Moebius strips in knitting

A Moebius strip is the formal name of the mathematical object whose discovery is now attributed to the German mathematician Moebius. Apparently, this was really discovered by another mathematician called Listing, but he was only Czech so he doesn't get the credit :-). Without splitting hair about the mysteries of attribution, this is a fascinating object, which has only one side and one surface even though it looks to have more at first glance. The neat part is that since it's the simplest of its topological category we can easily make a physical model of it. The next step up, the Klein bottle, requires some imagination and can only be modeled in our pedestrian world with a large measure of cheating. There are a variety of geek explanations of how easy it is to convince yourself of a Moebius strip's properties, which I highly recommend you do. Moebius strips have intrigued artists before our time, for instance Escher who did this ant picture.

Moebius strips were first popularized in knitting by Elizabeth Zimmerman, whose fertile and curious mind immediately seized on possible applications. She first came up with a free-standing Moebius strip as a scarf. There are many advantages to the Moebius approach to scarves, which make us personally fanatics about them. One is that it's practically impossible to lose a scarf that has no ends - you'd have to step over it to leave it behind, even if it had somehow managed to drop over your entire body. Another is that wear is completely even, since it gets worn subtly differently every time you pick it up, it's all the same so the actual position varies. You never get the thinned behind the neck part and splayed out ends that affect many knitted scarves after a short while.

From a purely esthetic point of view, they're also winners because of the nice twist and how they help the scarf behave. If you make a rather long one, say 50", it can be worn at least 3 ways. Just draped over, it hangs nicely, and the twist doubles as an instant muff for cold hands.

Doubled over so it goes around the neck twice, the twist helps it be organized and look much more interesting than a flat object. And finally if it's wide enough (say 8-10") one half can go over your head and the other half around your neck, so you get an integrated cowl that can come in handy if you get caught in a snowstorm.

A short one, about 24", also lies about the neck in a much nicer way than a simple tube would.

A medium one one, about 40", can also serve as a capelet.

One obvious technical point is that if you want a Moebius to look good you need to use a totally reversible stitch, which is a good thing in my book. It also opens up the field to the entire medium of crochet, which is rarely not reversible. Elizabeth's first impulse was to do a Moebius the way mathematicians describe it, using a strip of paper. This involves casting on a few stitches, making a long strip, and introducing a 180o flip before joining the ends. This is very straightforward, and has the advantage of leaving the decision of whether to Moebius or not to the end of the project. It looks much better if you use an invisible cast-on, and graft the ends together at the end so the seam disappears.

Other people have since figured out that it's very easy to make a Moebius strip lenghwise, sparing yourself the rigors of grafting. In fact, it's one of the major tricks of learning circular knitting, to learn to join a cast-on so that you'll either get a tube or a moebius strip deliberately. These are completely different mathematical objects, one will never be the same as the other, as many beginners have learned, things don't just shake out the other way in the end if you persist in your error after discovering it. So you can cast-on a sufficient number of stitches for the Moebius strip desired, flip it at the end of the first row, and knit on to get a perfectly decent strip, with the pattern going the other way than in the previous method.

Naturally every mathematically-inclined knitter has since gravitated to Moebius strips at some point or another. Elizabeth herself made some effort to include them in larger designs, in vest collars and so on. By far the most creative person with this has been Cat Bordhi, who recently published two books on the topic ("Magical Knitting"). She's worked with a mathematician's sensibility and expanded the basic topological form to produce all kinds of practical objects with unusual beauty and hidden technical features. If nothing else, she's completely convinced us that putting a twist in bag handles makes them much more likely to stay on the shoulder, something which we've faithfully stuck with to good effect. Check it out!