Field

The pattern itself

A seed head is a set of points, and nearly every claim about one is really a claim about how many spirals run through it. The counting can be done from the points alone, and the answer is not what the captions say.

A head of 200 primordia at a divergence of 137.51°. Nothing is placed by hand: the nth point sits at n·137.51° and radius √n. The closest any two points come is 1.60 of the mean spacing.

A head is a set of points

The nth primordium at n times an angle, and a radius of root n. Two lines of arithmetic produce a sunflower head, which is either remarkable or suspicious depending on how carefully the claim is stated — and stating it carefully is most of the work.

The two spiral families a counter finds between 0.68 and 0.92 of the radius. 34 spirals one way and 55 the other, found from the point positions alone — the counter is never told the divergence angle.

Counting the spirals

Almost every claim about phyllotaxis is a claim about how many spirals run through a pattern, and the count is almost never done. It can be done from the points alone, by a count that is never told what angle built them — and then a count of 34 is evidence rather than a restatement.

The spiral counts, band by band, in one head. The same flower gives 13/21, 21/34, 34/55, 55/89 at different radii. A caption saying "34 and 55 spirals" is a statement about one annulus.

The counts change with radius

The same head gives 13 and 21 near the centre, 21 and 34 further out, 34 and 55 beyond that, and 55 and 89 at the rim. The transitions are at computable radii, and a photograph captioned with one pair is a statement about one annulus rather than about a flower.

A round trip on four heads of 400 primordia: the divergence angle recovered from each. The counter is shown the points and nothing else. The worst recovery across the four is 0.035°.

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.

three stems: 1, 2, 3 primordia at a time. 1-jugate at 137.51° counts 2 and 3; 2-jugate at 68.75° counts 2 and 4; 3-jugate at 45.84° counts 3 and 6. Every count shares the factor k, and dividing it out leaves an ordinary lattice: what a k-jugate stem is, exactly, is k copies of an ordinary one wrapped k times round.

Two at a time

Every counter in these essays asks how far it is from element i to element i plus m, and that index is a claim that the elements arrived one at a time. For teasel, for Cephalaria and for a real minority of plants the claim is false — and what those patterns turn out to be is an ordinary lattice, wrapped twice.

Tracing one family: 4 chains. Every node is joined to the node one repeated displacement away, and the chains that result are drawn separately. There are 4 of them, which is the parastichy number of this family. No index of arrival was used anywhere, which is what lets the same count be made on a pattern where 2 primordia appear at once.

Counting without an index

A person counting spirals on a cone puts a finger on one scale, follows a family round, and counts how many distinct chains there are. That needs no order of arrival — and building it turns out to be strictly more general than the counter that reads the order of arrival, and to find a bug in the counting of a bijugate stem that nothing had caught.

The 2-jugate forks converge on 68.7539°. Every fork sits at a rational divergence, with denominator 4(m² + mn + n²) — 10/28, 30/76, 74/196 and so on. The limit is 68.7539°, which is 137.5078 divided by 2, and it is at none of them.

Half the golden angle

The forks of the van Iterson tree converge on 137.5078° and sit at none of them. Divide the whole tree by two and the same thing happens at 68.7539° — which is where teasel is, and where a bijugate sunflower counted 42 and 68 has to be.

A counter that sees no positions

This site has counted spirals two ways, and both were handed coordinates. A third counter is handed a list of angles and nothing else. It returns one number instead of two, it refuses more often, and where it refuses it would have been wrong every time.

Every family but two is the sum of two others. Four heads and the contact families each actually has. An arc arrives at every family that is the sum of two smaller ones; the two with no arc arriving are the generators, and on every head they are the two smallest. whorled, 144°: 2, 3, 5 — golden, 137.508°: 8, 13, 21, 34, 55, 89 — Lucas, 99.502°: 11, 18, 29, 47, 76 — rational, 137.5°: 8, 13, 21, 34, 55, 89 — 137.0°: 8, 13, 21, 29, 50, 71, 92, 113. So once a pair is counted the rest is arithmetic, and a third counted family is a prediction rather than a second measurement.

Every family but two is a sum

A seed head has six spiral families and everybody reports two. That looks like a convention hiding information and it is the opposite — every family but the two smallest is the sum of two others, so a third count is a prediction rather than a measurement, and a check that catches a wrong pair.

The angles name the branch

Seed the same rule at the Lucas angle and the readout returns 4 and 7, then 7 and 11 — the pairs the position counter finds, and not Fibonacci numbers. So a list of divergence angles carries not only how many spirals there are but which family of ladders the plant is on.

A lattice with an error that repeats every 8 organs. The autocorrelation of 759 divergence angles from a kinematic lattice at a divergence of 137.8261° and a rise of 0.005, with 0.5° of scatter on each azimuth. There is no placement rule anywhere in it: node i is put at exactly i times the divergence and then displaced. The only thing that differs between this figure and the control is the structure of the displacement — here, an error that repeats every 8 organs at period 8, weight 0.7. The largest comb mean is 0.592 against a sampling band of 0.073, and the readout returns 8/10.

A periodicity is not a lattice

Give a lattice's errors a period of eight and a comb appears at spacing eight, on an arrangement with no rule in it. But the partner it names is 10, then 12, then 11, then nothing — an accident of the disturbance rather than a measurement of the pattern. The forgery is caught by reading a second stem, and by nothing else.

