The brain does not fit neatly into Euclidean geometry. Its structures — cortical folds, vascular trees, dendritic arborizations — are irregular in ways that classical measurement fails to capture. Fractal geometry, introduced by Mandelbrot and progressively adopted in neuroscience over the past four decades, offers a framework suited to this complexity.
The central tool is the fractal dimension (FD): a non-integer value that quantifies the degree of geometric complexity and self-similarity of a structure across spatial scales. Unlike a smooth line (dimension 1) or a flat surface (dimension 2), the cortical surface has a fractal dimension between 2 and 3 — and that value is not constant across individuals, ages, or pathological states.
Self-similarity from dendrites to cortex
Fractal organization in the brain operates across multiple scales. At the microanatomical level, dendritic branching patterns exhibit self-similar geometry that can be quantified using box-counting algorithms. At the mesoscale, vascular networks distribute according to fractal rules that optimize coverage and efficiency. At the macroscale, cortical folding follows a universal scaling law across mammalian species.
Wang et al. (2024), publishing in eLife, demonstrated that the cerebral cortex of eleven primate species — including humans, chimpanzees, and marmosets — follows a universal fractal pattern, combining scale-invariance with self-similarity of folds. The authors proposed this as a universal blueprint for mammalian cortical architecture, with implications for understanding neurodevelopmental and neurodegenerative conditions.
Fractal dimension as a clinical biomarker
The clinical utility of fractal analysis in neuroscience is well-established and expanding. A comprehensive review published in January 2026 in Brain Sciences summarized the current state: FD reliably distinguishes white matter alterations in multiple sclerosis, reduced cortical complexity in Alzheimer’s disease, atypical maturation in intrauterine growth restriction, and altered neuroimaging patterns in schizophrenia.
Particularly relevant is the relationship between FD and consciousness states. Varley et al. (2020) showed that the fractal dimension of cortical functional connectivity networks correlates with the severity of disorders of consciousness — a finding that links structural complexity to the capacity for integrated experience.
From brain to biomedical imaging
Fractal analysis is not limited to neuroimaging. In pathology and biomedical image analysis, vascular fractal geometry has been applied to characterize tumor microenvironments, retinal vasculature as a diagnostic window into cerebrovascular disease, and prostatic tissue organization. The box-counting method, reviewed in detail by Karperien and Jelinek (2024), remains the most widely implemented approach for 2D imaging due to its computational accessibility and reproducibility.
In our own work, fractal analysis of vascular structures has served as a quantitative substrate for pattern recognition in biomedical images — an area where the intersection of classical morphometry and machine learning opens methodological possibilities that neither discipline reaches alone.
References
Mandelbrot, B.B. (1982). The Fractal Geometry of Nature. W.H. Freeman.
Wang, Y., Leiberg, K., Kindred, N., et al. (2024). Neuro-evolutionary evidence for a universal fractal primate brain shape. eLife. DOI: 10.7554/eLife.92080.3
Varley, T.F., Craig, M., Adapa, R., et al. (2020). Fractal dimension of cortical functional connectivity networks and severity of disorders of consciousness. PLoS ONE, 15, e0223812.
Di Ieva, A. (Ed.) (2024). The Fractal Geometry of the Brain, Advances in Neurobiology. Springer. DOI: 10.1007/978-3-031-47606-8
Karperien, A.L. & Jelinek, H.F. (2024). Box-counting fractal analysis: a primer for the clinician. Advances in Neurobiology, 36, 15–55.
Díaz Beltrán, L., Madan, C.R., Finke, C., et al. (2024). Fractal dimension analysis in neurological disorders: an overview. In Di Ieva (Ed.), The Fractal Geometry of the Brain. Springer.
— Daniela Di Marco, Operation Knowledge
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