Mobius Strips- How many sides does a strip of paper have?

Mobius strips are made by taking a length of paper, putting a twist in it, then joining the ends.  After looking at these strips, and having to answer some questions, the class began to question everything they thought we knew about a strip of paper. You see, Mobius strips have only one side!  If you follow the curve of the strip with your finger you end up on the other side of the strip from where you started from, which, mathematically, means you are still on the same plane (side).

They get even more interesting if you attempt to cut them in half (along the red line in the photo).  You would expect to end up with two separate, distinct loops but no!  Instead, you get one stip with two twists in it!  Try and follow a side with your eyes back to the start.

However, if you cut the strip close to one side and follow this around the Mobius strip, an even more curious thing happens- you end up with two strips linked together, one loop inside the other.

Repeat the process and, did you guess? Three loops, two as above and the third looped through the first two.  We stopped there but the process can continue for quite some time.

There's an algorithm which explains all of this mathematically.  I'm leaving that to the experts.