One wrecked stem, lag by lag — golden, rise 0.005, organ 6 back. How much each lattice hop moves from organ to organ in a stem that never repaired after a single removal, over its last 120 organs. The lag-one hop is the divergence itself and it swings by 55 degrees. The lag-4 hop swings by 0.12 degrees and sits 0.01 degrees from where the undisturbed stem put it, so that one family of the original lattice is still standing organ by organ. Its multiples inherit the same steadiness and nothing else comes within a factor of twenty. The block of angles this stem repeats has a period of 4, which is the surviving lag and not a coincidence.

A period that is not a count

Eighteen wrecked stems settle into a block whose period is one of their own spiral counts, and one settles into a block of four on a lattice counted 8 and 13. The odd one is not noise. It is the case that shows what the rule is actually conserving, and it is the reason this thread is about lattice steps rather than about spirals.

$Everywhere a cut of one to five organs can send a 5/8 stem. Every settled divergence reached by any arrangement of up to five organs removed from a stem at the 5/8 rung, on one axis. There are six of them and no more. three are slips of the lattice the stem was cut from: each keeps the lag-5 family intact and sits a whole number of turns of it from the next, which is the ladder marked below the axis with rungs 72.0 degrees apart. The other three keep no lag at all and sit near a fraction of a turn, marked above: 175.0 degrees near 1 of 2 turns, 190.0 degrees near 1 of 2 turns, 235.0 degrees near 2 of 3 turns. A stem that is cut either slides along the ladder it was on or leaves it for a lattice with files in it.$

A count with a factor in it

Cuts of several organs send a stem to six settled divergences and no more. Three of them are the old lattice with a family left standing. The other three are counted 2/6, 4/6 and 3/6 — and a pair whose numbers share a factor is the classic signature of a pattern that arrives several organs at a time.

$Two patterns a counter cannot tell apart — counted 2/6 against 2/6. On the left, the top 90 organs of a spiral stem that never repaired after two organs were removed, settling at 175.01 degrees. On the right, a stem grown by a rule that places two organs at a time on every node. A counter shown the positions returns 2/6 for the first and 2/6 for the second, and a pair whose numbers share a factor is the usual signature of a whorled pattern. Rotate each pattern and the answer separates them at once: the whorled stem maps onto itself at a half turn and the wrecked stem maps onto itself at no fraction of a turn at all. Its shared factor is a fact about where its divergence landed, 4.99 degrees from 1 of 2 turns, and not about how it grew.$

The symmetry that is not there

A wrecked stem counted 2/6 and a stem grown two organs at a time counted 2/6 are indistinguishable to a spiral counter. Rotate them by half a turn and one lands on itself and the other lands nowhere. A cut does not make a whorled pattern; it makes a pattern that counts like one.

Two readings of what a stem is, and one of them is wrong. The same runs read twice. On the left is the pair counted from the point positions by machinery that is never shown a divergence angle; on the right is the pair read from the divergence sequence, which is the column the depth thread has been quoting. Every exponent's stems are counted at 5/8 spirals from the points. Read from the angles, the deepest rule's stems come back as something else at every seed, and a pair with a 8 replaced in it is not a near miss but a family whose step is several times as long. The settled divergence moves by 0.118 degrees across all five exponents, so the rules are on one lattice and the disagreement is a defect in one of the two instruments.

The pair read from the angles

Checking that a jostled stem is still the lattice its panel is about turned up a disagreement. Counted from the point positions every rule's stems return five and eight spirals; read from the divergence sequence, the deepest rule's stems come back as five and seven at every seed. The points are right, and this site's founding rule is why.

All 19 destinations, counted at 17 windows from 20 to 800 organs. One row per destination, one cell per counting window, coarsest on the left. A dark cell is a window returning the same pair as the standing 200; a warm cell is a window returning a different one; a pale cell is a window that reaches below the run's own settling organ and was dropped. 5 of the 19 rows differ anywhere, and every difference sits left of the floor its own pair sets — 20 organs for a pair under 14 parastichies and the larger count plus 6 above it. The census read at 25 organs and at 800 is the same census.

The window nobody varied

Every parastichy pair reported for the settling table was counted over the top two hundred organs of a run, and that number had never been moved. Moved seventeen ways across a factor of forty, on all three hundred and eighty-four settled runs, it changes nothing at all.

What a 200-organ window allows against what the offset ceiling allows. Two settings decide what the counter can return, and at this window the offset ceiling is the tighter of them. A window of 200 organs holds a family of at most 194, since the counter scores an offset only where it has 6 hops of it inside the band; the largest offset it looks at is 60, a factor of 3.2 between them. The ticks are the members of the two sequences the essays are built on, and the largest counted number anywhere in the settling table is 19 — inside both bounds, which is what makes the table inert.

A plateau the instrument should have had

The counting window decides nothing on the settling table, and it has two bounds that decide everything outside it. A window one organ too narrow returns the previous rung of the same ladder, a family past the offset ceiling is reported as a coarser rung at every width there is, and neither failure produces a refusal, noise or a wide error bar.

The window 16 known pairs need, against the 20 organs the counter refuses under. Cylinders built at a stated divergence and rise, counted at every window from 20 organs to 200, so the pair is known before the counter sees it. The smallest window that reads it is 20 organs for every pair up to 14 parastichies and the larger count plus 6 above that — 13 of 16 measured exactly, on both branches, with nothing fitted. A pair of 76 and 123 needs 129 organs where 2 and 3 needs 20; the open marks are pairs the counter's own offset ceiling refuses at every window, read only once that ceiling is raised.

How many organs a pair needs

A count taken over too few organs does not fail. It returns the rung below, which is a perfectly good pair, and nothing anywhere says so. The window that avoids it is not a constant but the counter's own arithmetic, and 384 settled runs sit exactly where that arithmetic puts them.

