Phyllotaxis
the spirals are counted, not admired
Concept

Span of arc — where it appears

How much of a spiral a measurement covers, in turns, which almost no quoted figure about a fitted spiral names. A figure with no span attached is true at some spans and false at others: the centre-displacement figure published here holds above 4.25 turns and at no span a shell section offers.

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

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 growth factors surveyed, 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 the figure published earlier.

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 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 The two halves of a 3.2 spiral fitted about a centre 0.25 innermost radii off, towards 52°. A logarithmic spiral growing by 3.2 a turn over 4 turns, which does not change, split into an inner half and an outer half, each fitted about a centre displaced by 0.25 of the innermost radius towards 52°. The inner half returns 3.4193 and the outer half 3.2200, a split of −5.83 per cent. The panel on the right enlarges the first whorl, where the true centre and the assumed one can be told apart; across the whole spiral the displacement is 0.238 per cent of the outer radius.

A centre that invents a life history

The collection's advice for a shell that might have changed how it grew was to fit it twice, over different arcs, and compare. On a spiral that does not change at all, a centre displaced by a quarter of the innermost radius splits the two halves by 4.09 per cent — the split a genuine 8.35 per cent change from apex to aperture produces — in either sign, depending only on which way the centre is wrong. Point noise of the same size splits them by less than half as much, and averages away where the centre does not. The floor under the test is the centre, not the noise.

shells · Spiral fit Two diameters half a volution apart on a 3.2 spiral, aimed through the true centre. A logarithmic spiral growing by 3.2 a turn, its aperture 3.5 turns along. A line from the aperture through the true centre meets the outer wall half a volution back and a volution back. The conch diameter dm1 is 1.559017 of the outer radius, the diameter half a volution back, dm2, is 0.871517, and the apertural height between them 0.687500. Squared, dm1/dm2 is 3.200000 against the spiral's own 3.2, exact: both lengths are distances between wall points on one line, so the centre only aims it.

Three points on a diameter

Ammonoid workers measure a shell's expansion without a centre: two diameters half a volution apart, squared. On a logarithmic spiral that is exact, and the centre is needed only to aim the line. A quarter-radius aim error costs the fit 1.341 per cent and the diameters 0.0045, because the aim error enters as its square. Reading noise is another matter: at a thousandth of the outer radius the fit's four hundred points beat the calipers' three readings at every expansion up to the nautilus's, and which instrument is better depends on which error the section actually has.

shells · Spiral fit

Named alongside it

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

Claim testingGrowth factorLogarithmic spiralAssumed centreHonest limitsMeasurement errorDisplacementNegative resultResidualUntested claimWhorlModel scope

All concepts