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The thread: The round trip

Build the pattern from a stated number, forget the number, recover it from the pattern alone, and compare. It works for the divergence angle and for the growth factor, and the agreement is the measurement.
growth 3.20× per turnrecovered 3.200× Shells and growth

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.

21 and 34 spiralscounted, not assumed The pattern itself

Counting the spirals

Almost every claim about phyllotaxis is a claim about how many spirals run through a pattern, and the count is almost never done. It can be done from the points alone, by machinery that is never told what angle built them — and then a count of 34 is evidence rather than a restatement.

02468234exponent khow badly r₀ᵏ = Σ rᵢᵏ fails at that kk = 3127 junctionsfitted k = 3.000 Branching and transport

Fitting the exponent

Assuming the exponent is three and reporting the error tells you how wrong the data is. Fitting the exponent and reporting what it comes out as tells you what the network is doing — and the machinery has to be shown returning something other than three, or it is not a fit.

the first of the fourused to buildcountsrecovered137.508°55 · 89137.520°99.502°47 · 7699.500°151.100°31 · 81151.105°77.960°37 · 6077.960°worst error 0.012°counts in, angle outthe recovery never sees the angle The pattern itself

Recovering the angle from the counts

Build a head at a stated divergence angle, forget the angle, and get it back from the spiral counts alone. Four angles, worst error twelve thousandths of a degree — and the only thing that crossed between the two halves was a list of coordinates.

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