What is a Klein Bottle?
A Klein bottle is a type of non-orientable surface, which is often depicted as looking like a long-necked flask with a bent neck passing within itself to open as its base. A Klein bottle’s unique shape means that it has only one surface – its inside is the same as its outside. A Klein bottle cannot truly exist in 3-dimensional, Euclidean space, but blown glass representations can give us an interesting glimpse. This is not a true Klein bottle, but it helps one visualize what the German mathematician Felix Klein imagined when he came up with the idea of the Klein bottle. A Klein bottle is described as a non-orientable surface, because if a symbol is attached to the surface, it can slide around in such a way that it can come back to the same location as a mirror image. If you attach a symbol to an orientable surface, like the outside of a sphere, no matter how you move the symbol, it will keep the same orientation. The Klein bottle’s special shape allows you to slide the symbol in such a
Three glass Klein Bottles, available from Acme Klein Bottle. Ever hear of a Möbius Loop — a one sided, one edged surface? Give a strip of paper a half-twist, then tape the ends together. It’s one side and one boundary, with delightful properties dear to mathematicians. In 1882, Felix Klein imagined sewing two Möbius Loops together to create a single sided bottle with no boundary. Its inside is its outside. It contains itself. Take a rectangle and join one pair of opposite sides — you’ll now have a cylinder. Now join the other pair of sides with a half-twist. That last step isn’t possible in our universe, sad to say. A true Klein Bottle requires 4-dimensions because the surface has to pass through itself without a hole. It’s closed and non-orientable, so a symbol on its surface can be slid around on it and reappear backwards at the same place.You can’t do this trick on a sphere, doughnut, or pet ferret — they’re orientable. A true Klein Bottle lives in 4-dimensions. But every tiny pa
