1. A closely related 'strange' geometrical object is the Klein bottle.

2. The following square is a fundamental polygon of the Klein bottle.

3. The common physical model of a Klein bottle is a similar construction.

4. The real projective plane, like the Klein bottle, cannot be embedded in three-dimensions without self-intersections.

5. BS(1, 1) is the free abelian group on two generators, and BS(1, −1) is the fundamental group of the Klein bottle.

6. On the other hand, there are surfaces, such as the Klein bottle, that cannot be embedded in three-dimensional Euclidean space without introducing singularities or self-intersections.

7. Six colors suffice to color any map on the surface of a Klein bottle; this is the only exception to the Heawood conjecture, a generalization of the four color theorem, which would require seven.