Phyllotaxis
the spirals are counted, not admired
Series

Spiral fit — the series

10 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · shells
  2. 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.

    part 2 · shells
  3. 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.

    part 3 · shells
  4. The step floor derived, against the step floor measured — exact in 12 of 12, with 3.2 over 3.5 turns marked. One step of the dividers subtends half a turn at the inner end when it reaches the square root of the growth factor less one, which puts the floor at the factor to the power of the span, less one, over that. The smallest count at which the factor actually comes back exactly is then found by bisection, and the two agree in 12 of 12 cases: 74 steps for a 3.2 spiral over three and a half turns and 521 for a golden one. The three that appear not to agree are the ones whose floor falls below the fit's own nine-point minimum, where it cannot be observed.

    A measurement in steps

    Walking a pair of dividers along a shell's spiral is the oldest way to measure it and the best one available once there are enough steps, because it puts the points where the curve is. Under a count that follows exactly from the geometry it inflates the answer instead, and it is the only route measured here that pushes a nautilus towards a golden spiral.

    part 4 · shells
  5. 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.

    part 5 · shells
  6. 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.

    part 6 · shells
  7. 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.

    part 7 · shells
  8. A shell section square on and seen 30 degrees off. The same three turns of a spiral built at 3.2 per turn, drawn as a camera normal to the coiling plane sees it and as one 30 degrees away from normal does. The tilted view is the plane compressed by 0.8660 along one direction. A fit to the first returns 3.200000 with a residual of 1.8e-15; a fit to the second returns 3.19527 with a residual of 0.07763. Nothing in the second picture says it is not a shell.

    A section seen from the wrong angle

    A photograph of a shell section taken off the normal is the coiling plane compressed along one direction by the cosine of the angle, and nothing in the picture says so. The fit that recovers a growth factor is moved by it — half a turn seen twenty degrees off gives a band of answers 23.7 per cent wide as the span's starting point moves round the shell, centred almost exactly on the right answer, so it is a spread and not a bias. The caliper measure is exactly immune at every tilt and every aim, because a projection scales all three points on a line through the centre by the same factor. And the fit's residual names the tilt to three decimal places, which makes this the rare error a section reports about itself.

    part 8 · shells
  9. Half a turn of a shell section traced from a photograph 20 degrees off, and the same tracing with the tilt it names undone. 0.5 turns of a section built at 3.2 per turn, photographed 20 degrees off its plane across an axis at 37.5 degrees, every traced point off by a relative 0.005. The search over tilts and axes names 20.02 degrees across 37.6; undone, the tracing fits 3.2051 where the picture as photographed fits 2.9094. Over twenty-four views of this span the photographed fit spans a band 23.9% wide and the corrected one 0.86%.

    A tilt undone beats the calipers

    A shell section photographed off its plane reports its own tilt in the fit's residual, but not the axis the tilt was taken across, and a correction needs both. Read together from the whole outline — the tilt and axis whose undoing leaves the straightest line — they come back on half a turn of shell to a fifth of a degree and under a degree. Undone, the fit no longer merely matches the calipers that a tilt cannot touch: on a traced outline it beats them, by three times on half a turn and by twelve to forty from one turn up, because the calipers read three points and the fit reads all of them. Correcting a section that was square on costs a little, and a shell whose growth slows is mistaken for a tilted one without harm.

    part 9 · shells
  10. The tilt a search names on a section that has none, against the length of the error along its outline. An error of 2 per cent. 0.5 turns: 4.2 degrees at 0, 13.4 degrees at 0.02, 15.2 degrees at 0.05, 12.2 degrees at 0.1, 6.5 degrees at 0.25, 3.7 degrees at 0.5, 2.1 degrees at 1. 1 turn: 3.9 degrees at 0, 10.8 degrees at 0.02, 12.0 degrees at 0.05, 12.5 degrees at 0.1, 6.3 degrees at 0.25, 3.5 degrees at 0.5, 1.5 degrees at 1. 2 turns: 3.6 degrees at 0, 8.4 degrees at 0.02, 10.7 degrees at 0.05, 9.2 degrees at 0.1, 5.1 degrees at 0.25, 4.0 degrees at 0.5, 2.2 degrees at 1. The largest tilts come from a drift of a twentieth to a tenth of a turn, not from half a turn.

    A hand that drifts takes back what the fit won

    A tilt can be undone by searching for the view that makes a traced spiral straightest, and with an error independent from point to point the corrected fit beats the calipers twentyfold. A pen does not err that way. It wanders out for a few degrees and comes back. Give the same tracing error a length along the outline and the corrected fit's advantage is gone on one turn by a twentieth of a turn of drift, with the band as wide as the calipers', and on two turns falls from twenty-five times to about twice. The tilts it invents on a section with none peak at a twentieth to a tenth of a turn, not at the half-turn that looks like a tilt. The residual test still separates drift from a real tilt, on a turn or more.

    part 10 · shells

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