Historical Notes on Star Geometry in Mathematics, Art and Nature

Gamma: “I can. Look at this Counterexample 3: a star-polyhedron I shall call it urchin. This consists of 12 star-pentagons. It has 12 vertices, 30 edges, and 12 pentagonal faces-you may check it if you like by counting. Thus the Descartes-Euler thesis is not true at all, since for this polyhedron V − E + F = − 6 $V - E + F = - 6$ ”. Delta: “Why do you think that your ‘urchin’ is a polyhedron?” Gamma: “Do you not see? This is a polyhedron, whose faces are the twelve star-pentagons”. Delta: “But then you do not even know what a polygon is! A star-pentagon is certainly not a polygon!” In the above dialogue from Lakatos (Proofs and Refutations, Cambridge University Press, Cambridge, 1976), Imre Lakatos used the example of star polyhedra to describe the complex path definition–proof–refutation–new definition that mathematical thought threaded before reaching a consensus on fundamental points of the main mathematical topics. In this paper we will try to examine how in the history of polyhedra (and in particular star polyhedra), a long period of “discovery” of individual types due to the observation of natural objects or due to artistic imagination preceded (and was connected with) the mathematical solution fixing the “right” definitions. Such long period of discovery—we will argue—influenced further investigations on nature and art. The paper will start from the thirteen century and will end with the publications of Pappo’s work (Pappi Alexandrini, in Mathematicae Collectiones, ed. by Commandinus, Venezia, 1588) and Kepler’s Harmonices Mundi which provided solid mathematical foundations to the subject. We want to describe the geometric ideas and the theories of Adelard of Bath, Thomas Bradwardine, Luca Pacioli, Albrecht Dürer, Simon Stevin, Daniele Barbaro, Jan Brożek. We conclude with some short notes about the subsequent developments (Johannes Kepler, Louis Poinsot and Albert Badoureau).