Phyllotaxis
the spirals are counted, not admired
Concept

Goodness of fit — where it appears

How closely a fitted model reproduces the data it was fitted to, reported here as the largest departure of the points from the fitted curve. A close fit says the model can describe the data, not that it is the right model: a spiral whose growth changes by sixty-four per cent still fits as one.

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

The fit pointed at twelve curves, and the residual each one leaves. Every curve is handed to the same recovery from its own true centre with no noise anywhere, and every one of them returns a growth factor. A circle comes back at exactly 1.0000 with a residual of zero, and so does every ellipse tried, whatever its aspect. An Archimedean spiral read from its second turn comes back at 1.306 with a residual of 0.090 and is accepted; it is refused only when the arc includes its own first turn, at 0.1976. Across every growth factor and span the fit was tested at, the band inverts — a genuine spiral reaches 0.670 while archimedes-4 sits at 0.035 — so no threshold separates them.

The residual is not the test

The fit that recovers a growth factor also hands back a residual, and that residual has been read as what separates a genuine logarithmic spiral from something that merely looks like one. Pointed at twelve curves it fits a circle exactly, accepts an Archimedean spiral, and refuses a golden one that is right to three decimal places.

shells · Spiral fit How far a shell growing from 2.8 to 3.6 a turn departs from the one spiral a fit gives it. The shell's logarithmic radius along its 4-turn arc, less the straight line a single growth factor fits. The fit returns 3.17490 a turn, which is the geometric mean of the two ends, 3.17490. The largest departure is 0.0836 in the logarithm against the 0.15 the collection refuses a spiral past, so the fit accepts this shell; a change that is steady in its rate leaves ln(W₁/W₀) × turns/12 = 0.0838.

One number for a shell that changes

An animal is under no obligation to grow at one rate from hatching to maturity, and the fit that recovers a shell's growth factor returns one number whatever it is given. Handed a shell whose expansion rises steadily from 2.8 to 3.6 a turn, it returns 3.17490 — the geometric mean of the two ends, exactly — with a residual it accepts. A change of sixty-four per cent over three and a half turns passes as one logarithmic spiral, and at the aperture, where contact is decided, the one number and the last whorl give opposite verdicts.

shells · Spiral fit

Named alongside it

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

Claim testingGrowth factorHonest limitsLogarithmic spiralModel scopeResidualWhorlArchimedean spiralDiscriminationInstrument settingMeasurement errorNegative result

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