Auntie Ann Knits

Tuesday, August 08, 2006

Message in a Klein Bottle

Imagine my surprise to see this pattern for a Klein bottle hat in the summer issue of Knitty, since I had been working on my very own Klein bottle, sans pattern, on and off since spring.

What is a Klein bottle hat, you may very well ask. I refer you to this excellent website for a description of the 4-D topological shape known as Klein bottle. A Klein bottle hat is a mere 3-D knitted representation of a Klein bottle. From that same site you may even purchase your very own hand-knitted Klein bottle hat. That site is full of topological humor related to Klein bottles, too, if that amuses you (it amuses me, but obviously I come from a long line of geeks). Also a cut-away cross-section drawing of a Klein bottle hat (this is funny to me, too).

Back in May, I posted a progress pic of the hat I was working on for my math professor uncle, and now it is done:
Klein Bottle Hat 1

My model there is a cantaloupe. (I sure hope this doesn't end up on the new You Knit What?!.)

Here I've folded the opening back a little -- I chose the outer yarn because it's a good match to my uncle's Moebius scarf (or a "knitted representation of a Moebius band", as he put it), and the inner yarn because it was less itchy and wouldn't clash:
Klein Bottle Hat 3

And here's where they're grafted together, although normally this would be tucked away inside (except a Klein bottle has no inside):
Klein Bottle Hat 4

And a close-up of the grafting:
Klein Bottle Hat 5

The grafting has been "enhanced" by duplicate stitching the ends to make it appear more continuous, stitch-wise, but since there were 2 sets of 1x1 ribbing stitches knit in opposite directions meeting, a good clean graft was not really possible, the way it would be if it were stockinette. But hardly anyone will look at this, normally it shouldn't show.

I began at the lower edge with a provisional cast-on of 120 stitches in DK weight yarn on #6 needles in 2x2 ribbing. This is unlike the Knitty pattern, which begins at the point where I grafted mine. I knit as you would any other hat, decreasing to 18 1x1 ribbed stitches [ETA -- near the top, knit back a forth for a few rows to form the slit, needed since we don't live in 4-D], then knitting the narrow tube. When the darker gray part was done, I picked up the stitches at the lower edge with #4 needles and knit as any other hat, decreasing to 18 1x1 ribbed stitches, and then grafted.

Since there are 2 layers over the ears, this will be a very warm hat, appropriate for my uncle who lives in Vermont and Boston.

But mainly this should give him some bragging rights with the other math professors, who reacted to his Moebius scarf by showing off their Klein bottle hats. This is a gift for his 85th birthday party this coming weekend. Maybe I'd better block it before we eat that cantaloupe.


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