[Japanese]

# The Klein Bottle

You know "Moebius Strip", which is a half-twisted loop of a tape. When you walk round on the loop, you'll reach the reverse side of the start point. The trick is realized by bringing a 3-dimensional-twist into the 2-dimensional world (a plane) of the tape: the band is one-sided.

Imagine the Moebius Strip with one more dimensions, It is the "Klein Bottle". You can make a Klein Bottle from an inner tube of a tire. Cut the tube at any point you like, twist it a half in 4D-way, and re-join them. Now, you got a Klein Bottle! ...when you can't add the "4D-twist", well, make a hole on the tube. It is easier way to make it for 3D people. A sketch of Making a Klein Bottle (drawn by Dr. Hal Tasaki, comments are written in Japanese) will help you. It is an pseude-Klein Bottle. The real Klein Bottle cannot be embedded in our 3D space.

You can get the 3D-Klein Bottles from "Acme Klein Bottle". Even if you don't want to buy them, you should visit the site. It's very interesting. When I visited there, I feel 'I have to get them!'. And now, I do have two of them. ;-)
One of them is a Classical Klein Bottle, and the other is a something which is a very Klein Bottle in the topological sense, but is perfectly different in the cognitive-psycological way.

The Classical Klein Bottle (at the right) is a Klein Bottle, you know. If you are taking a walk on the outer-side of it, go through the hole (from left side of it in the figure), then you are now walking on the inner-side. Another item on the left is ... a beer mug? Yes it is. And it is also a Klein Bottle. Can you see the body is double-walled? It keeps your beer enough cool for hours... well, I mean, let's see the next photo.

You see, the handle is a tube. It is connected to the inner wall on the top, and the outer one on the bottom. The hole on the top corresponds to the hole of the Classical Klein bottle. The outer wall has a hole (it makes the bottle being in the 3D-world) to pass the handle, so you don't have to worry to lost your beer to the wormhole.

One more to tell you. Be careful not to pour your beer through the hole into the handle. Once you do it, it should be very difficult to clean-up the inside of the wall.

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