Inverting beauty

In this paper we give a simple application of spherical inversion, the most elementary among the non elementary geometric transformations, and of some of its generalizations. The principal motivation was an attempt to increase the interest for mathematics in high school students by proposing an easy but mathe- matically rigorous technique for creating new images, new shapes and, by means of 3D printing, new nice material objects. Also in order to put once again in evidence the possibility that mathematics can have something in common with Nature and the Arts. Amongst the generalizations of inversion (see [Bl], [H], [Ep], [S1]) , we find ideally more close to our point of view the hyperbolic inversion due to G. V. Schiaparelli1, an important Italian astronomer not as much known as a geometer, who in [S2], in 1898, tried to represent organic forms and the change from one species to another through geometry