Manifolds have a very important property called. Once back on their ship the planet express crew is once again attacked by the killer space whale which eats one of their engines. Mobius scarf, klein bottle, klein bottle ‘hat’. New york: penguin, 2004. This space has length and width but no thickness. And see if we can transform one to the other, under a continuous transformation over the surface. It should be obvious that this mobius strip does not have a physical surface.
Step 1: cut a strip off paper
Mobius strip wiki.
Introduction: mobius strip
A true mobius strip has only one side, an adjacent area that entirely surrounds it. For example, it would not act the same if you cut it in half lengthwise. But to understand this you have to first understand that the word side, like so many words, has more than one meaning and then you have to understand the meaning that applies to a mobius strip.