and I were chatting about knitted Möbius strips recently. There are several ways to knit a Möbius strip; despite the assertion of this page and other writings, making a mistake in joining up after casting on to circular needles is not one of them — that produces a strip with one twist (or an integral number of twists), not the half twist that a true Möbius strip requires.
The easiest way is to knit a flat strip and graft the ends together, putting a half twist in. Maybe too easy. Not that satisfying, it seems to me, and it does have a true defect: the single edge you end up with is half cast on, half bound off. Nicer to have a strip whose edge is the same all around.
A harder way involves casting on, picking up stitches from the “non working” side of the cast on, and then knitting outwards in both directions; the cast on “edge” becomes the middle of the strip, and the edge is 100% bound off.
What occurred to me in thinking about it is that there’s another way (and no, I don’t claim I’m the first to think of this… only that I did think of it): If you cut a Möbius strip down the middle, you end up with a single loop with two twists, right? And we know how to knit one of those — I mentioned it above: cast onto a circular needle and “make a mistake” joining up, i.e. put two twists into the stitches first.
Well, if cutting a Möbius strip down the middle make a double-twisted loop, and if we know how to knit a double-twisted loop, then we can knit a Möbius strip by knitting a double-twisted loop and instead of binding off the terminal edge, grafting it onto itself, right? This is kind of the reverse of the second method — the cast on edge provides the edge of the Möbius strip, and the terminal edge (the one that would have been the bound off edge, had we bound off) becomes the center.
Maybe someday I’ll knit a scarf like that, but for now, I just wanted to prove the principle, so I did a quick and sloppy job of casting on 60 stitches, knitting about 6 rows, and grafting it together. I’ve never done Kitchener stitch before, and I guess I did it wrong — it doesn’t match the garter stitch, in fact it came out as stockinette which of course doesn’t work well on a Möbius strip. Still, I think I’ve proved the idea is sound.
More pictures here.