Klein Bottle
The Klein bottle is a one-sided surface named after the great 19th-century mathematician Felix Klein, who was one of the first mathematicians to explore its unusual properties. The Klein bottle is formed from a square piece of paper by gluing opposite edges according to the following rules. The top and bottom edges are glued to each other by a reverse gluing; that is, the top edge is twisted 180 degrees before it is glued to the bottom edge. Performing only this gluing will turn the square into a surface called the Möbius strip. The left and right edges are glued in a very simple way: each point on the left edge gets glued to the opposite point on the right edge. Performing only this gluing will turn the sheet of paper into a cylinder.
Each of these gluings separately is easy to carry out. However, it is impossible to perform both gluings unless you allow the piece of paper to pass through itself, as in the depicted figure. Mathematicians have proven that there is no way to "embed" the Klein bottle in 3-dimensional space: no matter how much it is pulled or twisted, it will always intersect itself.
Although the Klein bottle cannot be constructed in space without self-intersection, mathematicians are still able to study the shape of the Klein bottle by looking at it intrinsically, considering what life would be like for a two-dimensional creature living on the Klein bottle. From this point of view, it is possible to ignore the specific way in which the Klein bottle is placed in space (the extrinsic point of view), and simply consider what a creature would experience living on a sheet of paper that has the appropriate gluings. Looking at the Klein bottle's intrinsic structure, it becomes clear that the Klein bottle is an example of a non-orientable surface, that is, a surface on which it is impossible to distinguish between left-handedness and right-handedness. Consider what happens if you draw a right shoe somewhere on the sheet of paper. If you move it to the right, when it gets to the edge of the paper it will come out on the other end, from the left edge, since these edges are glued together. It will eventually return to the spot where it started, just the way a person walking along the equator of the Earth will eventually return to his starting place. This process will not change the appearance of the shoe. If instead you move the shoe upward until it reaches the top edge, it will come out again from the bottom edge. However, since these edges are glued in reverse, the shoe will come out as the mirror image of a right shoe--that is, as a left shoe. Thus, there is no consistent way to define the difference between a left shoe and a right shoe on the Klein bottle.
The Klein bottle has another surprising property: although it is a closed surface, with no holes or boundary, it does not divide 3-dimensional space into an inside and an outside. If you stand on one side of the piece of paper and walk towards the top edge, then since the top edge is glued to the bottom edge with a twist, when you walk across the glued edge you will find yourself on the other side of the sheet of paper. Surfaces that have this property are known as one-sided. In its one-sidedness the Klein bottle resembles the Mobius strip, the simplest example of a one-sided surface. In fact, the Klein bottle can be formed from two Mobius strips, by gluing them together along their boundaries. The Klein bottle is also closely related to another non-orientable surface, the projective plane, which is formed from a round sheet of paper by gluing together opposite points on the boundary circle (like the Klein bottle, this surface cannot actually be constructed without the sheet of paper passing through itself). A Klein bottle is the "connected sum" of two projective planes: it can be formed by cutting a hole out of each projective plane and then running a tube from one hole to the other, connecting the two projective planes.
The Klein bottle cannot sit in 3-dimensional space without passing through itself, but in 4-dimensional space it is possible to build a Klein bottle that never intersects itself. In 3-dimensional space, the Klein bottle intersects itself along a circle. In 4-dimensional space, it is possible to "lift" one of the two circles of intersection off the other, into the fourth dimension. This is difficult to visualize, but it is similar to a lower-dimensional phenomenon: if you take a shoelace and lay it on the ground (a 2-dimensional surface) in a figure-8 pattern, the shoelace will cross itself at the center point. If however you are allowed to move part of your shoelace off the ground (into 3-dimensional space) then at the crossing point you can lift the top piece of shoelace up a bit, resulting in an "embedded" shoelace, that it, one that never touches itself. Mathematicians have used these techniques to show that every surface can be embedded in a higher-dimensional space.
This is the complete article, containing 857 words
(approx. 3 pages at 300 words per page).
