The Use of Continued Fractions in Botany (UMAP)

Students will 1) become acquainted with some of the most interseting aspects of continued fractions; 2) realize and appreciate the role of some mathematical concepts in botany; 3) develop observational capacities in the surrounding nature; and 4) acquire the fundamental notions that will eventaully allow them to investiagte an important field of research in biomathematics.

Table of Contents:

1. INTRODUCTION

2. FUNDAMENTAL CONCEPTS
2.1 Phyllotaxis of a System
2.2 Divergence and Phyllotactic Fraction
2.3 Convergents of a Continued Fraction
2.4 Klein's Diagram

3. BRAVAIS'S APPROXIMATION FORMULA FOR THE DIVERGENCE
3.1 Illustration of the Formula
3.2 Explanation of the Procedure and Jean's Theorem

4. A FIRST EXPLANANTION IN PHYLLOTAXIS
4.1 Bravais's Cylindrical Lattice in Botany vs. Klein's Square Lattice for Continued Fractions
4.2 Coxeter's Formula and the Hexagonal Scales of the Pineapple
4.3 Normal Types of Phyllotaxis

5. SPECIAL ASSISTANCE SUPPLEMENT

6. ANSWERS TO EXERCISES