Power scaling, vascular branching patterns, and the golden ratio

Abstract

The Golden Ratio (a ratio of ~1.618:1) appears repeatedly in nature including structural and functional traits of organisms (e.g. Fibonacci spirals of snail shells and certain seed heads), the spiraled shape of galaxies and hurricanes, and even in much cultural architecture and art. In the mid-19th century, branching structures in plant and animal vascular systems were found to follow the Golden Ratio; that is, successive branches in the vascular systems of plants and animals tend to follow a length ratio of about 1.618:1. Here we present a model that uses this empirical evidence as a branching ratio in theoretical vascular systems. We then use a defined mass of the model system as a predictor of log-log scaling of terminal units. In this model, log terminal units and log mass scale similarly with that of other models as well as empirical evidence, but with more parsimony and a perspective not yet offered among all available models of allometric scaling. This model invites novel and broad hypotheses on the influence of the Golden Ratio on power scaling in organisms.