The Fractal Geometry of the Brain: An Overview

Abstract

The Fractal Geometry of the Brain adopts an encyclopedic view of the multiple ways the notion of fractal enters into the clinical, as well as scientific, understanding of the human brain, with multiple papers addressing the brain’s form, function, and pathologies. This ambitious task is undertaken from the perspective of making available to the biomedical community a measure of complexity that can characterize both disease and wellness, a measure, the fractal dimension, which is fully consistent with the complexity of biomedical phenomena and with phenomena that have historically resisted being characterized by one or a few scales. A fractal is a geometric concept used to describe the scaling of physical objects in space, which cannot be characterized by one or a few scales. This is not an esoteric mathematical concept, but is a concrete practical measure of variability in complex phenomena. Thus, the folds of the brain’s surface, the regions of space that confine the flow of material through the brain, as well as tumor growth, are not described by smooth integer dimension paths, or surfaces, but have dimensions between the integers. These fractal dimensions entail a scaling across multiple scales, with the fractal dimension determining how the scales are tied together. This scaling gives rise to a self-similarity of the object in space, which like nested Russian dolls repeat the same structure at ever smaller scales. Multiple examples of such space-filling biomedical phenomena are presented throughout the book and used to explain how such scaling is related to robustness and its loss to disease.