Immersion of the Klein Bottle in Three-Dimensional Space

Abstract

We consider embeddings and immersions of surfaces (two-dimensional manifolds) into three-dimensional Euclidean space. Examples of embeddings for the torus, cylinder, and Möbius strip are given. For the Klein bottle, two immersions are constructed, each having a closed curve of self-intersection. The first immersion can be viewed as an embedding of two Möbius strips sharing a common boundary and intersecting along a common midline. This surface is obtained by a twisted rotation of a figure-eight around an axis.
The second immersion is obtained as a union of four embedded cylinders with horizontal bases. The embedding of two of these cylinders can be considered as halves of embedded torus.