A disc reads almost no Fibonacci heads by chance

A Fibonacci census asks whether the share of specimens whose counted parastichy pair is two consecutive Fibonacci numbers is the share the geometry gives by chance or the much larger share a grown history gives. The share by chance has always been a share of divergence angles: every angle on a fine grid, each read as the dominant pair of a cylinder at a stated rise, and the Fibonacci pairs counted. The high heads the geometry rarely makes drew that null at finer and finer rises, found it easier to beat at each, and found that a counter who aims a recount at the Fibonacci pair they expect, capped at twice their own spread, moves a few tenths of a per cent of it at most.

That essay ended on the null’s shape rather than its numbers. Every null in the census is a cylinder’s. A capitulum is a disc whose rise falls outward, and a surveyor counts it in a band, so a band can straddle a transition and read a mixture of the pairs on either side — perhaps a pair one count off a Fibonacci pair, which is exactly the head a capped aim was built to move. This essay reads the null the way a surveyor would.

A disc for each rise the census was read at #

The angles are the cylinder census’s own: 180·k/3600 degrees for k from 1 to 3,599. At each, a Vogel head is placed — organ $k$ at radius $\sqrt{k}$ and $k$ times the angle round — and its pair is counted by the counter that is shown only the points, over the band from 0.55 to 0.95 of the radius, the band several other readings here count in.

A disc’s rise is not one number. For organs at radius $\sqrt{k}$ it is $1/(4\pi r^{2})$ of a turn, falling outward, so a band holds a range of rises; at its middle, on a head of n organs, it is about 1/(7.6n). The head sizes are chosen so that the middle stands at the rises the cylinder census was read at: 40 organs near 0.004, 75 near 0.002, 150 near 0.001, 300 near 0.0005 and 500 near 0.0003.

The specified rise has no disc #

The census was specified at a rise of 0.008, and the disc whose band stands there has about twenty organs. Its band holds a dozen, and the counter refuses at every one of the 3,599 angles: there are not enough organs between 0.55 and 0.95 of the radius to trace two families through.

A forty-organ head refuses at 515 angles, one in seven; seventy-five organs at 154; 150 organs at 37; and the two largest at 36 and 71. So the coarsest census a surveyor can actually make with this band stands near a rise of 0.004, and the census as first specified describes heads too small to count. A person counting a twenty-organ head would use the whole head, or count organs rather than spirals, and neither is the reading the census assumed.

Almost no Fibonacci heads #

The disc’s null holds a fraction of the cylinder’s Fibonacci heads at every size. At forty organs it is 1.23 per cent of the readable angles, against 10.0 per cent on the cylinder at 0.004; at 75, 0.44 against 6.8; at 150, 0.17 against 4.7; at 300, 0.08 against 3.3; at 500, 0.06 against 2.5. A factor of eight at the coarsest size and of forty at the finest.

The rest of the null moves the other way and by much less. Whorled heads, whose two counts share a factor, are 46 to 49 per cent of the disc’s null against 39 to 46 of the cylinder’s, and the other spiral pairs make up the remainder on both. So the disc does not simply read every head differently; it reads one kind of head the cylinder called Fibonacci as something else.

Where the cylinder’s Fibonacci heads were #

Traced angle by angle, almost all of the cylinder’s Fibonacci heads are a single pair. At the rise matched to forty organs, 321 of its 359 Fibonacci angles read 1/2; at 0.001, 162 of 169; at 0.0003, 88 of 90. Every one of them sits at the smallest divergence angles — from 0.05° up to 16° at the coarsest rise and up to 4.4° at the finest — where each organ is placed a few degrees round from the one before.

The essay that drew these nulls first placed those 1/2 heads near a half turn, and has been corrected: they are at the other end of the range, and that matters for what they are. On a cylinder of fine rise, an angle of a few degrees makes successive organs nearly stack, and the two shortest lattice vectors are one step and two — a dominant pair of 1 and 2, consecutive Fibonacci numbers, from a head nobody would call a Fibonacci head.

What a disc reads there #

On a disc the same angles draw one arm wound tightly round the head, and a band crossing it is crossed by the arm once a turn. The counter there either refuses — at 278 of the 321 small angles on a forty-organ head and 56 of 88 on a five-hundred-organ head — or finds the two families of shortest chord among the arm’s successive turns and reports them: 45/89 at 4° on a head of 150, 81/95 at 2°, 45/90 at 8°. Not one of the 1/2 angles reads 1/2 on any disc.

