Klein Bottle Hat
Finished measurement: Circumference: 22 inches
Description from knitty.com:
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.