Phyllotaxis
the spirals are counted, not admired
Concept

Bijugate — where it appears

A pattern arriving two organs at a time, whose parastichy numbers therefore share a factor of two. A rule placing pairs grows one without being told to, and the resulting symmetry can be measured on the positions afterwards.

Named by 11 essays across 3 fields — each of them below, with the objects they name alongside it.

What the "whorled" bucket contains, at a rise of 0.008. 7 pairs, sharing 7 different factors, and every one of them is k and 2k. The bucket the previous census called whorled is the coarsest pattern the ladder has, repeated k times around the stem — not a residue of odd arrangements.

What "whorled" was hiding

The earlier work's census put 35% of divergences in a bucket labelled whorled and moved on. Opened, every pair in it is k and 2k — the coarsest rung of the ladder, repeated k times — and reading the census up to jugacy takes the Fibonacci share from 14.7% to 50.1% without describing a single extra plant.

wrong · Jugate census 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.

lattices · Jugacy 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.

lattices · Jugate limit 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.

lattices · Dose 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.

lattices · Jugacy 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.

lattices · Jugate limit 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.

lattices · Jugate limit 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.

lattices · Jugacy 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.

lattices · Jugate limit 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.

lattices · Jugate limit A bijugate stem answers in pairs, once the half turn is taken out. Removing one organ from a stem grown by a rule that places two at a time, at a rise of 0.0065, where the pattern counts 6 and 10 and has rotational symmetry of order 2. The open circles are the displacement of the next organ as it comes out of the arithmetic, which reaches 180° at offsets where the stem has demonstrably not been disturbed — the two organs of a whorl are interchangeable, so calling the other one "next" is a relabelling and not a movement. The filled circles are the same numbers read modulo 180°, which is the only way a 2-jugate divergence is defined. Read that way the response is a run of equal pairs — 68.0°, 68.0°, 42.9°, 42.9° — ending at 10, the larger parastichy number, with everything past it under 0.5°. Two organs of one whorl give the same answer to the last digit, which is the pattern's symmetry showing up in an experiment.

What a cut costs a whorl

A bijugate pattern is an ordinary lattice seen twice over, so the account that says a wrecked stem's repeating block is the repeat unit of the lattice underneath has a specific prediction here: three and five. It gets six and ten. And the thing a single missing organ does destroy on a whorled stem is the one property its counts cannot see.

cylinder · Jugacy

Named alongside it

The objects these essays reach for when they reach for this one.

JugacyHonest limitsRiseRotational symmetryClaim testingMeasurementControlParastichy pairThe placement ruleAblationCounting blindCensus

All concepts