Mandelbrot’s Fractals and the Geometry of Life: A Tribute to Benoît Mandelbrot on his 80th Birthday

Abstract

The concept of fractal geometry advanced by Mandelbrot since 1977 has brought new insight into the design of biological structures. Two fundamental geometrical forms abound: interfaces between different compartments with a very large surface within finite space, and branched trees that distribute blood and air into the tissue space. These structures show a level of complexity that is best described by fractal geometry. Thus, the surface area of cellular membranes as well as the gas exchange surface of the lung have a fractal dimension which is larger than 2. The design of the airway tree is described in quantitative terms and the functional consequences are discussed, both with respect to airflow in the bronchi and gas exchange in the acini. Similar conditions are described with respect to the blood vascular network. It is finally discussed whether fractal geometry plays a role in designing animals of greatly different body size from 2 g in a shrew to 500 kg in horses and steers. The scaling exponent of 3/4 for metabolic rate has been explained on a basis of two fractal models, but it is shown that this does not hold for maximal metabolic rate which is directly proportional to the surface of inner mitochondrial membrane that in turn has fractal properties. The concept of fractal geometry is valuable in understanding the design of biological structures at all levels of organization.