Athena International Publishing signed an agreement today to publish its first book in the field of Applied Mathematics on the geometric analysis of natural shapes and phenomena as can be seen in such areas as Biology and Physics. The book entitled “From Pythagoras to Fourier and From Geometry to Nature” is written by Paolo Emilio Ricci and Johan Gielis who also serve as Editors for the journal Growth and Form.
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. 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, it is shown 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.
The authors 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 under discussion for a transfer to Athena International Publishing in 2022. From Pythagoras to Fourier and From Geometry to Nature is expected to be published at the end of January 2022.
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