MODELING NATURAL SHAPES WITH A SIMPLE NONLINEAR ALGORITHM

Abstract

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 this work, 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.