Affine Unfoldings of Convex Polyhedra: Progress on Dürer’s Problem

Convex polyhedra are among the oldest mathematical objects. Indeed the five platonic solids, which constitute the climax of Euclid’s books, were already known to the ancient people of Scotland some 4000 years ago [1]. During the Renaissance, polyhedra were once again objects of fascination while painters were discovering the rules of perspective and laying the foundations of projective geometry. This remarkable confluence of art and mathematics was personified in a number of highly creative individuals including the German painter Albrecht Dürer who was based in Nüremberg at the dawn of the 16 century, and is credited with ushering the advent of Renaissance in Northern Europe. During extended trips over the Alps, Dürer learned the rules of perspective from his Italian contemporaries, and subsequently described them in his influential book, The Painter’s Manual [4]. Aside from being the first geometry text published in German, this work is remarkable for containing the first recorded examples of unfoldings of polyhedra.