A head's holes spell its angle

One over root two ended with a promise it did not keep. The largest empty circle of a head, read ring by ring, peaks wherever the lattice underneath flips from one pair of neighbours to the next, and the height of each peak is half the diagonal of a rectangle whose shape is set by the angle’s continued fraction. A noble angle’s flips all make squares. A partial quotient of four makes a run of long thin rectangles, and 137.3° shows it as a hole 0.862 of a spacing across near radius 13.5. That closed form was read in one direction: from the angle to the holes. The essay’s last section said it should run the other way, that a head’s ring profile carries its continued fraction in the heights of its peaks, and that this was a measurement a head could fail.

The measurement is an unusual one to attempt, because it asks more of a picture than anyone asks. A count of spirals leaves a band of angles open and that band is the whole of what a count can say. A largest circle is a different kind of reading. It comes out of every ring, it depends on the shape of the local cell rather than on which spirals the eye follows, and a head of 2,400 organs holds a hundred and sixty-seven rings a quarter of a unit wide. The question is how much of the angle those hundred and sixty-seven numbers hold.

The reading is not told the answer #

The head goes in as positions and nothing else. Delaunay triangles are built from the organs, each triangle’s circumradius is measured against the root of the interior’s mean cell area, and each ring keeps the largest. That is the same reading the gap that grows took of whole heads, kept per ring. Rings inside radius 4 are dropped, because there the lattice is not yet a lattice, and so are rings beyond 93 per cent of the radius, where the rim’s cells decide the largest circle.

Against that profile stands a lattice with one free number. A candidate angle has, at each radius, a rise and therefore a lattice; reducing that lattice to its two shortest vectors gives its largest empty circle exactly, by the same arithmetic as the closed form. The reading looks for the angle whose lattice comes closest to the rings. No peak is located and no flip is identified, so nothing about the answer can leak in through the choice of which feature to look at.

What the circles look like #

Drawn on the head, the holes are a surprise of a modest kind. They do not form rings in any visual sense: each is the largest in a thin annulus, so they scatter round the head wherever one triangle happens to win. Across the middle radii of this head they are visibly larger than at the rim, and the eye can pick out that much without being told why. There is no way to see the quotient by looking. What carries it is not where a circle sits round the head but how large it is against its radius, which is why a reading of this kind has to be a curve.

One curve, and the lattice it names #

Plotted against radius, the same circles become a curve with an unmistakable shape. A cusp near radius 8, a shoulder, three peaks at 11.1, 13.5 and 16.6, a long fall and a long trough to a final cusp past radius 43. The line through the dots is not a fit to that shape in the usual sense. It has one parameter, the angle, and the reading chooses it without knowing the shape existed. It returns 137.3003°.

That is three ten-thousandths of a degree from the angle the head was built at, and the quotients come back right as far as they were asked for: 2, 1, 1, 1, 1, 1, 4, 1, with the four where it belongs. Over the 167 rings the fitted lattice misses by 0.0047 of a spacing. The true angle’s own lattice misses by 0.0048, so the reading has found something slightly better than the truth, by an amount small enough to be the rings’ own unevenness. A ring’s largest circle is the largest triangle that happens to fall in it, and at a quarter-unit width some rings simply hold no triangle from the top of the range.

Moving the dial makes the same point from three other angles. At 137.0° the profile has a double peak near 0.89 and the reading returns 136.9998°, quotients 2, 1, 1, 1, 2, 5, 3, the five being what makes that peak. At 106.5° it returns 106.4999°. At 131.2°, with a nine among its quotients, the profile rises so far that the plot has to clip it, and the reading returns 131.2010°.

Why this is not a matter of a good guess #

A fit with one free number can always return something. What makes this one a measurement is the width of the valley it finds. Scanned across seventy-five degrees, a typical angle’s lattice misses the rings by 0.046 of a spacing and the true angle’s by 0.0048, ten times less. Most of the landscape is a noisy plateau, because two lattices at unrelated angles have their flips at unrelated radii. The minimum at 137.3° is a spike, and it stands well below every other dip in the range.

Those other dips are the interesting part, and they will come back. Several of them fall most of the way to the bottom. A dip means that another angle shares part of this head’s profile, and nothing about a flat scan would say which part.

The spike is also very narrow, and that had a consequence which should be recorded. The first version of this reading scanned the whole range at two thousandths of a degree and refined the best point. It read larger heads worse than smaller ones, with eighteen of forty wrong at 8,000 organs against eight at 2,400, which is the opposite of what more organs should do. Nothing was wrong with the heads. A head’s minimum gets narrower as it gets larger and the scan was stepping over it. The rings inside a radius $r$ can only separate angles about $1/r^2$ apart, so the reading now works outward: the rings inside radius 8 choose three hundred candidates at a coarse step, each wider set of rings narrows them at a finer one, and the last pass reads every ring.