Near the golden angle the two agree. On a forty-organ disc 8/13 is read at 37 of the 38 angles where the cylinder reads it; on 75 organs 13/21 at 14 of 15; on 500, 34/55 at both of the cylinder’s two. Where they disagree the disc reads the next pair up: at 150 organs the cylinder’s four 13/21 angles are read as 21/34 at three, because a band from 0.55 to 0.95 of the radius sits slightly outward of the rise it was matched by.

So the Fibonacci heads both nulls agree on are the near-golden sliver, and on either it is a few tenths of a per cent of the angles or less. What the cylinder census added on top was an artefact of reading a stacked arm as a lattice.

It is worth being exact about why the cylinder can call that arm 1/2 and the disc cannot. A cylinder of fixed rise has one lattice, and its two shortest vectors at an angle of a few degrees are one step and two steps along the arm — both nearly vertical, nearly parallel, and short only because the rise is small. A disc’s arm is not straight: its turns are a fixed distance apart in radius, and the band crosses each turn once, so the families a counter can trace are the turns themselves, every forty-fifth or ninetieth organ, and the step from one organ to the next runs along the arm rather than across the band. The cylinder’s reading is a property of a lattice the disc never forms.

The census needs three specimens #

The census arithmetic is unchanged: closing errors spread over 7.2°, the two counts of one head correlated at 0.9, one recount of an announced reading, and an alternative of ninety per cent grown plants at 34/55. Handed the disc’s null, the share of kept specimens reading Fibonacci by chance is 2.38 per cent at forty organs, 0.73 at 75, 0.24 at 150, 0.12 at 300 and 0.06 at 500 — against 16.0, 11.0, 7.5, 5.1 and 3.7 on the matched cylinders. The grown plants read Fibonacci on 63 per cent of kept specimens unaimed and 75 capped, on either null, since their own pair is the same.

The census needs five kept specimens at forty organs unaimed and four capped, and three at every larger size either way, where the cylinder’s needed nine, seven and five. At three specimens the census is close to its floor: with the null’s share under a per cent, three Fibonacci readings in three is already beyond chance, and the remaining cost is the power to see them when a fraction of the grown plants misread.

What an uncapped aim does to a small null #

The counter who aims every recount at the Fibonacci pair they expect, with no cap, was the one the recount essay found raising the null by reading whorled heads as the Fibonacci pair beside them. On the disc it multiplies the null fourteen-fold at 150 organs, from 0.24 to 3.46 per cent, and nearly thirty-fold at 500, from 0.06 to 1.75. On the cylinder the same aim took 7.5 to 10.6, less than half again.

Proportionally the uncapped aim is far more damaging on a disc, because there is so little Fibonacci by chance for it to add to. In absolute terms it still leaves the null under four per cent at every size from 150 organs, and the census needs four kept specimens instead of three. The cap does what it did before: 0.24 becomes 0.28, 0.06 becomes 0.10.

Bands do straddle #

The question was about straddling, and the disc does straddle. Each band is also counted in its innermost, middle and outermost tenths, and on heads too small for those narrow tenths to be read — every size up to 150 organs — nothing can be said. On 300 and 500 organs, where all four readings are made at about 3,450 angles, the two edges read different pairs at 1,635 and 1,724 of them: nearly half. A surveyor’s band from 0.55 to 0.95 of the radius usually crosses a transition on a large head.

What it reads there is almost never a mixture. At 91 to 92 per cent of the straddling angles the band reads one edge’s pair outright. At 113 and 145 it reads the pair its middle tenth reads, which is a legitimate reading — the band crosses two transitions and reports the pair between them. At 14 and 13 angles, under one in a hundred of those straddling, it reports a pair that none of its three tenths gives, and every one of those is an ordinary spiral pair far from golden, such as 18/53 or 23/70.

