# Carlo Séquin

I work on the boundary between Art and Mathematics. Sometimes I create artwork by using mathematical procedures; at other times I enhance a mathematical visualization model to the point where it becomes a piece of art. Recently I have studied the four different types of Klein bottles that cannot be transformed into one another by a regular homotopy (a continuous smooth deformation that creates no creases, cusps, or other singular points with infinite curvature). In this context I discovered a few new Klein bottle shapes with interesting symmetries, which also make pleasing abstract sculptures.

Boy's surface, a compact model of the projective plane, with a small disk removed is topologically equivalent to a Möbius band. Every Klein bottle can be composed of two Möbius bands that are glued together by their edges. In this model a Klein bottle is created by gluing together two mirror images of a 3-fold symmetrical Boy surface with a disk removed from its pole. The result is a Klein bottle with S6 symmetry, showing six of the “inverted sock” openings characteristic of the classical Klein bottle.