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 way that it takes on a different orientation - it can appear as its own mirror image on the same surface. This property of the Klein bottle is what makes it non-orientable.
The Klein bottle is named after the German mathematician Felix Klein. Felix Klein's work in mathematics made him very familiar with the Möbius strip. A Möbius strip is a piece of paper which is given a half-twist, and joined at the ends. This twist turns a regular piece of paper into a non-orientable surface. Felix Klein reasoned that if you were to attach two Möbius strips together along the edges, you would make a new type of surface with equally strange properties - a Klein surface, or Klein bottle.
Unfortunately for those of us who would like to see an actual Klein bottle, they cannot be constructed in the 3-D, Euclidean space in which we live. Joining the edges of two Möbius strips to build the Klein bottle creates intersections, which cannot be present in the theoretical model. A real life model of the Klein bottle must intersect itself as the neck of the bottle crosses through the side. This gives us something which is not a true, functional Klein bottle, but which is still quite interesting to examine.
Since the Klein bottle shares many of its strange properties with the Möbius strip, those of us who do not have the deep understanding of mathematics necessary to truly understand the Klein bottle's complexities can experiment with the Möbius strip to gain some insight into Felix Klein's fascinating discovery.
By Jeff Petersen