A narrower band straddles less #

The band is a choice, and a surveyor who wants to avoid straddling can narrow it. Counted from 0.75 to 0.95 of the radius instead, the five-hundred-organ head’s edges disagree at 15 per cent of the readable angles rather than 49, and the three-hundred-organ head’s at 13 rather than 47; the mixtures fall from 13 and 14 to 3 and 1. The narrower band costs readings on small heads — a forty-organ head cannot be counted in it at any angle, and a seventy-five-organ head refuses at 431 angles where the wider band refused at 154 — because a band a fifth of the radius wide holds too few organs to trace two families through.

What the narrower band does not change is the census. The disc’s Fibonacci share is 0.17 per cent at 150 organs on either band and 0.06 at 300 and 500; only at 75 organs, where the narrow band refuses more of the small-angle heads, does it rise, from 0.44 to 0.79 per cent. The census needs three kept specimens on either band at every size from 75 organs up. Straddling was never where the null’s Fibonacci heads came from, so taking it away moves nothing that matters to the test.

Where the capped aim’s heads come from #

The heads a capped aim can move are near-golden pairs that a consecutive Fibonacci pair is read as by a closing error within twice the counter’s spread. On the disc there are none at forty or seventy-five organs; at 150, 20/33 and 22/35, 0.28 per cent of the null; at 300 the same two, 0.17; at 500, 33/53 and 35/57, 0.14. The matched cylinders hold the same pairs at 0.17, 0.25 and 0.11 per cent.

Two of the disc’s are made by straddling: on the five-hundred-organ head, 33/53 at 54.45° is the band’s middle pair between an inner 20/33 and an outer 33/86, and 35/57 at 82.2° sits between 22/35 and 35/92. They are real readings of the middle of a head, and they are the same pairs the cylinder already had. The mixture the question imagined — two pairs combined into a near-miss — does not occur at any angle.

What this does to the census #

The census was specified on a cylinder at a rise of 0.008, and read on discs it is a different instrument in three ways. A surveyor cannot count the heads that rise describes. The null’s Fibonacci share, the number the grown plants must beat, is not 22.5 per cent or 15 or 7.5 but a few per cent at most, falling to a twentieth of a per cent on large heads. And the near-miss heads that drove two essays’ worth of worry about aimed recounts are a few tenths of a per cent of the null whichever way it is drawn.

None of that rescues a census that is badly counted: the habit that ties two counts of one head together was what cost the five per cent on the cylinder, and on a null this small a habit that adds a single Fibonacci reading moves the share proportionally further. It does mean the census is cheap, and that the question it asks is close to a question about whether plants read Fibonacci at all — which the geometry alone almost never does on a disc.

A Vogel disc is not a capitulum #

That a capitulum is a Vogel disc. A real head is grown, not placed; its centre is irregular and its rise may not fall as $1/r^{2}$; and every head here is exact, with no displacement of the organs at all. Nor that 0.55 to 0.95 of the radius is a surveyor’s band. A narrower band straddles less, and a band that included the centre would meet the tight arms of small angles in a different way.

It does not settle what the census’s null should be. A divergence angle drawn evenly from a half turn is a model of a plant that is not Fibonacci, and nothing here argues for it over any other.

What a disc would have to read to undo this #

A disc whose null’s Fibonacci share is not under a fifth of the matched cylinder’s. A disc band reading 1/2 at any of the small angles where the cylinder does. A reading that none of a band’s edges or middle gives, within a capped aim’s reach. A twenty-organ head counted in the band at any angle.

The null without its stacked arms #

Read on discs counted in a surveyor’s band rather than on cylinders, the geometry’s null holds a tenth to a fortieth of the Fibonacci heads, because nearly all of the cylinder’s were 1/2 at the smallest angles, a stacked arm a band never reads that way. The census then needs three kept specimens. Bands on large heads straddle a transition at half the angles, but read an edge’s pair or their own middle pair; true mixtures are fewer than one in a hundred of those, none near golden, and the heads a capped aim can move are the same few tenths of a per cent the cylinder had.

Still open: a disc that was grown #

Every disc here is placed by Vogel’s rule, exact at every organ. A head grown by a placement rule, or displaced the way a photographed head is, has its organs a fraction of a spacing off the lattice, and a displaced head’s band counts change where the lattice’s would not. The next measurement draws the null from discs displaced by a tenth and a quarter of a spacing, and asks whether the near-golden pairs a capped aim can reach grow with the displacement, since a displaced band can read a family that the lattice holds only marginally — the place a near-miss would come from if it comes from anywhere.