Figure 1 – Pine cone bottom showing clockwise and counter clockwise spirals. (c) DE Wolf 2019.

I was struck by the wonderful symmetries of the pine cone in Figure 1. This is a classic example of growth among plants, indeed in all of nature, where the Golden Ratio and its marvelous spirals dominates. It is everywhere, even buried, in approximation, in photography’s “golden rule of thirds! The golden ratio relates to Fibonacci series. 

The Fibonacci series of numbers is 0, 1, 1, 2, 3, 5, 8,13, 21,34 etc. It’s fundamental property is that each number in the series is the sum of the previous two numbers.

Now for the pine cone in Figure 1, you will note spirals that twist outward in a counter clockwise fashion. Here there are 13 of them. While these spirals dominate your eyes and perception, if you look closely you will also note that there are clockwise spirals. Here there are 8 of them. As is almost always the case with pine cones and such natural objects, such as sunflowers, the two numbers 8 and 13 are consecutive numbers in the Fibonacci series. Ain’t nature wonderful? Seriously however, this kind of growth is a consequence of the way that nature grows and packs petals. The golden ratio and its spirals follow the Fibonacci series and are ubiquitous in nature.