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Klein bottle


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The Möbius band has just one surface and has one edge. A sphere, on the other hand, has no edges, but it has two surfaces, the outside and the inside. An example with no edges and only one surface is the Klein bottle.

Consider, in the plane, the square of all points with Cartesian coordinates (x, y) such that −1≤x≤1 and −1≤y≤1. The operation of ‘identifying’ the point (x, 1) with the point (x, −1), for all x, can be thought of as forming a cylinder in 3 dimensions, with two surfaces, the outside and the inside, and two edges. The operation of ‘identifying’ the point (1, y) with (−1, −y) is like forming a cylinder but including a twist, so this can be thought of as resulting in a Möbius band. The Klein bottle is constructed by doing these two operations simultaneously. This is, of course, not practically possible with a square sheet of flexible material in 3 dimensions, but the result is nevertheless valid mathematically and it has no edges and just one surface.

http://www.kleinbottle.com/whats_a_klein_bottle.htm A fuller description and images of Klein bottles.

Subjects: Mathematics.


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