The Klein Bottle: Variations on a Theme

Together with the Moebius strip, the Klein bottle is one of the intriguing objects in the universe of geometry, present in topology courses of any level and sometimes appearing in non-mathematical contexts too. Until now, several parametrizations of it as a surface immersed in ordinary three-space have been found, some of which are very elegant and lead to nice and well understandable shapes. Nevertheless, these shapes are quite different from the object imagined by F. Klein in the late 19th century: a tube which passes through itself with the two ends glued together. Parametrizations for this particular shape of the Klein bottle do exist, but they are not fully satisfactory for some reasons. In this paper I collected and discussed some of the most interesting parametrizations of the Klein bottle in its various shapes and propose two new immersions of it in R^3 in the shape imagined by its first discoverer. They are intended to be a midstep, rather than an end of the route, towards a canonical expression of this surface in its first shape.