Gielis Transformations in mathematics, the natural sciences and technological applications

The Gielis Transformation (GT) defines measures and unit elements specific to the shape, extending Euclidean geometry and challenging current notions of curvature, complexity and entropy. Global anisotropies or (quasi-) periodic local deviations from isotropy or Euclidean perfection in many forms that occur in nature can be effectively dealt with by applying Gielis transformations to the basic forms that show up in Euclidean geometry, e.g. circle and spiral. Anisotropic versions of the classical constant mean curvature and minimal surfaces have been developed. In mathematical physics it has led to developing analytical solutions to a variety of boundary value problems with Fourier-like solutions for anisotropic domains.

GT have been used in over 100 widely different applications in science, education and technology. In the field of design and engineering they have been used, among others, for the optimization of wind turbine blades, antennas, metamaterials, nanoparticles and lasers.