It is hard for a plant to plan ahead, so it has to have a simple rule for how to grow. For simplicity, suppose a plant stem grows straight up and that it sprouts a leaf every few inches. The angle between successive leaves is always the same. The plant wants the leaves as spread out as possible so that as they get larger they won't shade the leaves below or be shaded by the leaves above. Also, the plant has no idea how tall it will be able to grow, so it must arrange its leaves so that the shade on lower leaves is relatively small no matter how tall it gets.

1. What angle should the plant choose between successive leaves? That is, should the next leaf sprout half way around the stem or 1/3rd of the way around the stem or 8/13ths of the way around the stem, from the previous leaf? Remember that the plant has no idea how tall it will eventually get, so must choose the angle between leaves to give relatively little shade for any height.
2. If you assume the leaves at the bottom of the plant are the largest, does this change your choice above?

We can model this problem by thinking of placing points on a circle, each one a constant fraction of the circumference from the previous point. We want to choose this fraction so that the points are as spread out as possible all the time. If we knew we were only going to put down N points then we would use 1/N as the fraction. However, if we put down N+1 points, each one 1/N of the way around the circle from the previous point then the N+1st point will be on top of the first. We want to choose a fraction so that no matter how many points we put down, they are relatively well spread out around the circle. Try different fractions. Can you determine a pattern to how the points appear around the circle? (Remember, we could use 2/3 or 5/8 or any number we want between zero and one to place the points.)

Communicated by G. Hall.