Phyllotaxis
the spirals are counted, not admired
Concept

Spiral lattice — where it appears

The pattern made by placing each organ at a fixed turn from the last while the radius or height rises steadily. Locally it is a lattice with two generating vectors, though a head holds a straight one only over a few dozen organs, and all of it only in stretched coordinates.

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

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.

lattices · Lattice A patch 3 spacings in radius at radius 30 of a golden head, against the straight lattice that fits it best. The 32 organs of a patch at radius 30, labelled by geometry alone with the nearest site of a straight lattice, and that lattice refitted to them. Filled: the organs. Open: the fitted sites, joined to their organs. The organs miss by 0.130 of a spacing at root mean square and 0.218 at worst; no two organs share a site.

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.

lattices · Lattice

Named alongside it

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

CylinderParastichyVogel's modelConeDivergence angleHonest limitsL-systemsLattice vectorsMeasurementMeristemPrimordiumRate

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