You can make a moebius strip out of a strip of paper by giving one of the short edges a half twist and attaching it to the other short edge.
I came to knitting as an adult and at some stage stumbled across the idea of moebius knitting. Using the strip of paper idea, you could knit a strip of knitting and then graft it to the twisted edge, right? But someone had to go one better. Various people have dreamt up ways of casting on stitches so as to make a moebius plane without any grafting. I loved the idea, but didn't get into working out how to do it. Then a few weeks ago my friend C came to me with a knitting puzzle. She'd bought a pattern for a Moebius wrap and it wasn't working the way she'd expected. I had another look at the geometry and studied her pattern. It wasn't a true moebius, just an interesting twisted loop. We puzzled over it for a while and then shelved the problem. Now it's holidays and I can finally get back to it. I found this Moebius cast on tutorial and started knitting. I can't wait for it to grow enough so I can cast it off and see what I've got.