Phyllotaxis
the spirals are counted, not admired
Series

Raup — the series

2 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A shell section at W = 2.40, D = 0.42. W·D = 1.01, so the whorls are free of each other — an evolute shell, like a ram's horn or a planispiral ammonite. Both are things animals grow.

    Raup's three numbers

    Nearly every coiled shell that has ever existed is a point in a three-dimensional space — how fast the whorl expands, how far the opening sits from the axis, how far it travels along it. That is a remarkable compression, and the most useful line in the space is the one where the whorls come apart.

    part 3 · shells
  2. The distance from the axis is one ratio on one ray, read from a centre displaced by 0.25 of a whorl. A section at W = 2.40 and D = 0.42, with four rays cast from a centre displaced 0.25 of the read whorl's outer radius to the right. Along each ray the reading is the inner wall's distance over the outer wall's — from the true centre that ratio is 0.42 at every azimuth, to the last bit a double holds. From the displaced centre the same four rays give 0.227, 0.351, 0.582, 0.347: the ray pointing at the displacement reads low and the ray opposite reads high, because subtracting the same length from both distances moves their ratio toward one. The spread is 0.227 to 0.582 on a shell whose distance from the axis is 0.42.

    What the axis distance costs

    Raup's contact boundary is a relation between two numbers and only one of them has ever been priced here. The second was expected to be the cheaper — a length against another length. It is not: an assumed centre costs it 79.3 per cent where the same centre costs the expansion 6.88, because a ratio of two distances is first order in the centre and a fitted rate is second. But a tilted camera costs it nothing at all, exactly, and averaging the reading round one whorl is free and worth a factor of 4.91. The two numbers fail at opposite ends, and they cross at 1.12 turns of arc.

    part 4 · shells

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