Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Curvature Changes in Biological Dynamics

Version 1 : Received: 13 October 2018 / Approved: 15 October 2018 / Online: 15 October 2018 (10:36:18 CEST)

How to cite: Tozzi, A. Curvature Changes in Biological Dynamics. Preprints 2018, 2018100299. Tozzi, A. Curvature Changes in Biological Dynamics. Preprints 2018, 2018100299.


Geometry deals both with analogical thinking and physical/biological observables. Naïve, common-sense descriptions of objects’ shapes and systems’ trajectories in geometric phase spaces may help experimental investigation. For example, very different biological dynamics, as the developmental growth patterns of the oldest known animal (the extinct Dickinsonia) and the human brain electric oscillations, display a striking analogy: when encompassed in abstract geometric spaces, their paths describe the same changes in curvature: from convex, to flat, to concave and vice versa. This dynamical behavior, anticipated by Nicholas de Cusa in his analogical account of “coincidentia oppositorum” (1440), helps to describe widespread biological paths in the manageable terms of concave, flat and convex curves on donut-like structures. Every trajectory taking place on such toroidal manifolds can be located, through a topological technique called Hopf fibration, into a four-dimensional space. We discuss how the correlation between Hopf fibration and Navier-Stokes equations allows us to treat the above-mentioned biological and neuroscientific curved paths in terms of flows taking place into a viscous fluid medium that can be experimentally assessed and quantified.


lines; brain; topology; curvature; Dickinsonia; fMRI



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