The five readings of 384 whose two counted families wind the same way. Each row is an additive sequence with the term the counter stepped over drawn open between the two it returned. 4/9 is 1, 4, 5, 9, 14 with 5 skipped, and the pair a crossing requirement would have returned is 4/5 — a sequence the same destination's other runs already sit on. Dropping these readings takes the census at 40 starting angles from 31 arrangements and 15 sequences to 28 and 12, on 5 of 384 runs.

A sequence that was a reading

One of the settling table's additive sequences rested on a single count that skipped a term, and the account of it was that the count had been taken over the wrong patch. It was not. The counter works out whether its two families wind opposite ways and the reading throws the answer away, on five runs in three hundred and eighty-four.

The pairs grown stems of one, two, three and four organs a whorl count as their rise falls, against the folded rise. Four stems grown by the same rule over the same range of folded rise, 0.4 to 0.0012 — the rise between whorls times the jugacy, which puts every jugacy on one lattice axis — each whorl folded to one node and counted in a sliding window. With 1 organ a whorl the counted pairs run 1/2, 2/3, 3/5, 5/8, 8/13, 13/21; with 2 organs a whorl the counted pairs run 2/4, 4/6, 6/10, 10/16, 16/26, 26/42; with 3 organs a whorl the counted pairs run 3/6, 6/9, 9/15, 15/24, 24/39, 39/63; with 4 organs a whorl the counted pairs run 4/8, 8/12, 12/20, 20/32, 32/52, 52/84. Every jugacy passes the same transitions at the same folded rises, with its pairs multiplied by its jugacy.

A grown stem halves its ladder

A bijugate stem at 68.754° was derived to pass the ordinary transitions at half their rises, because a lattice wrapped twice round is the ordinary lattice at twice the rise. No bijugate stem had been grown through them. Grown by the same rule that grows an ordinary shoot, stems of two, three and four organs a whorl walk the ordinary ladder with every pair multiplied by the jugacy, change pair at the ordinary rise divided by the jugacy between whorls — and by its square per organ — lag behind the static ladder by the same hundredth of a rung, and settle within three hundredths of a degree of 137.5078 over the jugacy.

Which rates keep a Lucas seed, on stems of one, two and three organs a whorl, placed by whorls a rung. Every rate each configuration was grown at, drawn at the number of whorls the stem places between one transition and the next: a dark cell keeps the Lucas ladder to the end of the rise, a light cell gives it up. 1 a whorl, 512: kept to 86.6 whorls a rung, lost from 87.6; 1 a whorl, 256: kept to 86.6 whorls a rung, lost from 87.6; 1 a whorl, 1024: kept to 86.6 whorls a rung, lost from 87.6; 2 a whorl, 1024: kept to 87.6 whorls a rung, lost from 88.1; 2 a whorl, 1024, imposed: kept to 87.6 whorls a rung, lost from 88.1; 2 a whorl, 512: kept to 87.6 whorls a rung, lost from 88.5; 2 a whorl, 2048: kept to 87.6 whorls a rung, lost from 88.5; 3 a whorl, 1536: kept to 87.9 whorls a rung, lost from 88.5; 3 a whorl, 1536, imposed: kept to 87.9 whorls a rung, lost from 88.5; 3 a whorl, 768: kept to 86.6 whorls a rung, lost from 88.2. On this axis every edge falls within about two whorls a rung of every other.

A Lucas seed counts whorls

An ordinary stem seeded on the Lucas lattice keeps that ladder when its rise falls fast and gives it up when it falls slowly, with the edge near ninety nodes a rung. A stem that grows two organs a whorl is an ordinary stem folded twice round, so the identity predicts its edge — once it says whether the edge counts placements or whorls. Grown across the edge, stems of one, two and three organs a whorl keep a Lucas seed to 86.6, 87.6 and 87.9 whorls a rung: one number to within a per cent in whorls, and one, two and three times as far out in organs. No grid moves it and imposing exact whorls moves it not at all, even where free trijugate whorls come apart completely as the seed is lost.

How far grown whorls of two to eight members miss exact symmetry, at four lattices. Whorls grown one member at a time at a fixed rise, on a grid of 1,680 azimuths that every jugacy divides, read at folded rises of 0.27, 0.18, 0.09, 0.036. With 2 members the largest miss is 0.00°, 0.00°, 0.00°, 0.00°; with 3 members the largest miss is 0.64°, 1.93°, 0.21°, 0.00°; with 4 members the largest miss is 0.00°, 0.00°, 0.00°, 0.00°; with 5 members the largest miss is 1.29°, 1.29°, 0.21°, 0.21°; with 6 members the largest miss is 0.86°, 0.86°, 0.43°, 0.21°; with 7 members the largest miss is 0.86°, 2.14°, 0.21°, 0.00°; with 8 members the largest miss is 0.00°, 0.00°, 0.00°, 0.00°. Whorls of two, four and eight members are exact at every lattice; three, five, six and seven miss.

A whorl that misses its share

A whorl of k organs is defined by its symmetry, and the rule that grows one places its members one after another, each against the members already there. Nothing tells it to put them a k-th of a turn apart. Grown that way, whorls of two, four and eight members sit exactly on their shares of the turn at every lattice measured, and whorls of three, five, six and seven do not — the pattern a mirror argument predicts, since only a power of two leaves every new member a position that mirrors every member already placed. A trijugate whorl misses by 6.5° at a coarse rise and not at all at a fine one, by an amount the rise sets almost everywhere, and none of it moves a single transition.

