A mathematical description of natural shapes in our nonlinear world

Concerning the problem of shape and pattern description, a compact formula (Gielis’ formula) has recently been proposed that generates a vast diversity of natural shapes. However, this formula is a modified version of the equation for the circle, so that it is expressed in terms of trigonometric functions which are inherent to linear phenomena but rarely appear in the description of nonlinear phenomena. In consequence, it generates highly idealized (Platonic) forms rather than real-world forms. Since natural shapes and patterns generally appear as a result of nonlinear dynamical processes, it would be natural that formulas which aim to describe them should be expressed in terms of related nonlinear functions. Here, two examples of simple mathematical formulas which are natural nonlinear modifications (one being a generalization) of Gielis’ formula are discussed. These formulas involve a comparable number of parameters and provide non-Platonic representations of a vast diversity of natural shapes and patterns by incorporating diverse aspects of asymmetry and seeming disorder which are absent in the original Gielis’ formula. It is also shown how diverse sequences resembling some natural-world pattern evolutions are also generated by such nonlinear formulas.— Although nonlinearity is an ubiquitous feature of real-world phenomena, some aspects of this fundamental property remain surprisingly poorly explored, partly because they are rarely incorporated into the mathematical tools and models which aim to describe even the simplest of such phenomena. In the context of the description of natural forms, such as biological shapes, a recent example is provided by Gielis’ formula: r (θ) = 1 n1 √