Two counts that slip together

A reported parastichy pair pins a plant’s divergence angle to a band about $221°/mn$ wide, which what a count is worth measured and which makes a report of 34 and 55 good to a little over a tenth of a degree. Two essays then asked what the report is worth when a count is wrong. A count that can be wrong by one found that a high count which is wrong is moved rather than blurred, and that two Fibonacci pairs in every six cannot be miscounted silently by a single error of one. A count that drifts by two found that the protection belongs to a drift of one only, and that the check which survives a drift of any size is a second annulus whose pair must continue the first.

In both of those readings the two counts erred independently: the 34-family could be miscounted while the 55-family was counted right. That is the convenient case, and it is probably not the common one. The two families are counted on the same head, by the same person, round the same circle, and many of the ways of getting one of them wrong get the other wrong too. This essay reads the same reports with the two errors coupled.

Two ways for both counts to move #

The first coupling is the one the drift essay named at its close. A counter who skips a spiral where the head is crowded skips the spiral crossing it as well, so both counts move by the same whole number: 34/55 becomes 33/54 or 35/56, and a slip of two makes it 32/53 or 36/57. Call that an equal shift.

The second is less obvious and probably more common. To count a family, a counter follows the spirals round the head from a starting mark and stops on returning to it. Stopping a little late means a sliver of the head has been counted twice, and the sliver holds spirals of both families in proportion to their counts. A counter who runs past the mark by a fraction $\varepsilon$ of a turn counts $34(1 + \varepsilon)$ and $55(1 + \varepsilon)$, rounded. Stopping early makes $\varepsilon$ negative. Call that a closing error. A mark misjudged by ten degrees is $\varepsilon$ of about 0.028.

The two models disagree about what a small error looks like. An equal shift moves both counts at once by a whole spiral. A closing error moves the larger count first, because 55 spirals cross a given sliver more often than 34 do: the 55-family gains a spiral once the overrun passes half a spacing of that family, a hundred-and-tenth of a turn, while the 34-family needs a sixty-eighth. Both are read below exactly, the second as a sum over the intervals of $\varepsilon$ on which the rounded pair does not change.

The difference decides every equal shift #

Whether a shifted pair shares a factor has a short answer. The greatest common divisor of $m + k$ and $n + k$ is the greatest common divisor of $m + k$ and $n - m$, since subtracting one number from the other changes neither’s divisors in common. So an equal shift is decided by the report’s difference, and on a Fibonacci report the difference is the Fibonacci number below the smaller count: 21 for 34/55, 13 for 21/34.

At 34/55 the shifts of one are 33/54 and 35/56. The first shares 3 and the second 7, both of them factors of 21, so a counter who skips a spiral pair at 34/55 reports a pair that announces itself as whorled. The shifts of two split: 32/53 shares nothing and reports 33.9°, while 36/57 shares 3. The shifts of three, 31/52 and 37/58, both pass silently, at 34.7° and 68.2°.

At 21/34 the difference is 13, a prime, and none of the six shifted smaller counts from 18 to 24 is a multiple of it, so every equal shift passes. Most land far from the truth — 20/33 reports 54.4°, 22/35 reports 82.1° — but a shift of three down, 18/31, reports 139.55°, only 1.73° from the edge of 21/34’s own band. That is the same pair the drift essay found nearest the truth at a drift of three with the counts moving independently, reached here by a different route.

The Lucas reports are worse, as they were for independent errors. 18/29 has a difference of 11 and every shift of 18/29 passes silently, while 11/18 announces only its shift of three up, 14/21, which shares 7.

Fourteen pairs in twenty-three #

The same question can be asked all the way up the Fibonacci sequence, since a difference is all it needs.

Of the twenty-three Fibonacci pairs from 3/5 to 121393/196418, seven admit no silent single miscount of one: the protected pairs of the one-step reading. Fourteen admit no silent equal shift of one. A counter whose two counts slip together is therefore protected at twice as many pairs as one whose counts slip apart.

From 21/34 upward the pattern repeats every six pairs without exception: two silent shifts at 21/34, none at 34/55, 55/89 and 89/144, two at 144/233, none at 233/377, and then the same again from 377/610. Four pairs in every six are protected. The independent miscount’s pattern repeats every six pairs too, with two protected in each, and the two patterns only partly overlap: 34/55 is protected against both kinds of error, 21/34 only against an independent miscount, and 55/89 only against an equal shift.

A period of six in the Fibonacci numbers usually means divisibility, and that is what it means here.

Cassini’s identity and the doubling formula #

Two identities account for every protected pair.

The first is Cassini’s: the product of the Fibonacci numbers either side of a third differs from its square by exactly one, with the sign alternating. Applied to a report with smaller count $m$ and difference $d$, it says that $m^2$ is one more or one less than a multiple of $d$, and that the sign flips from each report to the next. At 34/55, $34^2 = 1156 = 55 \times 21 + 1$. At 21/34, $21^2 = 441 = 34 \times 13 - 1$.

