Knitting One-Sided Surfaces
| Hello, and welcome to my one-sided mathematical surfaces page. The picture you see above is me sporting my latest topological creation, the Projective Plane Hat. But I didn't start off making such wild creations right away... | ||
We start off with the Mobius Scarf, which was the second thing I ever knit. (The first was a normal scarf - it turned out very lopsided and I unraveled it and made it into this.) The pattern for it is extremely simple. Just cast on about 30 stitches or so and work in garter stitch until the scarf is however long you want it. Then join the ends, but with a half twist (joining one side to the other). You'll have a scarf in the shape of a Mobius Band. It turns out a Mobius Band can be found in any non-orientable surface. |
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Next we come to the the Klein Bottle Hat. Making a Klein Bottle is basically the same idea as making a Mobius Band: You want to join one side to the opposite side, except you have a tube instead of just a flat band. To join the ends of the tube, you need to join the inside of the tube to the outside. In order to do this, it is necessary to make a hole in the tube (if you're in 3 dimesions) so you can connect the tube from the inside. A true Klein Bottle requires 4 dimensions to make, and has no hole in it. However, I recently realized it is in fact possible to knit a Klein Bottle without a hole in it since knit fabric already has lots of tiny holes in it. This makes it possible to literally pass the fabric right through itself. This is the closest thing to a "true" Klein Bottle you can get in three dimensions. I've written a pattern for the klein bottle hat if you'd like to make one for yourself. If you want to add the words "Klein Bottle" to it, as I've done with this hat, see here. |
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Finally we have the Projective Plane Hat. The pictures below show that hat from various angles, and the picture on the bottom left is of a model I built in order to get a pattern for the hat. A projective plane also has it's inside connect to it's outside, but in a different way. There is no loop like there was with the Klein Bottle. Instead, think of a hemisphere where you need to connect opposite endpoints along the equator (refer the the picture on The Math of Non-Orientable Surfaces). The model is based off of a particular immersion of the projective plane known as Boy's Surface. (An immersion is a way of of placing a surface in a lower dimension that allows it to intersect itself, but still requires it to be smooth. In other words, the surface cannot have any "pinch points"). While there are simpler representations of the Projective Plane, I find that the lack of pinch points an immersion provides makes it much easier to visually understand what the surface is doing, which is why I chose Boy's Surface over the usual Cross-Cap representation (not an immersion). If you want to get a feel for the shape of Boy's Surface, I've produced some animations that I hope are enlightening. |
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| I've also written a pattern for the projective plane hat if you'd like to make one of your very own (It involves some pretty advanced knitting, so I don't recommend attempting it unless you're an experienced knitter). If you want to see what the hat looked like as I was making it, I have some pictures of the unfinished hat. The original pattern did not have the two-color effect - it was something I added later on, but after making it I discovered the new pattern was far easier to knit, so I scrapped the old pattern. It's basically an alternating checkerboard pattern, but the shape of the Boy's Surface makes for an interesting twist when the squares meet up! (Can you guess what happens?) If you want to add the words "Projective Plane" to it, as with the Klein Bottle Hat, see here. |  Side view |
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 Projective plane model |
 Top view |
 Extreme close-up |
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