**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.

- Why do the families of spirals in a sunflower always come in consecutive Fibonacci numbers?
- Why do different sunflowers contain different Fibonacci numbers of spirals?
- 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.)

**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**

- Amherst College MAA Student Chapter, April 2001
- Ross Program, Ohio State University, July 2001
- Radical Pi, Ohio State University, October 2002
- Math and Computer Science Club, St. Mary's College of Maryland, November 2004

- Young Mathematicians Conference, The Ohio State University, August 2004
- Amherst College MAA Student Chapter, February 2005
- Hampshire College Summer Studies in Mathematics, July 2005