How often three counters read a band's own pair, against the width of the band, on an ordinary stem. Bands of 40 to 320 organs slid up an ordinary stem grown at T = 300 over a rise from 0.05 to 0.0005, each read three ways. The counter without an index reads the pair at 65%, 66%, 87%, 88%, 88%, 88%, 89%, 100%, 100%, 92%, 54%, 23%, 8%, 2%, 1%, 0%, 0%, 0%; the counter families required to cross reads the pair at 65%, 66%, 87%, 88%, 88%, 88%, 89%, 100%, 100%, 92%, 54%, 22%, 8%, 2%, 1%, 0%, 0%, 0%; the counter with the index reads the pair at 100%, 100%, 100%, 100%, 100%, 100%, 100%, 100%, 100%, 100%, 100%, 100%, 100%, 100%, 100%, 100%, 100%, 100%. The index-free counter reads nine bands in ten or more only from 80 to 100 organs.

A counter that cannot be slid

The counter that needs no order of arrival follows each family into chains and counts them, and on an ideal lattice it agrees with the counter that does. On a stem whose rise falls it works only inside a band of widths, and outside the band it returns a pair rather than refusing: the rung below when the band is too narrow for the larger count, a pair on no rung when the band spans more than about a third of a rung of rise. The upper edge moves with the rate, so a width that is right on one stem is wrong on another, and on the fastest bijugate stem measured no width works at all.

The band the counter chooses for itself, against how slowly the stem grows. The counter is given positions and no width. It fits the decay of the rise through the spacings of its own band's whorls and takes the band that spans a third of a rung of rise: 49 organs on the stem that falls over 150; 97 organs on the stem that falls over 300; 193 organs on the stem that falls over 600. A stem grown four times as slowly chooses a band 3.94 times as wide, because a third of a rung is a statement about rise and a width is a number of organs. That is exactly why no one width fits the seven stems, and why the previous reading found a stem for which none worked.

A band that follows the rise

The index-free counter reads a growing stem only inside a band of widths, and the upper edge is a third of a rung of rise rather than a number of organs — so a width right on one stem is wrong on another. A counter that fits the decay of the rise through the spacings of its own band's whorls, and takes the band spanning a third of a rung, chooses 49, 97 and 193 organs on stems falling over 150, 300 and 600. Given no width at all it reads within eight points of the best of eighteen fixed widths on six of the seven stems, matching it exactly on one and beating it on two. Refusing any band too narrow to have shown the rung above the pair it counted removes every reading of the rung below, on every stem — and costs between five and fifty-one points of correct reading to do it.

Read as whorls, the three jugacies fall on one line. The rate at which a stem seeded on the Lucas lattice gives that ladder up, against how long the seed is, for one, two and three organs a whorl. The three lie on top of each other at every seed length: 15 whorls gives 30.80, 30.80, 30.80; 20 whorls gives 42.35, 42.35, 40.42; 30 whorls gives 63.52, 63.52, 63.52; 40 whorls gives 85.66, 87.58, 85.66. Every departure is one or two steps of the grain the edge is bisected to, which is two whorls of rate — 1.925 whorls a rung at every jugacy and every edge drawn.

A seed measured in whorls

A Lucas seed's length counts whorls, and the number published earlier for the rate edge counts nothing at all. Grown from seeds of fifteen to eighty whorls at one, two and three organs a whorl, stems of every jugacy lose the seed at the same rate in whorls a rung for the same seed length in whorls — 30.80 at fifteen, 63.52 at thirty, identically across the three — and at the same seed length in organs they differ by a factor of 3.24. So the seed is measured in whorls, as the edge is. The other half is worse for the earlier reading: the edge is not a constant but 2.24 times the seed less three, straight to within 2.7 whorls a rung over a fivefold range, so the eighty-seven whorls a rung reported everywhere is a property of the forty-whorl seed nobody varied.

The rate at which a Lucas seed is lost, read through four counting windows. Eighteen stems — one, two and three organs a whorl, seeded with 15 to 80 whorls of Lucas lattice — each with the edge found four times, through counting windows of 20, 26, 39, 52 folded nodes, the four dots at each seed drawn side by side. 17 of the 18 stems give the same organ through every window. The exception is three organs a whorl with a 30-whorl seed, where the widest window finds the edge at 207 organs against 201 for the others. A window twice as wide as the one every earlier reading used moves no other edge by an organ, so the window is not what makes the ratio of edge to seed change with the seed.

The window was not carrying it

The ratio of a Lucas seed's rate edge to its length rose from 2.05 at fifteen whorls to 2.20 at eighty, and the suspect was the counting window, which is most of a short seed. Read through windows of 20, 26, 39 and 52 folded nodes, seventeen of the eighteen stems lose the seed at exactly the same organ of rate, so the window carries almost none of it. Part of the rise was the grain of rate the edge was found on, worth up to six hundredths of the ratio. What is left rises by a tenth below thirty whorls and has stopped by sixty, at a level an ordinary stem reaches about one and a half per cent lower than a bijugate or trijugate one — and a trijugate edge is not always a line.

What the round trip returns as the organs are displaced: golden angle, 300 organs. Thirty heads at each size of displacement, from none to two and a half spacings, each run through the round trip and sorted by what came back. The undisplaced head counts 21 and 34. At 0.25 spacings, 97% the undisplaced pair, 3% a neighbouring pair; at 0.5 spacings, 50% the undisplaced pair, 50% a neighbouring pair; at 0.75 spacings, 30% the undisplaced pair, 70% a neighbouring pair; at 1 spacing, 13% the undisplaced pair, 80% a neighbouring pair, 7% a straddling pair; at 1.25 spacings, 7% the undisplaced pair, 57% a neighbouring pair, 7% a straddling pair, 30% refused; at 1.5 spacings, 40% a neighbouring pair, 3% a straddling pair, 57% refused; at 2 spacings, 10% a neighbouring pair, 90% refused; at 2.5 spacings, 3% a neighbouring pair, 97% refused. 10 recovered intervals in all exclude the true angle.

