Phyllotaxis: the Mathematical Study of Plant Pattern Formation

Many plants show regularly arranged rows of leaves, seeds, florets etc. The eye wants to join the neighboring elements into repeating curves – spirals around the center, or helices around the stem. These spirals, known as “parastichies”, usually come in two families, winding in opposite directions.

Fibonacci Phyllotaxis

One central phenomenon in Phyllotaxis is that many plants display numbers of parastichies that are pairs of successive numbers in the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55

This pinecone displays 8 parastichies in one direction and 13 in the other.
The florets on this sunflower display 34 parastichies in one direction and 55 in the other.

Once Fibonacci phyllotaxis was observed in the early 1800’s, scientists of all persuasions attempted to explain it physically, mathematically, and biologically. Our understanding progressed alongside technology: microscopes, computers, and modern molecular biology. Despite many efforts to understand this phenomenon, some questions remain unanswered still. How do patterns transition between different Fibonacci numbers as they grow? What can we say of all the plants that do not exhibit Fibonacci phyllotaxis?