24 February 2021

Golden Maths — Advanced arithmetic (Ridley's Theorem)

 Among the many questionable claims concerning the golden number is this one: that φ is a fundamental ratio that is almost as ubiquitous as π (Gardner1961 p.89). While π does impose itself everywhere, it is quite possible to graduate in mathematics without ever running into the golden number —I did. In secondary school it had been mentioned in the construction of the regular decagon, and that was it. (And even then, Pythagoras would have done as well.) So, where are all those golden problems forcing φ upon you? I'm proud to present the one I eventually was able to dig up: 

James N. Ridley, Packing Efficiency in Sunflower Heads. Mathematical Biosciences 58 (1982), pp.~129-139. (Here.) 
As the title shows, it deals with a certain mathematical model of a sunflower head. The main theorem, stripped to the pure mathematics, may be stated as follows:

(DIPAS stands for Discrete Parabolic Spiral.) Golden numberists may feel somewhat disappointed, because for finite spirals (like real sunflowers) the golden angle is defeated, even by rational numbers! The success, therefore, is restricted to theoretical mathematics. Also, Ridley emphasises the fact that some magical divergence angle can not be the explanation for the observations in sunflower heads.

Ridley's Theorem is very skilfully composed out of four lemmas, the key tools being provided by the theory of continued fractions. Find it, somewhat elaborated, illustrated and restricted to the purely mathematical aspects, here. Graphs and computations are courtesy of GeoGebra 5; I couldn't have done without.