The (spatial) fractals and (temporal) power laws are ubiquitously displayed by large classes of biological systems. Nevertheless, they are controversial phenomena with still unexplained genesis. From the far-flung branch of topology, a helpful concept comes into play, namely the Borsuk-Ulam theorem, shedding new light on the scale-free origin’s long-standing enigma. The theorem states that a single point, if embedded in just one spatial dimension higher, gives rise to two antipodal points that have matching descriptions and similar features. Here we demonstrate that, when we introduce into a system the proper fractal extra-dimension instead of a spatial one, we are able to achieve two antipodal self-similar shapes, corresponding to the distinctive scale-free’s higher and lower magnifications. By showing that the elusive phenomena of fractals and power laws can be explained and analyzed in a topological framework, we make clear why the Borsuk-Ulam theorem is the most general principle underlying their pervasive occurrence in nature.
Nonlinear chaotic dynamics are widespread, both in physical and biological systems. This form of dynamics is frequently studied through logistic maps equipped with bifurcations, where intervals are dictated by the Feigenbaum constants. In such a multifaceted framework, a concept from the far-flung branch of topology, namely the Borsuk-Ulam theorem, comes into play. The theorem tells us that a continuous mapping from antipodal points with matching feature values on an n-sphere to the same real value can always be found. Here we demonstrate that embracing nonlinearity in the framework of the Borsuk-Ulam theorem means that bifurcation transformations (the antipodal points) can be described as paths or trajectories on abstract spheres equipped with a Feigenbaum dimension. Such an approach allows the evaluation of nonlinear systems through linear techniques. In conclusion, we provide a general topological mechanism which explains the elusive chaotic phenomena, cast in a physical/biological fashion which has the potential of being operationalized.