Phyllotaxis
the spirals are counted, not admired
Concept

Chain counting — where it appears

Counting a parastichy family by tracing each short lattice step from point to point and counting the chains. It needs no order of arrival and no divergence angle, which is what makes it usable on a photograph and on a point set whose history is unknown.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

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 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.

lattices · Counting 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.

lattices · Chain counting 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.

lattices · Chain counting 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.

Counting blindHonest limitsParastichy pairJugacyMeasurementNearest neighbourRefusalRiseAblationClaim testingInstrument settingLattice

All concepts