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The Klein bottle can be constructed (in a mathematical sense, because in reality it cannot be done without allowing the surface to intersect itself) by joining the edges of two Möbius strips together, as described in the following limerick by Leo Moser:1.
There is a 2-1 covering map from the torus to the Klein bottle, because two copies of the fundamental region of the Klein bottle, one being placed next to the mirror image of the other, yield a fundamental region of the torus.
Note that this is an "abstract" gluing in the sense that trying to realize this in three dimensions results in a self-intersecting Klein bottle.
To construct the Klein Bottle, glue the red arrows of the square together (left and right sides), resulting in a cylinder.
It is also homeomorphic to a sphere plus two cross caps.After the figure has grown for a while, the earliest section of the wall begins to recede, disappearing like the Cheshire Cat but leaving its ever-expanding smile behind.By the time the growth front gets to where the bud had been, there’s nothing there to intersect and the growth completes without piercing existing structure.While the Möbius strip can be embedded in three-dimensional Euclidean space R, as follows: one takes the square (modulo the edge identifying equivalence relation) from above to be E, the total space, while the base space B is given by the unit interval in y, modulo 1~0.
The projection π: E→B is then given by π([x, y]) = [y].This immersion is useful for visualizing many properties of the Klein bottle.For example, the Klein bottle has no boundary, where the surface stops abruptly, and it is non-orientable, as reflected in the one-sidedness of the immersion.Like the Möbius strip, the Klein bottle is a two-dimensional manifold which is not orientable.