The role of continued fractions in phyllotaxis (Q1271019)

Phyllotaxis is the study of the arrangement of botanical units such as leaves, scales, and florets around a stem. From the very beginning of the scientific study of this subject the numbers observed in phyllotaxis were seen to be connected to simple continued fractions. The reason for this connection, however, was not immediately understood. Because of this, one leading botanist (J. Sachs) rejected it outright as meaningless playing with numbers, and one well-known mathematician (P. G. Tait) was trapped into an erroneous argument leading to a false conclusion. It was not until 1974 that the real connection between continued fractions and the numbers significant in phyllotaxis was finally clarified in a few rigorously established theorems about cylindrical point-lattices. But first it was necessary to clarify the concepts that arise in phyllotaxis. This is the subject of Section 9. It was also necessary to seek out the underlying geometric meaning of a simple continued fraction. This is taken up in Section 12. The rest of this paper traces the history of the connection between phyllotaxis and continued fractions from the time it was first observed to the time it was finally understood.