Klein Bottles without Holes, or How to Crochet Through
After I crocheted Whitney Umbrellas, I got excited that I figured out how to crochet through an existing layer of fabric. I decided to use my skills to crochet “correct-er” Klein bottles.
A Klein bottle is a cool mathematical surface with only one side, similar to the Möbius strip. Unlike the strip, the Klein bottle has no border making it a non-orientable manifold. The problem in making Klein bottles is that the Klein bottle can’t be embedded into 3D space. Thus, all 3D models of Klein bottles have to self-intersect. But all the models that I saw, including glass models and crocheted hats that you can buy at ACME Klein Bottle, have holes.
I realized that my method of crocheting through might allow me to make more accurate models of Klein bottles, ones without holes.
Now it is time to spill my secret. The idea is easy, the implementation is not. The two pictures show the same yellow cylinder crocheted through a green square, viewed from different angles. I crocheted the green square first, then half of the yellow cylinder. Afterwards, I had to pull the whole ball of yellow yarn through a tiny hole in the middle of the green square. Then, with my hook, I pulled each yellow stitch from one side of the green square to the other side through its own tiny hole and finished the stitch on the new side. In the third picture, you can see my Klein bottle made in two colors. You might be able to see the second color inside instead of a hole.
I invented this method while crocheting Whitney’s umbrellas. I had to pull the whole ball of yarn through a tiny hole once per row. I still remember the tediousness of it with dread.
After the bottles, I decided to try projective planes. In the fourth picture, you can see two projective planes and two projective planes with holes. For the former, I started with the bottom hemispheres, and for the latter with cylinders. I didn’t need to crochet through or pull yarns through tiny holes. I just crocheted one row across the other. I left the easiest crochet task for last!
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