A head displaced before it is counted

The round trip from a head's spiral counts back to its divergence angle was tested on heads whose every organ sat exactly where the rule put it. Displaced by a normal error of up to two and a half spacings, heads of 900 organs keep counting a pair from their own sequence and return intervals holding the true angle to a spacing and a half; heads of 300 organs move to the neighbouring pair by half a spacing and then refuse, nine in ten of them by two spacings. Every moved count brings in the family whose chord was third shortest. Of 898 heads recovered, 19 intervals miss the true angle and 17 of those by about a tenth of a degree — displacement makes the reading coarser and then silent, not confidently wrong.

A head of 300 organs as the rule places it, and twisted by 2 rad. Two heads at the golden angle, 300 organs each. On the left every organ is where Vogel's rule puts it, and the round trip counts 21 and 34 and recovers 137.515°. On the right the same organs turned about the centre by 2 radians times their radius over the head's, so the rim turns by 2 radians and the centre not at all; the round trip counts 34 and 47, recovers 137.80°, missing, with an interval of 0.230° that excludes the true angle.

A twist is a divergence

Recovering a head's divergence angle from its spiral counts survived independent displacements of whole spacings, moving to a neighbouring pair and then refusing rather than misleading. Displacements with a direction are harder on it in only one case. A head pressed to an aspect ratio of 2.25, spread at the rim by eighty per cent or sheared with a slope of 1.6 is still counted as its own pair or its neighbour, and recovered inside its interval. A head twisted — each organ turned about the centre in proportion to its radius — is not: past a turn of the rim of about a radian and a half the counts leave their sequence and the recovered angle misses, by up to sixty-one degrees, because a twist changes the angle between one organ and the next. The round trip is not fooled. It is reporting the angle the twisted head has.

A 900-organ head twisted by 0.5 radians at the rim, with the bands its spirals are counted in. Every organ turned about the centre by 0.5 times its radius over the head's. The rings mark the inner annulus, 0.35 to 0.6 of the radius, and the outer, 0.7 to 0.95. The inner annulus counts 34 and 55; the outer 34 and 89 — skips one; the single band, 0.55 to 0.95, 34 and 89 — skips one. A count that is not two consecutive Fibonacci numbers flags the twist; on this head that happens from half a radian, and the single band is misled only from six.

The count sees the twist first

A twisted head recovers a changed divergence, and the check proposed for it was two annuli: the twist's extra angle falls with radius, so an inner and an outer annulus should disagree. Read on golden heads of 900, 2,400 and 9,000 organs, they never do in time. Their intervals separate at eight radians on 900 organs and never on the larger heads, always after the ordinary reading has been misled — from six radians on 900 organs and from two on 2,400 and 9,000. What catches the twist first, at every size and on every seed, is the count: at half a radian to three quarters some band stops returning two consecutive Fibonacci numbers — 34 and 89, 89 and 233 — which no untwisted golden head, clean or displaced, ever does.

The flag reads the angle, not the twist

A twisted seed head is caught first by its counts: at half a radian some band stops returning two consecutive Fibonacci numbers, which an untwisted golden head never does. Twist the head in proportion to the square of the radius instead and the organs land exactly where a head grown at the golden angle plus a/(N − 1) puts them — to a billionth of a spacing — so the two read the same pairs in every band, and the flag fires on both. The flag detects a divergence that is not golden, and it has a resolution: 0.015° on 900 organs, 0.01° on 2,400, 0.003° on 9,000. Every twist shape from a half to four is flagged exactly when its change to the divergence in the outer annulus passes that resolution. So a flagged head is not golden, and nothing in its counts says whether it was twisted or grown that way.

The divergence read from the positions ring by ring on a 900-organ head twisted half a radian at the rim, for six shapes of twist. A 900-organ golden head twisted by 0.5 radians at the rim as a·(r/R)^p, displaced by 0.1 of a spacing, read in six rings from 0.35 to 0.95 of the radius by the organs' phase sums within the counts' interval. Above the golden angle, ring by ring: p 0.5: 0.0371°, 0.0212°, 0.0179°, 0.0126°, 0.0114°, 0.0091°; p 1: 0.0451°, 0.0306°, 0.0271°, 0.0214°, 0.0192°, 0.0176°; p 1.5: 0.0426°, 0.0324°, 0.0320°, 0.0277°, 0.0264°, 0.0253°; p 2: 0.0374°, 0.0318°, 0.0334°, 0.0310°, 0.0316°, 0.0321°; p 3: 0.0241°, 0.0237°, 0.0306°, 0.0334°, 0.0385°, 0.0437°; p 4: 0.0148°, 0.0156°, 0.0249°, 0.0313°, 0.0414°, 0.0530°. The square law reads flat, 0.0310° to 0.0374°, as a head grown at another angle does; every other shape falls or rises across the head.

