Phyllotaxis
the spirals are counted, not admired
Concept

Assumed centre — where it appears

The point a spiral fit measures its radii from, chosen rather than given because no shell carries its mathematical centre marked. Displacing it by a quarter of the innermost whorl's radius moves the recovered growth factor by 4.56 per cent at two turns, about five times the figure published here.

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

Also named here as span of arc — the same set of essays touches all of them, so they are one junction rather than several.

What a quarter-radius centre error costs a 3.2× spiral, against what the collection publishes. Root-mean-square error in the recovered growth factor when the assumed centre is displaced by a quarter of the innermost whorl's radius, against how much arc is measured. It is 4.56 per cent at two turns, 2.52 per cent at two and a half, 1.91 per cent at three and 1.34 per cent at three and a half. It first falls under one per cent at 4.25 turns — and at 4.25 turns at all six of the factors the collection draws, so the span rather than the factor is what decides it.

What the centre costs

The fit that recovers a shell's growth factor needs a centre, and no shell has one marked. Displacing it by a quarter of the innermost whorl's radius moves the answer by 4.56 per cent at two turns, which is about five times what this collection has been publishing.

shells · Spiral fit Every assumed centre from 0.01 to 500 innermost radii, at 19 spans, against a spiral drawn at 3.2. One row per span of arc, one cell per assumed displacement on a logarithmic grid from 0.01 to 500 innermost radii, shaded by the highest growth factor any of 180 directions returns there. A displaced centre reaches 6.854 at every span up to 1.15 turns and at no span from 1.2 upward, so the boundary is a span rather than a displacement. The dashed rule is the two-turn span floor, and the cheapest golden fit anywhere leaves a residual of 0.163 against a threshold of 0.15 — so a golden reading is refused twice over.

How far a centre must move

Four hundred and eighty-two thousand assumed centres, at nineteen spans and a hundred and eighty directions each, asked whether a spiral drawn at 3.2 can be made to read as the golden 6.854. It can, at every span up to 1.15 turns and at none from 1.2 upward, and every centre that manages it is refused twice over.

shells · Spiral fit

Named alongside it

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

Claim testingDisplacementGrowth factorLogarithmic spiralResidualSpan of arcφ, the golden ratioHonest limitsMeasurement errorNegative resultParameter spaceRefusal

All concepts