Mathematics of the Natural World

Patterns in nature are visual regularities of form that occur in the natural world. These patterns can be modeled with mathematics and physics. Natural patterns can include symmetries, fractals, spirals, meanders, waves and dunes, foams and bubbles, arrays, cracks, and stripes (some examples are shown below):

Romanesco Broccoli in fractal form

radial symmetry

bilateral symmetry in a zebra's stripes

radial symmetry in a kiwi

logarithmic spiral of a Nautilus

dune meanders

tessellation array of scales

tessellation array of scales

sand dunes at equivalent angles

crack patterns

inelastic crack patterns

meanders

phyllotaxis Fibonacci spirals

phyllotaxis arrangement

Philosophers, mathematicians, and physicists have applied their skills to the natural world across the ages. Early Greek philosophers Plato, Pythagoras, and Empedocles often studied natural form, hoping to explain the ordered occurrence of patterns. In the 19th century, Joseph Plateau developed the theory of minimal surface area as shown in soap bubble films, and was able to mathematically model the concept. Ernst Haeckel painted thousands of Radiolaria (small marine organisms) to show their symmetry in detail. D'Arcy Thompson extensively studied and modeled the growth patterns of flora and fauna, applying mathematics to spiral growth. Alan Turing established methods for predicting morphogenesis in embryos that would eventually become spots and stripes. Benoit Mandelbrot and Aristid Lindenmayer developed the concept of fractals that can be used to approximate plant growth patterns.

While the models are not always spot on, the conceptual process of predicting the patterns of the natural world has broadened our horizons and increased our appreciation of the beauty of nature.