The positions read the twist's shape

A skipped Fibonacci count says a seed head's divergence is not golden, and cannot say whether it was twisted or grew that way; only the radial profile of the angle between neighbours can, and the counts are too coarse to see it. The organs' positions are not. The birth order is not in them — consecutive organs at the rim of a 900-organ head are a hundred and sixth of a spacing apart in radius, so a hundredth of a spacing of displacement scrambles a quarter of them — but each counted family is, and the sum of its phase angles across the organs peaks at the local divergence with no organ named. Displaced by a tenth of a spacing, a 900-organ head's outer annulus reads its divergence to 0.0001°, a hundred times finer than the flag resolves it, and the two annuli tell every twist shape from a changed angle at a quarter of a radian or less, no later than the flag fires. The square-law twist stays inside the untwisted heads' spread at every size and displacement: it is a changed angle, organ for organ.

A seed head with half its rim missing, and three places its centre could be put. A 900-organ golden head, every organ displaced by 0.1 of a spacing, with the organs beyond seven tenths of the radius removed over half the head. The centroid of what is left lies 3.099 spacings from the centre the head grew about; the centre about which the inner annulus's two counted families are most coherent lies 0.0132 spacings from it. The inset magnifies half a spacing around the true centre.

The centre the spirals give

Reading a seed head's divergence from its organs' positions needs the head's centre, and a photograph does not give it. A misplaced centre turns every organ's angle by an amount that varies round the head and grows toward the middle, so it could read as a twist. It does not, until it is large: a 900-organ head tolerates a centre a sixth of a spacing off, a 2,400-organ head a quarter. The centroid of the organs finds a whole head's centre to a hundredth of a spacing, and read about it the positions separate every twist shape exactly as well as with the centre given. The radius law, which looks like the natural fit, does not find it at all. And when half the rim is missing the centroid moves three to five spacings and turns an untwisted head into a twisted one, while the centre about which the counted spirals are most coherent still lands within a hundredth of a spacing.

A golden seed head photographed off its axis, with the squash its second moments find. A 900-organ golden head, every organ displaced by a tenth of a spacing, photographed 20° off its axis: every coordinate along a bearing of 7° shortened by the cosine of the tilt. The head's second moments about its centroid put the short axis at 7.7° and imply a tilt of 19.75°. Read square on, the inner annulus gives a divergence of 137.5080° and the outer 137.5080°; stretched back along the short axis first, 137.5078° and 137.5077°, against the golden angle's 137.5078°. The bands count 34 and 55 spirals inside and 55 and 89 outside.

A head photographed from the side

A seed head photographed off its axis is squashed by the cosine of the tilt across one bearing, and the squash turns every organ's angle by an amount that goes round the head twice. Read square on, a 900-organ golden head tolerates twelve to fourteen degrees of tilt before its two annuli disagree by more than any untwisted head's do, and the spiral counts never notice at all. Past that, the reading is not a smooth drift but a scatter of heads misread one at a time. Stretching the head back along the short axis its own second moments find restores the square-on reading exactly to thirty degrees, even though the moments cannot see a tilt under three or four degrees. And the square-law twist stays hidden: it is a changed angle organ for organ, so no camera, at any angle, sees it differently.

The edge-to-seed ratio of ordinary stems that count one to five images of each neighbour. The ratio of the rate edge to the seed, on a fine rate, for ordinary stems whose repulsion counts each neighbour at one to five images a whole folded turn apart, at seeds of 15 to 80 whorls. one image: 2.076, 2.133, 2.175, 2.166, 2.186, 2.184, level 2.1848; two images: 2.065, 2.141, 2.175, 2.197, 2.215, 2.216, level 2.2155; three images: 2.075, 2.135, 2.178, 2.205, 2.224, 2.224, level 2.2237; four images: 2.073, 2.137, 2.176, 2.204, 2.222, 2.223, level 2.2226; five images: 2.074, 2.136, 2.177, 2.205, 2.223, 2.223, level 2.2231. One image is the ordinary stem and two the bijugate one; the step between them is the one and a half per cent the jugacies differ by, and the images beyond the second move the long seeds' level by less.

A whorl feels the far side of its stem

An ordinary stem keeps a long Lucas seed to an edge-to-seed ratio of 2.185 and a bijugate stem to 2.216, and the question was whether the difference follows the stem's angle or its timing. Folded, it is neither. A bijugate stem grown by the free rule is, node for node, an ordinary stem whose repulsion counts every neighbour twice round — once where it is and once a whole folded turn away — and that second copy is the whole of the difference. It carries an eighth of the repulsion where the seed is won or lost and a fortieth where the stem ends, and turned up from nothing it does nothing for its first quarter and then carries the ratio steadily from one level to the other.

A head seen in perspective needs a camera, not an ellipse

A seed head photographed off its axis from far away is squashed, and stretching it back along the short axis of its own second moments restores its reading exactly. Photographed from close up, the near side of the head is larger than the far side, and the stretch stops working: at twenty degrees off the axis it reads every head right from ten head radii and none from five. The organs' centroid drifts towards the near side as the camera comes in, but that is not why — the same stretch about the head's true centre fails at the same distances. What fails is the taper. A camera fitted from the organs alone, its tilt, bearing, distance and the head's centre chosen so that the head it un-projects is round, centred and evenly packed, reads seven to ten heads in ten right at every distance down to two radii.

The edge-to-seed ratio of ordinary stems counting one, two and three images, as the repulsion's range shortens. The level of the ratio of the rate edge to the seed, over Lucas seeds of sixty and eighty whorls, for ordinary stems whose repulsion counts each neighbour at one, two and three images, every term cut off by a gaussian of the stated range in spacings. none: 2.1848, 2.2155, 2.2237; 6: 2.1901, 2.2209, 2.2280; 4: 2.1950, 2.2282, 2.2337; 3: 2.2025, 2.2389, 2.2424; 2.5: 2.2199, 2.2500, 2.2521; 2: 2.2474, 2.2717, 2.2724; 1.75: 2.2806, 2.2921, 2.2924; 1.5: 2.3332, 2.3274, 2.3275; 1.25: 2.4027, 2.4004, 2.4004; 1: 2.5507, 2.5476, 2.5476. With no cut-off the three are the ordinary, bijugate and three-image levels; every level rises as the range shortens, and below two spacings the images stop mattering.

