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I thought a klein bottle is supposed to have walls with zero thickness -it's made of an idealized plane...

Ok, it's an immersion in R^3 of a tubular neighborhood of an embedding of the Klein bottle in R^4.

But be like Michelangelo and see the immersed Klein bottle inside the glass.

... but with an imaginary klein bottle you can own one without buying it; that's cheaper, isn't it?

You'd think so, but it's actually more expensive because you don't get the experience of buying a Klein bottle from Cliff Stoll.