Forty angles nobody chose #

Four hand-picked angles prove little, particularly ones already known to have interesting profiles, so the test that matters takes forty angles from a seeded generator, to four decimals, between 100° and 175°. None was looked at before the reading ran. Of the forty, 28 come back within a thousandth of a degree and three more within half a degree. The median error is a quarter of a thousandth of a degree, and the median reading gets five partial quotients right before its first wrong one, with several reaching ten.

The plot’s shape deserves a look because it is not a smooth spread. Most of the angles sit in a band around $10^{-4}$. Then there is a gap of four orders of magnitude, with hardly anything in it, and nine points at the top that are wrong by between one degree and sixty-eight. A reading of this kind does not degrade gradually. It either finds the angle or it finds a different one, and the reason is the most useful thing the measurement turned up.

The misreadings keep the end #

Look at what the nine wrong answers actually are. The head built at 147.4363°, whose expansion runs 2, 2, 3, 1, 3, 1, 3, 2, is read as 171.25°, which runs 2, 9, 1, 3, 1, 3, 2. The head at 125.3306°, running 2, 1, 6, 1, 5, 7, 17, is read as 172.06°, running 2, 10, 1, 5, 7, 13. The silver angle, whose expansion is all twos, is read as 99.16°, running 3, 1, 1, 1, 2, 2, 2, 2.

The wrong answers are wrong at the start and right further along. Each of these three shares a run of quotients with the truth, shifted by however much its first few differ. Across all nine the longest shared run has a median of three and reaches six, where two of the forty angles picked at random share a median of two, and share five or more less than one time in two hundred. Three of the nine carry a run of five or six. That is not an accident of these examples. Two angles whose expansions end alike are related by a change of basis on the lattice, so they make the same lattices at different rises, and the same sequence of cells a fixed factor apart. What a head can mean by most irrational found the arithmetic side of this, that over the counts a head shows, an angle is only as good as the run of counts it adds up from, and forty-six angles had nothing to choose between them there.

The geometric side is that a continued fraction is laid out across a head from the centre to the rim. A quotient belonging to denominator $q$ has its flips at the radius where the family of $q$ spirals is forming, about $q$ spacings’ worth of circumference out. The first quotients belong to denominators of 1, 2, 3 and 5, and so they have their flips inside radius 4, in the part of the head that one over root two found was decided by a single triangle and which this reading had to drop. Nothing outside that radius records them directly. What records them is only the position of the later flips, the offset of the whole pattern along the radius.

So the head tells the reading the end of the fraction loudly and the start of it quietly. Of the nine misreadings, five actually fit their head’s rings better than the true angle does, so no search would have saved them. The other four are angles the outward reading settled on after losing the truth at the innermost rings, where a head is furthest from a lattice. The two kinds together are a statement about how little of the profile the start of an expansion controls.

What a larger head adds #

Reading the same forty angles at five sizes separates the two kinds of result cleanly. From 600 organs to 8,000 the median error falls from $3.5 \times 10^{-3}$ of a degree to $3.3 \times 10^{-5}$, a factor of a hundred for a factor of thirteen in organs. That is the arithmetic of denominators again, since more radius admits larger denominators and an angle pinned by a larger denominator is pinned more tightly. A head of 8,000 organs knows its angle to about a thirty-thousandth of a degree, which is better than any protractor.

The misreadings do not move. Nine at 1,200 organs, nine at 2,400, nine at 4,800, nine at 8,000. Twelve hundred to eight thousand adds outer rings, and outer rings are not where the wrong readings went wrong. That flat line is the clearest evidence in this essay that the failure belongs to the centre: a resource that can only be spent at the rim does nothing for it. At 600 organs the count is thirteen, and the reading to a thousandth drops to twelve, because a head that small has too few flips outside radius 4 to fix anything finely.

What a protractor is worth #

If the failure is that the start of the fraction is not in the profile, the repair is to supply the start from somewhere else. The obvious source is the crudest possible measurement of the divergence, read off the first few organs with a protractor. Told the angle to within five degrees either way, the reading misidentifies five of the forty, and told it to within one, two. Readings to a thousandth rise from 28 to 31 and 32.

That is a good trade in a specific sense. A measurement good to a degree and a profile with no idea where it is combine to give an angle known to two ten-thousandths of a degree. Neither carries the other’s information. The coarse measurement contains the first quotients, and only those; the rings contain everything after. A count of spirals at the head’s centre would serve the same purpose, since a count names a convergent. The two misreadings that remain are rivals within a degree, 132.1492° read as 131.36° with three quotients shared, and 165.348° read as 164.63° with none. A window of a degree still holds other angles, and every window holds expansions ending in anything.

