Sunflower math

A golden rule of biology is that individuals attempt to leave as many offspring (i.e., genes) as possible in the next generation. Plants, like the sunflowers growing in my backyard right now, try to maximize their seed production by packing seeds into the flower head in the most optimal way. And that’s where the math comes in.

The sunflowers have grown quite tall (the fence posts are five feet), and some of the flowers are dinner-plate size (that’s a bumblebee on the flower head below for size reference).

But have you ever really inspected the interior of one of these complex flower heads?

The outer, yellow petals are really individual infertile ray flowers. The center, disk flowers open from outer toward inner rows, a few layers at a time. The yellow-tipped anthers stick up from the central part of each flower, presenting a disk of pollen for the lucky pollinators.

Once fertilized by the many bees that visit these flowers, each of the individual flowers produces one seed. The entire flower head can produce as many seeds as there are individual disk flowers. So the question is, how to pack as many of those flowers as possible into the circular head.

This pattern is not random. Each disk flower (or potential seed) lies at an angle of 137.5 degrees from its neighbor. Lined up from center to outer rim, the flowers (or seeds) describe two sets of spirals, one curving to the left, and one curving to the right. This is the formula for maximal packing into circular space, and has been termed the “golden angle”.

As explained by mathematicians, divide a circle into two sections, such that the ratio of the large arc to the small arc is the same as the ratio of the entire circle to the large arc. The golden angle is that created by the smaller of the two arcs, exactly 137.5 degrees.