The program on the previous page glued a Mobius band to a disk to form the real projective plane. Here, we’ll glue from the previous page to a second Mobius band to form a Klein bottle. We construct from so that the two Mobius bands meet along ‘s boundary, which is (topologically) a circle. Specifically, to form , we invert across the ellipsoidal surface that contains this boundary curve. In the following program, and are named band1 and band2 respectively.

Yet another Klein bottle

We’ve seen that we can construct a torus from a cylinder by glueing together its two circular boundary curves. Yet-another-Klein-bottle can be understood in this way, except that the cylinder passes through itself in a line of self-intersection before its two boundary curves are glued.