Cutting the far side in half keeps its step

A whorled stem differs from an ordinary one by the far copies of its neighbourhood, a whole folded turn away, and the inverse cube that felt them had no range. Give the repulsion a gaussian range of a few spacings and the far copies' share of the repulsion falls with it — halved at three spacings — while the one and a half per cent they make in a Lucas seed's edge does not fall at all. At three spacings it is a fifth larger than uncut. It shrinks only below two and a half spacings and is gone at one and a half, where the far copies carry under two per cent; and on the way the ordinary stem changes how it loses its seed, from a whorled pair to the Fibonacci pair a bijugate stem always left by.

A golden seed head laid on a dome and photographed twenty degrees off its axis from infinitely far. A 900-organ golden head, every organ displaced by a tenth of a spacing, laid evenly on a paraboloid dome 0.2 head radii high and photographed by a camera infinitely far away, tilted 20° across a bearing of 7°. The dome lifts the centre of the head towards the camera, and seen from the side every organ moves along the bearing by its height times the sine of the tilt, more at the centre than at the rim. Of ten such heads, 0 are read as untwisted after the moments' ellipse and 0 through the camera fitted for a flat head, which puts the camera 7.4 head radii away at a tilt of 20.2° (median).

A dome is read as a nearer camera

A camera fitted from a photographed seed head's organs alone restored the reading of a flat head seen in perspective from as close as two radii. Lay the same heads on domes and the camera is fooled in a particular way: it finds the tilt, its median within about a degree, and puts itself nearer than it is, by an amount set by the dome's height, so that a dome a fifth of the head's radius high photographed from infinitely far is read as a flat head photographed from seven radii. The reading it hands on is wrong. Off the axis, a domed head is misread once its height times the sine of the tilt passes about four hundredths of the radius — the same number at which a flat head's perspective broke the old correction — and only a photograph taken square on is safe.

The far image's step against what the cut-off leaves of the far copy: both shapes lose it at about three hundredths. Each range of each shape placed at the weight its cut-off leaves on the far copy of the nearest neighbour, a turn round, at the first organ above an eighty-whorl seed. Exponential: 0.585 0.0341, 0.447 0.0286, 0.343 0.0274, 0.201 0.0205, 0.119 0.0091, 0.078 0.0038, 0.042 0.0134, 0.027 0.0004, 0.015 -0.0118, 0.006 -0.0033, 0.002 -0.0104. Gaussian: 0.749 0.0308, 0.521 0.0332, 0.314 0.0364, 0.191 0.0301, 0.076 0.0243, 0.035 0.0115, 0.011 -0.0057, 0.002 -0.0023. On this axis the two shapes lose the step together: the gaussian's last surviving step is at a weight of 0.035, the exponential's at 0.042, and both are gone below three hundredths — the dashed line.

The far copy's weight sets the step, whatever the cut's shape

A whorled stem keeps a Lucas seed longer than an ordinary one because its repulsion feels each neighbour's far copy, a whole folded turn away, and a gaussian cut-off removed that step only below one and a half spacings. Grown again under an exponential cut-off — the profile of a substance that diffuses and decays — the step goes at a range of its own, between one spacing and seven eighths. But it goes at the same weight of the far copy: under both shapes the step survives while the cut-off leaves the nearest neighbour's far copy more than about three hundredths of its weight, and is gone below. The exit from the seed changes at about half that copy's weight under both. The far side's share of the whole repulsion lines up with neither.

Whether a stem keeps its Lucas seed at twenty-four rates, under a hard cut-off of 3.5 spacings. Ordinary stems on an eighty-whorl Lucas seed, their repulsion counting every neighbour within 3.5 spacings in full and none beyond, grown at twenty-four rates, the rise falling by a factor of e over 56 to 240 organs, counting each neighbour once (top) or twice round (bottom). A green cell kept the seed's pair to the end of the stem; an orange one lost it. Once round: 3 changes along the rates, 1 rate lost below the last one kept; Twice round: 3 changes along the rates, 1 rate lost below the last one kept. Keeping the seed is not a matter of where the rate sits against one edge: there is no single rate edge.

A hard edge has no step to read

Under a gaussian and an exponential cut-off the far image's step lasted while the nearest neighbour's far copy kept about three hundredths of its weight, and the exit from a Lucas seed changed where it kept about a half — which predicted that under a hard edge both would change exactly where the far copy crosses it, 3.2 spacings out. Grown, neither does. The exit changes between six spacings and four and a half, with the far copy counted in full. The step is 0.029 at six spacings, under 0.002 at four and a half and 0.041 at four, the far copy counted in full at all three. And from four spacings down, before the edge reaches the far copy at all, whether a stem keeps its seed stops depending on its rate in one direction: kept, lost and kept again as the rate changes. A hard edge cuts through near neighbours and half-turn images first, and a ruler built on the far copy has nothing left to read.