When the sign is plus, $d$ divides $m^2 - 1$, which is $(m - 1)(m + 1)$, so every prime of the difference divides one of the two neighbours of $m$. Whether both neighbours get one depends on whether the difference has two primes to hand out, and the second identity settles it. On the reports where the sign is plus and the difference is odd, the difference is a Fibonacci number of even index, and those factor as $F_{2j} = F_j L_j$ — a Fibonacci number times a Lucas number. At 34/55 the difference is $21 = 3 \times 7$, and 3 divides 33 while 7 divides 35. At 89/144 it is $55 = 5 \times 11$, and 11 divides 88 while 5 divides 90. All six such reports from 8/13 upward split exactly that way. The one exception is at the foot of the sequence: 5/8’s difference is 3, which is $F_2 L_2 = 1 \times 3$, and a factor of one protects nothing, so 5/8 keeps one silent shift.

When the difference is even, which happens every third report, $m$ is odd and both its neighbours are even, so both shifts share a factor of two whatever the sign: 55/89 is protected that way, with a difference of 34. When the sign is minus and the difference is odd, as at 21/34, neither neighbour of $m$ shares anything with $d$, and both shifts pass. Plus-and-odd and even together make up four reports in every six, which is the period the count found.

A closing error overcounts a sliver #

The closing error is a different kind of object, because it is continuous, and its first effect is on the larger count alone. At 34/55 the reading is exact for any overrun or underrun of less than a hundred-and-tenth of a turn, 3.27°. Past that the 55-family gains or loses a spiral while the 34-family does not, and the reading becomes 34/54 or 34/56 — both even, both announcing themselves.

Further out the 34-family moves as well, and the readings become 33/54 and 35/56, which share 3 and 7: the equal shifts of the previous section, reached now by overrunning a sixty-eighth of a turn. Only at an overrun of three hundred-and-tenths of a turn, 9.82°, does the 55-family reach 57 or 53, and the readings 35/57 and 33/53 are the first that share no factor. Those report 82.2° and 54.4°, some fifty-five and eighty-three degrees from the truth. Across the whole range read, fifteen per cent of a turn either way, 34/55 passes through twenty-six wrong readings; ten announce themselves and sixteen are silent.

55/89 behaves the same way, and first goes silent at the same 9.82°, when its 55-count reaches 57 or 53 and pairs with 91 or 87. It is exact over a narrower interval, since its 89-family moves at an overrun of two degrees, and every reading between two degrees and 9.82° shares a factor.

Where the protection runs out #

Lower down the sequence the closing error is not so obliging, and the reason is again arithmetic.

21/34 stays exact for a closing error under 5.3°, and its first wrong readings, 21/33 and 21/35, share 3 and 7 just as 34/55’s do. The first silent readings arrive at 8.57°, as 20/33 and 22/35, reporting 54.4° and 82.1° — the same two angles 34/55’s first silent readings report, from the other side of two sequences each built by adding the last two terms — 13, 20, 33, 53 and 13, 22, 35, 57 — whose consecutive pairs all sit near those angles. 13/21 has no announcing stage at all. It is exact out to 8.57°, and the two readings just past that, 13/20 and 13/22, are both silent.

So the high Fibonacci reports have a margin, and the margin is about ten degrees. A counter who closes the circle within ten degrees of the starting mark can report a wrong 34/55 or 55/89, but the wrong report will share a factor and look like a whorled plant rather than an ordinary one. It is the same statement the census of whorled readings should be read against: a shared factor in a survey of heads is sometimes a jugate plant and sometimes a count closed a few degrees late, and the pattern that arrives two at a time is the check that separates them.

Higher counts go wrong more often, and say so #

A margin of ten degrees is only worth something if the counter’s closing error is usually smaller than that, and whether it is cannot be read from a head. What can be read is what each report does for a stated spread of closing errors, taken as normal about zero.

With a spread of 3.6°, 13/21 is read right 98.3 per cent of the time and 34/55 only 63.7, and 55/89 only 42.6. A high count is the more fragile to a closing error, simply because it has more spirals per degree to gain or lose. But almost all of its errors announce themselves: 34/55 is wrong 36.3 per cent of the time and silently wrong 0.64, and 55/89 is wrong 57.4 per cent and silently wrong 0.63. 13/21’s 1.7 per cent of errors are silent almost to the last reading.

The Lucas reports do not share the high Fibonacci pairs’ margin. 29/47 is silently wrong 20.4 per cent of the time at the same spread, because its first readings past exact — 29/46 and 29/48 — share no factor and report 148.8° and 37.4°. More than a quarter of its readings are wrong in all. That is the difference between the two branches of counts the independent readings found, repeated for a coupled error.

Silence arrives with the spread #

A wider spread reaches the silent readings past the ten-degree margin more often, and the share of errors that pass silently climbs.

