  Together with the Moebius strip, the Klein bottle is one of the intriguing
objects in the universe of geometry, 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 version of the Klein bottle do exist, but they are
not fully satisfactory for some reasons. We discuss some of the existing
representations and propose two new immersions of the Klein bottle in R^3,
which are intended to be a step towards a canonical expression of this surface
in the shape imagined by its first discoverer.
