In his Habilitationsschrift Riemann mentioned the geometric tangent 2D-indicatrix as a relevant extension of the Euclidean unit circle, thus conceiving the simplest Riemann-Finsler metrics.  However, in 1818, Gabriel Lamé (1795-1860) had already provided a systematic treatment of supercircles and superellipses, with the aim of describing crystals geometrically.

The generalization of Lamé curves to the Superformula, a generic geometric transformation generalizing conic sections, turns out to be an excellent scientific tool for the study of natural shapes and phenomena, from the very small to the very large.  These developments have found their way in numerous applications in technology and science.  It also connects many fields of science in a simple way, e.g. shape and form (constant anisotropic mean curvature surface), and inequality in economics and biology.

In a dynamic and kinematic setting, recent developments include a complete differential geometry apparatus, adapted to these shapes.  This aligns with Lamé’s Unique Rational Science: find the best fitting coordinate systems and then solve the relevant boundary value problems.

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