The Klein bottle in its classical shape: a further step towards a good parametrization

TL;DR

This paper reviews existing Klein bottle parametrizations and introduces two new immersions aiming to better represent Klein's original conceptual shape in three-dimensional space.

Contribution

It proposes two novel immersions of the Klein bottle in R^3 that move closer to Klein's original conceptual shape, improving upon previous parametrizations.

Findings

Two new immersions of the Klein bottle are introduced.

The new representations better approximate Klein's original conceptual shape.

Existing parametrizations are analyzed and compared.

Abstract

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.