Winter Hats, Part III: Mathematical Shapes (Mobius and Klein bottle)

Part III of winter hats is here! And this time it's math-based. Remember Part I? It featured a blue hat? Well, I had yarn left over, in prodigious amounts. So I decided to make something for myself. I had seen a Mobius scarf, which I had really liked the idea of. As you'll see, mine didn't turn out quite the way I'd pictured, but it works! I also made a knitted math hat. Peruse the pictures below!

Garment Data:
- Type: Hat, scarf
- Date made: Probably summer of 2008?
- Pattern: None. I looked at other hats/scarves in the math shapes I wanted, but then just went from there on my own.
- Fabric/Materials: Acrylic yarn (or maybe an acrylic/wool blend?)
- Trim: none
- Time to finish: I know it took a while! A few weeks, maybe?

The hat and scarf set. They don't look like much, but I promise that they are mathematically cool! And yet warm to wear ;)

Here you can see them together.

The scarf is actually a Mobius strip! In fact, instead of knitting a strip, putting a twist in, and having a seam, I actually took several tries to put the twist into my first row, so that as I knit, I actually went twice around and added to "both" edges before coming back to the beginning of my row!

Here you can see the twist in the scarf. This means that though it is "two-dimensional" (flat), the twist makes it 3D.

It's ribbed, 5 rows each of knit and then purl. Unfortunately this means it curls up into a narrow strand.

I tend to wear it looped 3 times around my neck.

The hat, though, is even cooler. It is a Klein bottle, which is a tube that connects to itself - but not at the ends, to make a donut ("torus"). Nope, it goes through itself!!

Here, from the outside, you can see the outer surface narrow and go in to meet the inner surface.

Here's a close-up of where the outside goes in. Instead of going through its own surface like a proper Klein bottle (explained below), since knitting is a "porous" surface, I had the small end of the tube go through a stretched hole. Then, from the other side, I widened the tube by knitting expansions into it, so that it could meet the inside tube.

Here you can see the inside of the hat, where the outside comes through and joins (you can see the seam about 3 inches down from the crown). Because of the way a Klein bottle works, the hat is actually two layers. You can also kind of see that the top of the hat was knitted on smaller gauge needles. I go back and forth on whether that was a good decision or not - it made it harder to work, and also made it less "loopy" and thus less obviously a Klein bottle.

 This video has a pretty good visual explanation of what my hat does:

Video credit to bothmer for the excellent visual explanation of how a Klein Bottle is basically a 3D Mobius strip.

As to how it compares to my hat, the curly bottom part in the video is the top of my hat. And doubled over, almost donut-like top in the video is the two-layer part of the hat that goes on my head!

Here you can see the ribbed brim I added to the part of the shape that I decided would be the best "hat" part (the top of the shape in the video). This made the shape into a true hat, since now you can see which part to put on your head!

 Most sincerely yours,
~ Sarah