Math in… Flowers

The Fibonacci numbers are a sequence where each number is the sum of the previous two, starting with 0 and 1:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . .

When you count the spirals of seeds and leaves on plants, you’ll typically find a Fibonacci number.

If you want to efficiently pack seeds or leaves in a disk, it turns out a good strategy is to place each one a little further away from the center and offset it by about 137.5°. This angle is called the golden angle for its relationship with the golden ratio, φ:

Nature seems to have stumbled upon this efficient packing trick long before mathematicians! Since the golden ratio is well-approximated by ratios of Fibonacci numbers, the spiral counts are Fibonacci numbers.

You could try other angles, but the packings aren’t as good. Here are some packings with offset angle θ = x ⋅ 360° for some interesting values of x:

If x is a (reduced) rational number x = a/b, the spiral devolves into b rays, like the 9 you see when x = 8/9.

If x is irrational, then there are still arms, but their count morphs between the denominators of best rational approximations for x, spiraling clockwise or counterclockwise based on if the approximation is high or low.

π has approximation 22/7, and you can count 7 spiral arms in the center of its packing. The next approximation isn’t until 333/106, which is why the 7-arm spiral takes up so much of the center. Contrast this with e, which has approximations 19/7 and 87/32. The chunk of its packing with 7 spiral arms is comparatively smaller.

The square root of 2 does a pretty great job of not letting any spiral get too large before the next spiral takes over, but φis the champ with approximations like 8/5, 13/8, 21/13, 34/21, 55/34, 89/55, ...