Wednesday, January 11, 2012

Mathy Arts and Crafts

One of my favorite hobbies, apart from riding my bike and lighting things on fire, is knitting. I like knitting because there is a lot of math involved. Math is used to figure out how to make stitches create the shapes you want, be they sweater sleeves or cozy wool socks. This holiday season I decided to use the math of knitting to create something, well, mathy; a Klein Bottle.

Most people have heard of a Mobius Strip. I remember hearing about it long ago on Mr. Wizard. For those of you unfamiliar with this neat surface, it is a loop that looks like it should have two sides, but only really has one. It's twist means that if you were an ant on the strip and you started at say, a little sugar cube, you could walk straight ahead, never turning around and come right back to your tasty treat. It is pretty easy to knit a Mobius scarf. It is a pretty standard scarf design and there are a lot of patterns available. When the scarf is first started, a neat little technique, and some ability to visualize how to make strange surfaces makes the half twist needed for this scarf to be a Mobius strip. Though, it is also possible to knit a regular scarf, put a half twist in and sew (or for the knitters, graft) one end to the other.

Klein bottles are just a Mobius strip only a little more involved. They still only have one surface and are made by gluing two Mobius strips together along their edge. Think about that. How would you do that? I hope that at least some of you reached for the paper and tape, made two Mobius strips and tried to figure out exactly where the edges were and how to glue them together. Its about as easy as kissing your elbow (I now hope at least some of you have let go of your mouse and are trying to kiss your elbow). Klein bottles can't really exist in three dimensions without going through themselves. They will happily sit in 4 dimensions with no self intersections. It is interesting and challenging to think of something sitting in 4 dimensions and being different. Try thinking about taking your coffee cup and putting a book on top of it and flattening it to two dimensions. How is it different than it was in 3? For one thing, it can't exactly hold coffee, for another, its now a circle. Very different from its former self. A Klein bottle is much the same. It is a surface that would look one way in 4 dimensions and quite different squashed into three. Though it is a bottle, it, like the smashed coffee cup can't really hold coffee. The inside is the outside and the outside is the inside.

I wanted to make a Klein Bottle hat but I also wanted one that was a continuous surface that that could be sliped through itself to demonstrate that it really is only one sided. Knitting such a Klein Bottle poses a bit of a challenge. There is a company that makes knitted Klein Bottle hats if you don't want to take on the challenge yourself. Though they do not list their pattern. They also make fun blown glass bottles, one of which I have sitting on my mantle right now. There is a pattern for a hat, but the top tube is so narrow that it is impossible to use your new hat to show that the bottle only has really one side. That, and I didn't realize this pattern existed till after I made this hat. So using what I know about both surfaces and knitted things, I created my own pattern. The video above shows me pulling the hat through itself. As you can see, it is only one sided! For those of you interested in the pattern I used, it can be found here in a separate blog post.

Even though this hat has one side fewer than most, it is quite toasty!

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