Phyllotaxis
the spirals are counted, not admired
Series

Spiral — the series

6 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A logarithmic spiral growing by 3.20× per turn. Fitting log r against angle on the drawn points returns 3.2000, and the model's worst residual is 9e-16 in log r.

    Growth as a rule

    A logarithmic spiral is not a shape somebody admired. It is what a thing grows into when it adds material at its opening without changing shape, its one parameter is how much it grows per turn, and that parameter can be recovered from any drawn curve to the last digit.

    part 2 · shells
  2. One spiral at 3.20× per turn, marked at equal intervals of time under four rate laws. The curve is identical in all four panels — every mark lies on r = W^(θ/2π) exactly, whichever clock put it there — and the marks are not. Under a constant angular rate the three whorls hold 14, 13, 14 marks; under a constant length added the three whorls hold 3, 9, 29 marks; under a constant area added the three whorls hold 1, 3, 37 marks; under a constant volume added the three whorls hold 1, 1, 39 marks. The growth factor is a rate per turn of the shell's own coiling, and a turn is not a unit of time; the whole of what an animal's growth rate means is in the spacing of these marks and none of it is in the curve.

    A spiral with no clock

    The growth factor a shell's curve gives up is a rate per turn of the shell's own coiling, and a turn is not a unit of time. Four clocks — the aperture advancing at a constant angular rate, adding a constant length, a constant area, a constant volume — trace the identical curve: every mark one of them leaves lies on r = W to the power theta over two pi exactly, so a fit through any of them returns the same factor. What differs is where the marks are, and the difference is enormous: at 3.2 per turn the outermost of three whorls holds 33.4, 71.0, 90.3 and 97.0 per cent of the record. It also breaks the instrument. The routine that recovers a growth factor unwraps the angle by assuming successive points advance less than half a turn, and five of the twenty readings here leave gaps past that — returning 3.33 where the curve was built at 3.20, and 8.19 where it was built at 6.85.

    part 3 · shells
  3. The ratio of two whorls' line counts is the growth factor raised to the clock's own power. Each curve is W^p for one law: flat for a clock that advances the angle at a constant rate, W for one that adds a constant length at the opening, W² for a constant area and W³ for a constant volume. At 3.20× per turn those are 1.00, 3.20, 10.24, 32.77. So a count of lines in two successive whorls, divided, and read back through a growth factor the curve already gives, names the law — and the four are further apart the faster the shell expands.

    What the growth lines carry

    A shell's curve says nothing about how fast the animal grew, and its growth lines say all of it. Under a law that holds the pth power of the radius constant per unit time, the time spent crossing one whorl is proportional to the change in that power across it — so the lines in successive whorls stand in the ratio of the growth factor raised to p, and each whorl holds the count the closed form predicts to within the one line rounding can move. Dividing two counts and taking the logarithm against a growth factor the curve already gives returns p: 17 of 20 readings name their own law, the furthest 0.012 from a whole number. The other three are not wrong, they are uncountable — at 4.5 per turn under a volume clock the inner whorl of the pair holds one line.

    part 4 · shells
  4. A shell whose deposition law changes part way through, beside one that never does. Both panels are the same logarithmic spiral at 3.20 per turn, marked at equal intervals of time. On the left the animal holds a constant angular rate for the first 3.50 whorls and a constant length added after that; on the right it holds a constant length added throughout. The curves are identical to the last bit a double holds, because a curve records no clock at all. The counts per whorl are not: 56, 57, 56, 67, 191, 613, 1960 against the unchanged shell's, and the change is in where the marks crowd rather than in where the shell goes.

    A shell that changed its law

    An animal that grew as a juvenile under one deposition law and as an adult under another leaves a sequence of whorl ratios rather than one, and the sequence says where the change happened. The ratio across the change is a closed form that is neither law's — 6.72 between a length clock and an area clock at 3.2 per turn, exactly the average of 3.2 and 10.24 — and it is monotone in where inside its whorl the change sits, so it inverts. On a seven-whorl shell of 18,466 lines a change at 3.5 whorls comes back at 3.5001, in a band 0.027 whorls wide that holds the true position. The reading refuses a change in the outer three whorls or the inner three, because a plateau it will trust is two agreeing ratios and two ratios need three untouched whorls.

    part 5 · shells
  5. A step and a drift between the same two laws, as sequences. Two shells, both starting at a constant angular rate and ending at a constant area added over 6 whorls. The stepped one changes at a single position and its sequence is flat, crossed, flat. The drifting one changes evenly and its sequence is a straight ramp. The largest difference between them is 0.624 in power, against a rounding of 0.0055 — so what separates a step from a drift is the shape of the sequence and never any one of its ratios.

    A law that never stopped changing

    A shell whose deposition law moved evenly from one end to the other gives a sequence of whorl ratios that is a straight ramp rather than a plateau, a crossing and a plateau, and the two are separated by more than the counts' own rounding on every shell holding three countable ratios. Each ratio on the ramp reads the law at the boundary it straddles — 0.3486, 0.6865, 1.0320, 1.3755, 1.7180 against 0.3333, 0.6667, 1.0000, 1.3333, 1.6667 — so the reading is local where a fit to the curve is global, and a fit handed the same shell returns the geometric mean of its ends with no warning. The reading that locates a single change refuses a drifting shell at every size, naming the number of ratios that agree with neither end.

    part 6 · shells
  6. Every law but one is pulled towards a length clock. The power a shell's outermost countable pair names, against how coarsely its lines can be told apart. A length clock sits flat at one however much of its record is lost. Every other law bends towards it: an angular clock reads 0.993 where the shell had 0 and a volume clock 1.083 where it had 3. The reason is a closed form: where the limit binds completely the surviving count in a whorl is that whorl's arc over the limit, and a logarithmic spiral's whorl arcs stand in the ratio of the growth factor exactly. So a shell too worn to read reports the law of its own geometry.

    A count that is not exact

    Reading a deposition law off two whorls' growth-line counts divides one by the other, so a miscount that is the same in both divides out: four lines in five missed at random moves the answer by five thousandths and costs only scatter. What biases it is a miscount that varies along the shell, and there is one that always does. The arc between successive lines rises or falls with the radius according to whether the law is shallower or steeper than a length clock, so a section's resolution limit eats the inner whorls of a shallow shell and the outer whorls of a steep one, and eats evenly at exactly p = 1. Where the limit binds, a whorl's surviving count is its arc over the limit, and whorl arcs stand in the ratio W — so a shell too worn to read reports a length clock whatever law it had.

    part 7 · shells

All series