## Sunflowers and the Least-Rational Number

Sample Abstract
This talk will explain the surprising mathematical patterns that may be observed in the centers of sunflowers and in many other plants. In particular, the following questions will be answered.

1. Why do the families of spirals in a sunflower always come in consecutive Fibonacci numbers?
2. Why do different sunflowers contain different Fibonacci numbers of spirals?
3. Are spirals in a sunflower the same shape as the spiral in a chambered nautilus? (If you think you know the answer, you're probably wrong. The correct answer resolves the previous question.)
Mathematical souvenirs will be distributed.

Description
The central idea of this talk is the mathematical effect of phyllotaxis, the growth process in plants whereby leaves (or seeds or petals) of a plant emerge with a fixed angle between consecutive leaves. We explore why the phyllotaxis angle in sunflower seed heads and in pinecones and in many other plants is the golden secton of the circle and why this angle produces Fibonacci patterns in the spirals of such plants. Continued fractions play a key role in unraveling the mystery.

I also have a more advanced version of this talk for a mathematically savvy audience that I developed as a plenary address at the Young Mathematicians Conference at the Ohio State University, and which ultimately lead to my paper “Dancing Elves and a Flower's View of Euclid's Algorithm.” That talk is available as a streaming video; the PowerPoint slides are improperly synched (but can be advanced manually).

For source materials, see R. Knott's excellent discussion of Fibonacci Numbers and Nature, page 1 and page 2, and the references at the bottom.

Level
Early High School and up.

The audience should be able to cope with examples of continued fractions and other basic arithmetic formulas. The limits used in the talk are all hand-waved, so prior knowledge of limits is not necessary. In the discussion of the shape of sunflower spirals, polar coordinates appear, but this is a small portion of the talk.

Mechanics
Part of the lecture is on the blackboard and part is on transparencies, but as these parts are disjoint, it is fine if the projector screen covers the blackboard.

It is likely that I will not give out the "mathematical souvenirs" in the future, especially for large audiences. However, at the very least, I will have some nice pinecone-based visual aids.

Appearances

As “Phyllotaxis, Dancing Elves and a Flower's View of Euclid's Algorithm”