At 34/55 the silent share of wrong readings is nil at a spread of 1.8°, 1.8 per cent at 3.6°, 12.2 at 5.4°, 22.9 at 7.2° and 34.3 at 10.8°. 55/89’s climbs a little more slowly, to 32.7 per cent. 21/34’s starts higher, at 12.2 per cent at 3.6°, because its announcing interval is narrower. 13/21’s starts at all of them and falls as a wider spread reaches its announcing readings further out, 12/20 and 14/22.

For comparison, the independent reading put 34/55’s silent share of wrong reports near one per cent when a count drifts only by one, and at a half when drifting by two is as likely as by one. A closing error spread over seven degrees sits between them. So coupling does not make a report safe; it moves the danger outward, to closing errors of about ten degrees, and a counter who is not good to that is in about the same position as one whose counts drift independently.

The relation passes a shared closing error #

The check that rescued the independent readings is a second annulus. Counting (a, b) in one annulus and (c, d) in the next, a golden head’s pairs continue each other when $c = b$ and $d = a + b$. Every one of the four counts appears in that relation, so an independent error in any one of them breaks it. It caught every single error of any size, and let a wrong reading through about one time in a thousand when combinations of errors conspired.

A closing error does not break it, and the reason is the one thing that distinguishes it from an independent error: it multiplies every count by the same $1 + \varepsilon$. The relation is linear, and a linear relation between four numbers survives multiplying all four by the same factor. $c = b$ holds by construction, since both are the 55-family counted round the same overrun, and $d = a + b$ holds whenever rounding the sum agrees with summing the rounded parts — which it does about half the time.

At a spread of 7.2°, a counter who closes both annuli at the same wrong mark gets 17.6 per cent of wrong readings of 34/55 past the relation, against 1.25 per cent for a counter whose two annuli have closing errors of their own. At 13/21 the relation accepts 76.8 per cent of wrong readings, which is almost every silent one. A counter who uses one mark for both annuli has not checked the count at all for the errors most likely to occur.

It also raises false alarms. The second annulus’s larger count at 34/55 is 89, which moves at an overrun of two degrees, so at a spread of 3.6° a right reading of 34/55 often fails the relation: 63.7 per cent of readings are right and only 42.6 per cent are right and pass, so one reading in five is right and flagged. A count carries no error asked that a published count say how it could be wrong, and a closing error is a way to be wrong that the most-recommended check both misses and misreports.

What catches it is a rough angle #

The check that survives is the one the drift essay treated as the weaker: an estimate of the divergence angle from something other than the counts, good enough to say which of the candidate bands the reading belongs to.

For a closing error it has to be much less good than for an independent one. Every silent reading 34/55 produces from a closing error under eight per cent of a turn — 29° either way — reports an angle at least 24.8° from the true band: the silent readings nearest the start are 33/53 at 54.4° and 35/57 at 82.2°, and those further out sit at 33.9°, 48.8°, 56.4°, 109.9° and 162.4°. The first silent reading near the truth is 31/49, at 139.5°, 1.86° from the edge of the true band, and it needs an underrun of 36°. A protractor laid on a photograph and good to twelve degrees settles every closing error up to 29°. At 21/34 the same margin is 13.3° for closing errors up to 29° — a coarser protractor, but still one a photograph supplies, as a round trip through the counts would at a lower count.

So the two checks trade places. Against independent errors the second annulus was the whole of the protection and the rough angle a help at the high pairs only. Against a shared closing error the second annulus is half blind, and the rough angle catches nearly everything a counter is likely to do.

What the closing-error model leaves out #

It does not measure how large real closing errors are. The spread is a parameter swept from 1.8° to 10.8°, and where a person counting spirals by eye sits in that range is a question about people, which a head cannot answer.

It treats a family’s spirals as evenly spread round the head, so that a sliver holds its fair share of each family. On an ideal lattice they are. On a real head the counts change with radius, and near a transition an annulus holds spirals of three families unevenly, which would couple the two counts in a way neither model here does.

And it assumes both families are closed at one mark. A counter who starts the 34-family at one organ and the 55-family at another has two closing errors, correlated to whatever extent the two marks are chosen alike — the case between the independent readings and these.

Three findings that would overturn it #

A Fibonacci report among the first twenty-three whose difference is odd, whose Cassini sign is plus, and which still admits a silent equal shift of one. A closing error under 9.82° that produces a silent reading of 34/55. A silent reading of 34/55 within twenty degrees of the true band from a closing error under 29°. Any one of them would mean the arithmetic above is wrong, not merely incomplete.

Still open: two marks rather than one #

Every closing error here is shared by the two families and, in the relation’s case, by the two annuli. The realistic counter is somewhere between that and the independent readings: two marks, chosen by the same eye, whose errors are correlated but not equal. The measurement is the same exact integration with the two families’ closing errors drawn from a joint normal distribution whose correlation is swept from nought to one — asking at what correlation the second annulus stops being the better check and the rough angle starts to be, and whether any Fibonacci report keeps its ten-degree margin once its two families can be closed at different marks.