Changes in Approximate Symmetries of a Parametrized Turing Pattern

Liam D. O'Brien

The Ohio State University

"Changes in Approximate Symmetries of a Parametrized Turing Pattern"

Organisms exhibit a dazzling array of symmetries, from the rotational symmetries of flowers to the fractal symmetries of trees and even bilateral symmetries in humans. Symmetry is fundamental and is often a predictor of survivability, fecundity, and evolvability. Although it is intuitively clear that symmetry exists in nature, the symmetries are typically imperfect, making it difficult to apply mathematical tools that were built to understand idealized versions of symmetry. In 2021, Gandhi et al. proposed a real-valued operator that can quantify approximate symmetries by evaluating how much an object changes under a transformation. When one parametrizes the transformation and considers the operator’s graph on the parameter space, the symmetries of the object appear as local minima. I consider the rotational symmetries of a Turing pattern, showing that if we treat minima and maxima of the graph as stable and unstable equilibria (respectively), the changes in extrema are qualitatively similar to changes in equilibria that we observe in classical local bifurcations. Studying relevant properties of the operator may allow us to apply the tools of bifurcation theory to understand how approximate symmetries form in development.

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