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The application of mathematics to physical and natural phenomena in almost all cases makes use of pre-existing mathematical structures, such as Riemannian geometry and tensor calculus for general relativity, and matrices and Hilbert spaces for quantum mechanics. According to René Thom (Fields Medal 1958): “There is only one authentic counterexample to the thesis supporting the preformed, almost a priori character of the mathematical structures applied progressively to theoretical physics or to other branches of knowledge of reality: Fourier’s wave theory. It is very clear that the Fourier series theory was really inspired by physics, more precisely by the study of vibrating strings or the theory of heat.” The current book deals with a second example of a mathematical development inspired by natural shapes and phenomena. Gielis curves were inspired by botany and various symmetries in nature, expanding Gabriel Lamé’s proposed use of superellipses to model crystals to a very wide range of natural shapes and phenomena.
In this book, these two methods are combined by showing how the original Fourier projection method can be used to solve boundary value problems on normal polar domains, in particular Gielis domains. Moreover, since each specific instance or curve comes with its proper trigonometric functions and Pythagorean theorem, this opens up new possibilities and connections in mathematics. In specific cases like the diamond, this leads to generalizations of Fourier’s work to deal with piecewise linear functions.
The key observation in this book is that Lamé-Gielis transformations provide for a new way to study nature in which many different fields of science can be unified. By generalizing Lamé’s work, the authors arrive at a 21st-century version of the Pythagorean theorem. Studying the world through these glasses we see more structure than chaos, more redundancy than entropy and continuous transformations between shapes. Circles and squares, ellipses and polygons, starfish and flowers, are no longer different, but one family of geometrical shapes. Gielis transformations (which have found hundreds of applications in technology ranging from antennas to lasers, from data processing to nanotechnology, from virtual reality to sounds, and more) are an effective geometric approach to deal with some of the global anisotropies in many forms that do occur in nature and with imperfections or certain kinds of repeated local deviations from Euclidean perfection in such forms.
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Paolo Emilio Ricci (UniNettuno International Telematic University, Rome, Italy) and Johan Gielis (University of Antwerp, Antwerp, Belgium) have written numerous journal articles on (Differential) Geometry and the applications of Geometry in reputable scientific journals such as the American Journal of Botany (which published the original paper introducing Gielis transformations), Symmetry and PLOS One. In addition to this, they have authored several books on these topics. Moreover, both also serve as Editorial Board Member and Editor-in-Chief, respectively, for the journal Growth and Form which is published by Athena International Publishing.