by Marie-Christine Mahé

We're all pretty familiar by now with the uses of the Moebius strip in knitting, and they delight my mathematical heart. But did you know that Moebius strips are just a special case, and that this category of non-orientable surfaces, ie 'objects with a twist, so with a single surface', actually exists in any dimension? The one most accessible to our limited imagination is the 4-D version, which is called a Klein bottle. It sort of looks like a bottle at first glance, except that it only has one surface, and the liquid that's in it is also outside.

It doesn't seem entirely practical for knitting. There would be room for only one sleeve in a sweater using this concept, the 2-arm version would be another topological object altogether. There'd be no place to put your hand out, irrelevant you might argue since your whole body would technically be outside. For a bag version, things you would put in would also be outside, and start by falling in between the walls, which would be rather inconvenient.

But by folding a Klein bottle right you can make a useable hat. I'm providing here a very easy version - the stitch is elementary 1x1 rib, the construction no more complicated than for a normal hat, only the assembly takes off from the ordinary. Alas, it's not perfect, it's only a 3-D representation, the surfaces should completely meld together in a moveable join, but that's always the problem with math: mere reality interferes with beautiful theory.

Still, when made in a light fuzzy yarn it also happens to looks very much like the great hat Marc Jacob is showing for next Fall, the big ball, and that should be very good for our Northern friends as it's also very warm.

Moebius strips were first introduced to knitters by Elizabeth Zimmermann in 1981, and Cat Bordhi, a Canadian mathematician, has recently published a couple of 'Magical Knitting' books that really ran away with the concept. I highly recommend them all.

model: Rosemarie Lewis photos: Marie-Christine Mahé

[Knitty's list of standard abbreviations and techniques can be found here]

Push short tube inside out and to inside of work, folding work inside itself until short tube emerges from side opening.

Use yarn tail to sew held sts to CO end of tube. Be sure each st is secured. (If you chose to use a provisional CO method, graft beginning and end of work together.) Weave in ends.

Voilà! Arrange cap and handle to suit yourself, and be prepared to explain the mathematical concept to curious onlookers.