Math as Myth: What looks like the golden ratio is sometimes just fool's gold
nautil.us
nautil.us
https://github.com/drostie/essay-seeds/blob/master/misc/fibo...
I might expand it into a proper document at some point, but until then, here is how you draw sunflowers with the golden ratio; you can do it with any spreadsheet document that lets you draw scatter-plots:
http://tmp.drostie.org/sunflower.png
You have an index k which goes from 1 to N, and you define a radius R for each of these points as R(k) = sqrt(k / N), which keeps an "even spacing" of the points as they spiral out. (It's a simple argument: the sub-circle has area proportional to R^2, so to get the number of points proportional to the area, the radius must go like the square root of the number of points.)
You choose a ratio `s` and define the angle as `angle(k) = 2 * pi * s * k`. Finally, the actual points are located at `x(k) = R(k) * cos(angle(k))` and `y(k) = R(k) * sin(angle(k))`, which you scatter-plot as points. Then you can play with this graph for different values of `s`. Indeed, the "spirals" which you count correspond to the denominators of the best rational approximations to those values; try using s = pi and you'll see 7 branches at first, then 113. It gets this magical appearance when you choose s = phi, where the points don't "stack on top" of each other, because the golden ratio is the most irrational number. You can also use the Python scripts in the document above to search for other "highly irrational" numbers (ones with low numbers in their continued fraction representations; the square root of 3 is pretty good for example).
If I turn the above document into a proper essay, I'd probably also create an HTML5 canvas which draws parametrized sunflowers, to save the effort of creating that whole darn spreadsheet.