One very cool thing not mentioned in the article about Fibonacci spirals on plants:

The golden ratio is the continued fraction 1/(1 + 1/(1 + 1/(1+ …))) With that pattern repeating forever, which is cool on its own. The fraction is self similar, and it’s an irrational number (or else it would end somewhere).

When you look at ratio of a full rotation around a stem, any rational/fractional ratio will eventually repeat, so eventually a leaf will cover another leaf which might reduce sunlight capture for a plant. With irrational ratios we know that will never happen.

But is there a best irrational number to reduce overlap of leaves? If there was we would want one where the ratio is furthest from any whole number fraction (otherwise you could approximate the angle by that fraction). It turns out, the golden ratio is exactly that, because the continued fraction always has the largest denominator possible.

So the golden ratio giving a Fibonacci spiral actually results in the maximum spacing between overlapping leaf cover.