Phyllotaxis
the spirals are counted, not admired
Concept

Rotational symmetry — where it appears

How many times a pattern maps onto itself in one turn, measured on the positions rather than assumed from the counts. It is the property a whorled stem has and a spiral one does not, and spiral counts alone cannot decide it.

Named by 10 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 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.

lattices · Chain counting Every open question here needs under 34 specimens. The sample size at which each comparison reaches 90 per cent power at a 5 per cent false-positive rate, from the exact binomial rather than a normal approximation. The census question — do plants show consecutive Fibonacci pairs far more often than the geometry does — needs 4: 14.7% is the share of divergence angles giving a consecutive Fibonacci pair at a fine rise; 90% is what a grown history gives.

The survey this site cannot do

Four rounds of asking for a dataset, and it is still not here. What the work here can do instead is specify it — the fields, the sampling, the sizes, and which of this collection's claims each one would settle. Two of the four fields asked for turn out to be worth less than the asking implied, and one was never asked for at all.

wrong · Survey spec 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 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 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 The twelve wrecked runs that finish at a half turn. Wrecked runs whose final divergence is a half turn, which is two files of organs rather than a spiral. Every one is on a rung at the coarse end of its branch, where the front is short enough that removing one organ reaches past it. The value is within 0.35 degrees of 180 on eight of the twelve.

Two files, and a way back

Twelve wrecked runs finish at exactly a half turn, which is a pattern with no spiral in it, and every one is at the coarse end of the ladder. Three finish at the divergence they would have had anyway, after being thrown a hundred degrees off it.

cylinder · Both walls

Named alongside it

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

JugacyBijugateHonest limitsRiseAblationCounting blindClaim testingMeasurementParastichy pairWhorlWhorledCensus

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