Irregular Shapes: Randomness in Fractal Constructions

Abstract

Self-similarity seems to be one of the fundamental geometrical construction principles in nature. For millions of years evolution has shaped organisms based on the survival of the fittest. In many plants and also organs of animals, this has led to fractal branching structures. For example, in a tree the branching structure allows the capture of a maximum amount of sun light by the leaves; the blood vessel system in a lung is similarly branched so that a maximum amount of oxygen can be assimilated. Although the self-similarity in these objects is not strict, we can identify the building blocks of the structure — the branches at different levels.

Why is geometry often described as ‘cold’ and ‘dry’? One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline, or a tree. Clouds are not spheres, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line. [...] The existence of these patterns challenges us to study those forms that Euclid leaves aside as being ‘formless’, to investigate the morphology of the ‘amorphous’.

Benoit B. Mandelbrot¹