Klein bottles are an entertaining mathematical idea–a shape with no volume. A Klein bottle is basically a tube where the inside is connected to the outside. Making a Klein bottle in our 3D world requires a bit of cheating to work, by adding a hole in one of the walls of the tube to provide a place for an intersection.

The most common physical realization is a glass Klein bottle, which you can ogle and buy at Acme Klein Bottles. They also sell wonderful knit Klein bottle hats which can be bought with a matching mobius strip scarf. I was lucky enough to be given a set as a gift, and it is cozy and bright and wonderful. My only complaint is that the narrow neck of the Klein bottle makes it hard to pull it inside out (or right side out, since it is the same thing) to play with it.

I have found that the concept of a mobius strip is more understandable when you can hold it in your hands and turn it around and around, and I thought the same would be true with a Klein bottle, if only it were a little more flexible than my hat. With that in mind, here’s how to make a simple fabric Klein bottle you can play with from two sleeves of a worn out shirt.

You’ll want to use something like a soft old long sleeved t-shirt. The jersey knit gives it enough stretch to be able to play with it, and starting with sleeves means you know that you can fit your arm inside.

Cut the sleeves off, and cut off any cuff binding as well.

Cut a notch out of one of the elbows, and sew around the edges to keep it from fraying.

Slide one sleeve inside the other and stitch around the shoulders to join them together. Then slide the uncut sleeve through the elbow hole in the other one.

Turn the sleeve that you didn’t cut a hole in inside out so that the other sleeve is inside it. Line up the wrists and sew around the edges to complete the tube.

Reach in and pull out the center and you can see the familiar shape. Then keep turning it inside/right side out just for fun!

I’d seen somewhere that Clifford Stoll had started knitting Klein hats; didn’t know he was selling them though. The money from The Cuckoo’s Egg must have run out.

Several years ago I made a möbius bag, a Klein bottle change purse made from a sock. Nice to see the concept on a larger scale. Nathaniel Hellerstein’s sock directions are here:

http://www.meru.org/Advisors/klein.html

So, if I remember it right, you can now cut your klein bottle in half and have TWO mobius strips.

It’s even better than one size fits all: it’s one size fits EVERYTHING.

There is a glass Klein bottle at the Hayward gallery at the moment, very impressive.

Mathematical note: if it exists in 3-dimensional space, it’s NOT a Klein bottle.

A better way to understand (the topology of) a Klein bottle is to try making a real one. It’s easy to make a Möbius strip from a strip of fabric with a sewing machine: twist once and sew the ends together. Stitching two Möbius strips together along their boundaries produces a Klein bottle — which, of course, can’t exist in the space we live in. It’s illustrative to try the experiment, however.

I talked my sister (who’s an artist) into trying to make one. It was priceless to see the look on her face when she realized, empirically, that 3 dimensions constitute a genuine constraint to what one can sew.

Coming to the catwalks soon: the Calvin Klein bottle