Chapter 8: Implications for Everyday Systems

Section 6: Growth of Plants and Animals

Mathematics of phyllotaxis

A rotation by GoldenRatio==(1+Sqrt[5])/2 turns is equivalent to a rotation by 2-GoldenRatio==GoldenRatio-2≃0.38 turns, or 137.5°. Successive approximations to this number are given by Fibonacci[n-2]/Fibonacci[n], so that elements numbered Fibonacci[n] (i.e. 1, 2, 3, 5, 8, 13, ...) will be the ones that come closest to being a whole number of turns apart, and thus to being lined up on the stem. As mentioned on page 891, having GoldenRatio turns between elements makes them in a sense as evenly distributed as possible, given that they are added sequentially.

Image Source Notebooks:

From Stephen Wolfram: A New Kind of Science [citation]