One cone, one a whorl stem read band by band at a fixed width of 49 organs and by the band that sizes itself. Every band top of the cone, one a whorl stem, from organ 326 to 1926, coloured by what the count returned: right, the rung below, a pair on no rung, or refused. Top row: the chain counter at a fixed width of 49 organs, right on 68% of bands — 100%, 100% and 4% in the stem's first, middle and last thirds — with 32% the rung below and 0% a pair on no rung. Bottom row: the counter that sizes its own band, right on 94% and refusing 6%, the rung below on 0% and a pair on no rung on 0%. Its width runs from 84 organs to 272 along the stem. Below, how fast the stem's rise falls at each band, from 1/261 to 1/847 per organ.

A band that follows a curving rise

The index-free counter that sizes its own band was built on stems whose rise falls by a constant factor per organ, so the decay it fits through its band's spacings was one number and the fit a straight line. On stems whose rise falls as a cone's does, fast near the base and slowing sixfold, or as an ogive's, slow and then speeding up, the straight line still recovers the local decay to within half a per cent at every band, and every band it chooses spans a third of a rung of rise, 0.324 to 0.333, with no excess at either end. Its width runs from 84 organs to 272 up a cone and from 139 to 58 up an ogive. A fixed width reads one end of such a stem and fails at the other: 49 organs returns the rung below on almost all of a cone's tip, 193 a pair on no rung on almost all of its base. Where the self-sized band fails, at an ogive's fastest end, it refuses, and it never returns the rung below.

A head is a lattice two dozen organs at a time

Everything said about a seed head leans on the head being a lattice, and nobody states how much of one it is. Fitted by geometry alone, a straight lattice misses a patch of radius L at radius r by L squared over four r, to within a few per cent, whatever the divergence. So a lattice good to a tenth of a spacing holds about three quarters of r organs: twenty-three at organ 900, thirty-seven at the rim of a 2,400-organ head. At about twice that, two organs land on one site and the patch has no straight lattice at all. The coordinates that straighten a cylinder do not help. Azimuth against radius squared makes the whole head exact, to the last bit a double holds.

A golden head of 1200 organs whose radius grows as the organ's index to the power 0.5, with the radii where its counted pair changes. The same divergence at every setting: only the exponent of the radius changes. With organ i at i to the power 0.5, the pair 21/34 arrives at organ 65, radius 8.1; the pair 34/55 arrives at organ 187, radius 13.7; the pair 55/89 arrives at organ 517, radius 22.7. A head that crowds its rim has its transitions earlier in radius than one that spreads it.

A growth law moves the counts

A head's counted pair changes at radii a factor of φ apart, and that factor has only ever been read on heads whose radius is the square root of an organ's index. Free the exponent and the arithmetic says two things: the same pair arrives at an organ in proportion to the exponent, and consecutive changes of pair are a factor of φ to the power twice the exponent apart in radius. Measured on six exponents from 0.4 to 0.7, both hold, the second to about a hundredth of the exponent, and it needs no organ counted from the centre. It also survives displacement of a fifth of a spacing, where a reading from the cells has long since failed.

What each counter returns, band by band, along a stem falling thirtyfold at the base rate and climbing back. One mark per band. The counter as built is right on 92 per cent of bands on the falling side and 12 per cent on the climbing side, where it refuses 60 per cent and returns a pair on no transition on 28 per cent. Sized from the magnitude of the decay, it is right on 94 per cent of the climbing side. The widths 49, 97 and 193 are on the rows below.

A counter built on a falling rise

A shoot starting to flower has a rise that falls and then climbs again. Read through that turn, the index-free counter that sizes its own band is right on nine bands in ten going down and one in eight coming back, and on a quarter of those it returns a pair on no transition at all. The fit it depends on is fine. It reads a negative decay perfectly and treats it as no decay, so it holds its widest band over more than a whole step of rise between transitions. Sized from the magnitude of the decay it reads the climb as well as the fall, at 94 per cent on every stem. And the stem itself goes back down the transitions it went up, at the same rises, to the width of a band.

The cells of a 2400-organ head grown 2 organs at a time that are not hexagons. Organs in pale points; five-sided cells and seven-sided cells marked. 330 of 1596 interior cells are not hexagons, lying on rings at 22.5, 36.5, each ring with its pattern repeated 2 times round the head.

A whorl moves a ring out and multiplies it

A head grown k organs at a time is an ordinary lattice wrapped k times, which predicts its defect rings with no measurement at all: each of an ordinary head's rings should appear at k times the radius, holding k times the cells, one copy in every sector. On eight-thousand-organ heads of two to five organs a whorl every ring read holds exactly that count, 110 fives where 55 would be and 105 where 21 would be, at the predicted radius to a tenth of a unit. Stagger the members of each whorl by the single organ's radius that Vogel's rule would give and the counts do not change, while a fifth to a quarter of the defects lose their partner one sector on.

A stem at a rise of 0.003 unrolled, with each organ moved by 0.5 of a spacing. Two hundred and forty organs of an ideal lattice, azimuth across and height up, each moved in both directions by a gaussian with a standard deviation of 0.5 of the lattice's shortest vector. Open circles mark where each organ belonged, and lines join it to where it went. The pair 8/13 is the lattice's own.

A wider window buys back a shaken stem

On exact stems the counting window decides nothing once it clears a small floor, which leaves it looking like a setting nobody needs. Move every organ and it becomes the instrument. A pair that a twenty-organ window reads on an ideal lattice needs one or two hundred organs with each organ moved by half a spacing, four to eight hundred at four fifths, and up to twelve hundred at a whole spacing, where the organs have left their places altogether and the pair is still there to be counted. The price is steep, close to the two-and-a-half power of the error, and at one and a half spacings no window up to sixteen hundred organs holds the pair. When it goes, it goes upward: the wrong answer is the next pair, not the one below.

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