Coprime — where it appears
Named by 9 essays across 2 fields — each of them below, with the objects they name alongside it.
Recovering the angle from the counts
Build a head at a stated divergence angle, forget the angle, and get it back from the spiral counts alone. Four angles, worst error twelve thousandths of a degree — and the only thing that crossed between the two halves was a list of coordinates.
A count that can be wrong by one
A reported parastichy pair pins the divergence angle to a band 221°/mn wide only if both counts are right. Allowing either to be off by one adds the bands of every neighbouring pair whose counts share no factor, and those bands sit where their own lattices live: for 2/3 they swallow the whole range, and for 34/55 they are two bands as narrow as the true one at 109° and 113°, twenty-five degrees away. So a high count that may be wrong is not a blurred reading but a short list of sharp candidates, costing log₂ 3 bits. And on the Fibonacci pairs, two in every six — 21/34 and 34/55 among them — cannot be miscounted silently by one count at all, because every such miscount shares a factor.
A count that drifts by two
A reported pair of 34 and 55 that may be wrong by one allows three sharp bands; allowed to drift by two it allows thirteen, and by three, twenty-nine — and the information lost is exactly the logarithm of that count, because every band is as narrow as the true one. The nearest wrong band stays twenty-four degrees away until a drift of three brings one to eleven. What does not survive is the protection: 21/34 and 34/55 could not be miscounted silently by one, but every Fibonacci pair can be by two, so a counter who drifts by two as readily as by one reports 34/55 silently wrong 9.5 per cent of the time rather than 0.13.
Two counts that slip together
A counter who closes the circle a few degrees late counts a sliver of the head twice, in both families at once, so the two counts of a reported pair drift together rather than apart. Coupled that way the count is safer than it was: fourteen Fibonacci pairs in twenty-three admit no silent equal shift of one, against seven that admit no silent single miscount, and 34/55 announces every closing error short of 9.82°. The check is what breaks. Two annuli closed at the same wrong mark pass 17.6 per cent of wrong readings of 34/55 and 76.8 per cent of 13/21's, because a linear relation survives multiplication — and what catches them instead is a protractor good to twelve degrees.
Two marks chosen by one eye
A counter traces each family of spirals from a starting organ of its own, so a reported pair carries two closing errors, correlated because one eye chose both. Letting them differ costs 34/55 its ten-degree margin — 33/56 and 35/54 share no factor, and marks that err 5.3° in opposite directions reach them — while 21/34 keeps its margin whatever the marks do. And it decides the second annulus. At a spread of 7.2° the relation passes right readings 2.8 times as readily as silent ones when the marks are independent, 1.25 times at a correlation of 0.9, and stops telling them apart at 0.98; where it does work it keeps one reading in forty-six.
A chosen mark errs by a third of a spiral
A counter who starts every family of spirals at the organ nearest one place on the head, and stops when the count comes back round to that place, makes a closing error the head decides: the gap between the place and where that organ's spiral crosses the counting circle. It is three tenths of a spiral spacing in every family of every pair, so in degrees it falls as one over the count — 3.2° on a 34-family, not the 7.2° assumed — and the two families' errors are correlated by where on the head the place is, from +0.35 at the inner edge of a pair's band of radii to −0.5 at the outer. From the geometry alone a reading is wrong one time in eleven and almost never in both counts, so 34/55 is silently wrong once in six thousand. An eye that misplaces the mark by a tenth of a spacing is enough to turn the correlation positive.
A pinned count errs where the spirals crowd
A counter who closes each family on a chosen place is wrong one time in eleven from the geometry alone. Pin the starting organ instead and the closing costs nothing; the only error left is a slip on the way round, and a slip is likeliest where neighbouring spirals cross the counting circle closest. They cross evenly round any one circle and much closer at the inner edge of a pair's band, and the larger family's always cross closer by the ratio of the counts — so it slips about four times as often per count. A pin is best at the outer edge, where a place is worst, and beats a place there for an eye that slips less than once in 156 crossings at an organ's spacing; to beat a place a third of the way in it must slip less than once in 2,400. And a pin makes an error a place never does: a count two out, silent on 34/55.
A shared factor has two readings
A report whose two counts share a factor has been read two ways. One account says it is a miscount announcing itself, since a true ordinary pair cannot share one. The other says it is a plant grown k organs at a time, and that reading took a census's Fibonacci share from a seventh to a half. It is one piece of evidence, and nobody has compared the rates. Above the smallest pair, at least two of the four reports one count from a Fibonacci pair share a factor and carry a family's name: 21/35 reads as seven whorls of 3/5, and 33/55 as eleven. At a chance of 5 per cent that a count is off by one, with 5 per cent of plants truly whorled, a report with a shared factor is a true whorl less than half the time.
A slip that remembers is caught by a second pass
A pinned count errs only by slipping between spirals on the way round, and on 34/55 only a count two out is silent, which independent slips make rare. An eye that has lost its place once is likelier to lose it nearby, and runs of slips ought to make that silent error common. Holding the mean rate of slipping fixed, runs do raise it, by more than five times when each slip makes the next hundred times likelier. It still never reaches the whole error of a count taken at a chosen place at the rim, which is fifteen in a hundred against under half of one. Bunching the same slips together leaves fewer counts wrong in all. And two passes that must agree take the wrong readings kept from 5 per cent to a tenth of one, with runs or without.
Named alongside it
The objects these essays reach for when they reach for this one.
FibonacciHonest limitsMeasurement errorParastichy pairSilent failureDivergence angleDiscriminationIdentifiabilityJoint distributionLucas numbersSurveyNearest neighbour