The first number is the one that is lost #

The noble angles make the same point in its purest form. Golden is read as 137.5080°, two ten-thousandths out and every quotient right. Lucas, whose expansion differs only in its first term, is read as 99.5015°. But those are two angles whose tails are identical and whose flips all make squares. The rings of those heads hold no quotient but one, and the reading is choosing between them purely on how far along the radius the identical pattern falls. It manages this on a 2,400-organ head, and the margin it does so by is small enough that the silver angle, the same kind of case with twos instead of ones, fails: it reads as 99.16° with a misfit of 0.0063 where the truth’s is 0.0068. There is no error in that. A head in this size range genuinely does fit a different start slightly better.

How little it takes to end this #

Everything above is a head placed exactly. Moved organ by organ, the reading holds up for almost no distance at all. At a thousandth of a spacing all fifteen heads still come back within half a degree. At two thousandths, likewise. At five thousandths eight of fifteen do; at a hundredth seven, and only two to a thousandth of a degree; at a twentieth, none.

That is far more fragile than the other readings of a head here. A defect ring keeps its count to a displacement of about three over its family, which is five hundredths of a spacing on the ring where 55 enter, and a count of spirals is a median over every organ in a band, built to ignore a displacement this size. The reason for the difference is the statistic. A largest circle is an extreme value, drawn from whichever triangle a displacement has opened most, so it responds to the tail of the noise, and in every ring at once. Measured on the golden head beyond radius 8, a displacement of a hundredth of a spacing lifts the largest circle in the average ring by 0.011 of a spacing above where it sat, and a displacement of a twentieth by 0.058: the bias is about the displacement itself, always upward, and the differences between candidate angles are in the third decimal.

What the head has said, then #

For an exact head the conclusion is striking and it can be stated without hedging. Five or six figures of a divergence angle are in the holes of a head of a couple of thousand organs, and the quotients can be read off them one by one, starting at the fourth or fifth and running to where the head ends. The closed form runs both ways, as the essay that found it said it should.

For anything that grew it is a different claim, and it should not be allowed to pass for the same one. A real organ is not placed to a two-hundredth of a spacing. So what this establishes about plants is nothing, directly. What it establishes about the model is sharper: the largest circle of a head is a fingerprint, carrying more of the angle than a spiral count can, in a form that is destroyed by the first noise any physical specimen would have. An instrument that reads to six figures in the absence of noise and to none in its presence is not a rival to a count of spirals.

Where a reading like this could still work #

The instrument is not useless outside the exact case; it has a narrower use than it first appears to. It can test a model. A stem or a head grown by a placement rule rather than by Vogel’s formula has no reason to sit on one lattice, and a profile read from a grown head and compared with the lattice of its own settled angle would say how far the grown thing departs from the idealisation, flip by flip. A departure there would be a statement about the rule, because the rule’s output has no measurement noise in it at all.

It could also average. Noise moves one circle in a ring and the statistic takes the largest, which is the worst way to treat noise. A reading of the whole distribution of circumradii in a ring, a high percentile instead of the maximum, might give back some of what the maximum gives up, at the cost of some of its sharpness. Nothing here measures that. It belongs with the things a next reading could try.

What this does not establish #

It does not show that the first quotients are unreadable from a head. They are unreadable from the rings outside radius 4 by this reading, and a reading that used the central triangles too might do better, though on a golden head a single triangle decides that region. It does not show that nine misreadings in forty is a property of angles in general. Forty seeded angles between 100° and 175° is a sample and a different forty would give a similar but not identical count. It does not say what the reading does on a cylinder, on a cone or on a head whose radial law is not a square root, where the rise at a given radius differs and every flip moves. And nothing here compares the rings’ information to that of the whole triangle population, of which the rings keep one triangle per quarter unit.

How this could turn out wrong #

If an angle whose quotients sit outside the first four were misread at 8,000 organs in a way that a window round the truth did not fix, the account that the failure belongs to the centre would be wrong. If a larger sample of misreadings shared runs no longer than two arbitrary expansions do, the account of why the misreadings have the angle they have would be wrong as well. If a displacement of a thousandth of a spacing could be shown to have left the peak heights intact and the reading failed anyway, the fragility would belong to the search rather than the statistic. Each of those is a check on the argument and none has been made to fail so far.

Still open: what the rule’s own heads have in their holes #

Every head here is placed exactly. A head grown organ by organ has no measurement error, and it does have the settling of its angle, its first few organs out of place, and whatever difference there is between what the rule does and a lattice at the same rise. The next measurement reads the same ring profile from a head grown by a placement rule, compares it with the lattice of the angle that head settled on, and asks where along the radius the two part: whether a grown head’s holes spell its settled angle to five figures, as a placed head’s do, or whether the departures from a lattice are large enough to make the reading fail in the same way a hundredth of a spacing does. A failure would be informative, since it would put a number on how far a grown head is from a lattice, ring by ring.