Phyllotaxis
the spirals are counted, not admired
Concept

Coprime — where it appears

Two whole numbers are coprime when no number larger than one divides both. An ordinary lattice's two spiral counts are always coprime, so a reported pair that shares a factor describes a whorled pattern or a miscount rather than a slightly different spiral.

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

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.

lattices · Recovery For a high pair a miscount does not blur the report: it moves it. The divergence axis from 50° to 145°. For 21/34 and 34/55, the dark band is what the report allows if both counts are right and the pale bands what it allows if either is wrong by one. 21/34 allows 0.308° exactly and 0.931° in all, in 3 pieces: 20/33 at 54.37° and 22/35 at 82.13° beside the true band at 137.48°; 34/55 allows 0.118° exactly and 0.354° in all, in 3 pieces: 33/56 at 109.23° and 35/54 at 113.27° beside the true band at 137.52°. Each wrong band is as narrow as the right one and tens of degrees from it, so the reading does not widen; it becomes a short list of sharp candidates, and neither wrong candidate contains the golden angle.

A count that can be wrong by one

A reported parastichy pair pins the divergence angle to a band 221°/mn wide only if both counts are right. Allowing either to be off by one adds the bands of every neighbouring pair whose counts share no factor, and those bands sit where their own lattices live: for 2/3 they swallow the whole range, and for 34/55 they are two bands as narrow as the true one at 109° and 113°, twenty-five degrees away. So a high count that may be wrong is not a blurred reading but a short list of sharp candidates, costing log₂ 3 bits. And on the Fibonacci pairs, two in every six — 21/34 and 34/55 among them — cannot be miscounted silently by one count at all, because every such miscount shares a factor.

wrong · Count value

Named alongside it

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

Divergence angleFibonacciAnnulusBranchCylinderHonest limitsIdentifiabilityInterval estimateLattice chordLucas numbersMeasurement errorParastichy

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