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In mathematics, the Klein bottle is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down. Other related non-orientable objects include the Möbius strip and the real projective plane. Whereas a Möbius strip is a surface with boundary, a Klein bottle has no boundary (for comparison, a sphere is an orientable surface with no boundary). The Klein bottle was first described in 1882 by the German mathematician Felix Klein. It may have been originally named the ''Kleinsche Fläche'' ("Klein surface") and then misinterpreted as ''Kleinsche Flasche'' ("Klein bottle"), which ultimately led to the adoption of this term in the German language as well.〔, (Extract of page 95 )〕 ==Construction== The following square is a fundamental polygon of the Klein bottle. The idea is to 'glue' together the corresponding coloured edges so that the arrows match, as in the diagrams below. 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. To glue the ends of the cylinder together so that the arrows on the circles match, you must pass one end through the side of the cylinder. Note that this creates a circle of self-intersection - this is an immersion of the Klein bottle in three dimensions. Image:Klein Bottle Folding 1.svg Image:Klein Bottle Folding 2.svg Image:