The Mathematical Lives of Plants

Many plants including cactus, sunflowers and pine cones grow in spiral patterns. The same spiral pattern was discovered way back when when the Greeks were developing geometry. The angle of the spiral is consistent ( 137.5′) and found in nature frequently not just in plants, but also in many sea shells and other life forms.

The spirals tend to appear both clockwise and counterclockwise and the number of spirals is usually two sequential Fibonacci numbers ( 1, 1, 2, 3, 5, 8). The Fibonacci numbers appear when ever these ( 137.5′ ) Golden Ratios appear. The ratio between Fibonacci numbers is very close to the Golden Ratio.

Scientists have wondered for centuries, why the golden ratio? And why Fibonacci numbers?

In 1868 Hofmeister was studying the plant equivalent of stem cells and noticed that the spot in which the new ones form is furthest from the old ones. In 1992 Douady and Couder dropped magnetized fluid into a magnetized dish. The droplets were repelled from each other and from the edge of the dish. At higher speeds earlier droplets effected newer ones landing and the new drops would move in a different direction, 137.5′ away from the previous drop. The new patterns formed spirals.

Forces on the plant and magnetic drops when all 3 dimensions were combined naturally form golden ratio spirals.

Some plants use 99.5′ pattern which produces the Lucas numbers ( 1, 3, 4, 7… )

More information:
Plants and Phi