On 13/21 the same reading gives 0.57 at its best slice and 0.76 at the outer edge.

So a chosen mark has a radius at which its error is, very nearly, as even as an error can be — and at that radius it is right because it cannot be more than half a spiral out, not because it is small.

That turns into advice a counter could use. On a head that shows one pair over a broad band of radii, counting a third of the way out from where the pair first appears gets a chosen mark right ninety-nine times in a hundred, and counting at the rim, where the next family is about to take over, gets it right only about five times in six. The window nobody varied found the radius a pair is read at to be a choice with consequences for which pair is read; this is the same choice with consequences for whether that pair is read right.

The eye’s own error #

A real counter does not find the place perfectly every time. Suppose the eye misplaces it by a normal error with some spread, the same error for every trace that closes on it — one place, judged once. That error is shared by both families with the same sign, and it adds a positive part to their correlation of whatever size the eye allows.

Measured, the eye takes over quickly. On every pair the correlation rises steadily with the eye’s error, crosses nought when that error is between six and twelve hundredths of a spacing of the smaller family, and climbs towards one once it passes about a third of a spacing. In degrees the eye needed is smaller the higher the pair: 3.6° turns 13/21’s correlation to +0.03, 1.8° turns 34/55’s to +0.24, and 0.9° turns 55/89’s to +0.16. At the 7.2° the drawn-mark essays used, the correlations are 0.42, 0.70, 0.87 and 0.95.

Measured in spacings, the four curves nearly coincide, which is the answer to the question the earlier essay ended on. The correlation is set by two lengths, both measurable on a photograph: the eye’s error in finding the place, and the spacing of the smaller family’s spirals at the counting radius, whose three tenths is the geometry’s own contribution. A survey that recorded how far out each count was made and how reliably the counter found the same place twice could compute its correlation head by head rather than assume one.

The second annulus under chosen marks #

The second annulus is the same procedure on a larger circle, in the band of the next pair: the counter reads the larger family again and the sum family, closing every trace on the same place round the head, and accepts the first reading only if the second repeats its larger count and makes the other the sum. With the geometry alone it passes 91 per cent of right readings and none of the silent readings of 13/21 or 34/55 — there are almost none to pass — and 3.6 per cent of 55/89’s silent ones.

With an eye error the silent readings arrive, and the relation’s worth falls fast. At 3.6° the relation passes right readings 8.5 times as readily as silent ones on 13/21, 7.6 times on 21/34, 2.4 times on 34/55 and 1.65 on 55/89. At 7.2° those are 2.0, 1.55, 1.27 and 1.33. The high pairs lose the relation first, because at a given eye error in degrees their correlation is higher, and a correlated pair of errors is the kind the relation passes.

Whether the drawn model can be fed the head’s numbers #

The useful question for a survey is whether the drawn-mark model, given the spread and correlation measured off the head, predicts what the head does. It is a model with two numbers in it; if those two are enough, a survey can print them beside every count and use the earlier essay’s tables.

Fed each pair’s spread and correlation at each eye error, the model predicts the share of exact readings to within two hundredths everywhere. It predicts the silent share to within about two hundredths at large eye errors and less well at small ones. And it predicts the relation’s worth to within a third at 7.2°, where the eye carries most of the error — 2.17 against 2.03 on 13/21, 1.32 against 1.27 on 34/55 — but at 3.6°, where the geometry still carries much of it, it misses by up to a factor of two, in both directions: 11.5 against 8.5, 4.0 against 7.6, 4.5 against 2.4. A chosen mark’s error is not normal — it is bounded at about seven tenths of a spacing and its two parts are shaped by the lattice — and the relation reads the tails of the joint error, which is exactly where two numbers stop being enough.

What a chosen mark changes #

Three things move. The spread is not a property of the counter but of the count: three tenths of a spacing, so the high pairs are closed more accurately in degrees than the low ones without anyone trying. The correlation is not a property of the counter either, until the eye’s error passes about a tenth of a spacing; below that it is set by the counting radius and runs from positive to negative across a pair’s band. And the readings are nearly all single miscounts, which makes two counts that slip together and the drawn-mark essay’s two-count failures rare unless the eye is poor.

What does not move is the arithmetic. Which single miscounts announce themselves is a property of the pair, and a chosen mark only decides how often each is made. 21/34 and 34/55 stay protected; 13/21 and 55/89 stay exposed.

What this reading assumes #

A Vogel head at the golden angle, exact, with organs at radius $\sqrt{k + \tfrac12}$. A counter who closes each trace on a place, not on the organ — a pin in the starting organ itself removes the geometric error entirely, and then only the eye’s error is left. One place for both families and both annuli. An eye error shared by every trace, normal, of one spread. Places spread evenly through each pair’s band; a counter who prefers the rim would see the outer-edge numbers. A real head’s organs sit off the lattice, which adds to the offsets this reads, and a head displaced before it is counted measured how much that moves a band’s reading.

Readings that would overturn it #

A family whose chosen-mark error, in its own spacings, departs from the others’ by more than a fifth. A pair whose two families’ geometric errors are positively correlated over its whole band. A silent reading of 21/34 or 34/55 from the geometry alone at more than one in a thousand, or one whose two counts err on the same side. An eye error that lowers the correlation. Each is checked on the readings above.

Still open: a pin rather than a place #

The counter modelled here closes on a place and lets the organ fall where it does. The careful counter closes on the organ — a pin through the starting floret, a dot of ink — and then the geometry contributes nothing and the only error is losing the organ on the way round. That error is not a gap at the closing but a slip partway round the circle, where a spiral is skipped or counted twice as the eye moves between neighbours, and it should depend on how far apart the spirals are where the eye crosses them rather than on where the trace began. The measurement is a trace that follows each family round from a pinned organ with a stated chance of slipping at each crossing, set by how close the neighbouring spirals are there: whether slips are commoner on one family than the other, whether they ever move both counts, and whether a pinned count at the rim is better or worse than a placed count a third of the way in.

Measured, a pinned count errs where the spirals crowd: the larger family slips about four times as often, both counts move together only by coincidence, and a pin at the band’s outer edge beats a place a third of the way in only for an eye slipping less than once in 2,400 crossings at an organ’s spacing.