Phyllotaxis
the spirals are counted, not admired
Ladder

Spiral fit — the ladder

4 distinct arguments against 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 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.

    rung 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.

    rung 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. Over the whole ladder 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 this collection has been reading that residual 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.

    rung 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.

    rung 4 · shells

All ladders