

Math as Myth: What looks like the golden ratio is sometimes just fool's gold - joewee
http://nautil.us/issue/0/the-story-of-nautilus/math-as-myth

======
drostie
I've long loved the Fibonaccis and the Golden Ratio; here are some of my more-
readable notes on what exactly is going on in their connection, and why the
golden ratio is the most irrational number:

[https://github.com/drostie/essay-
seeds/blob/master/misc/fibo...](https://github.com/drostie/essay-
seeds/blob/master/misc/fibonacci_and_golden_ratio.md)

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.

------
twiceaday
Relevant: <http://www.lhup.edu/~dsimanek/pseudo/fibonacc.htm>

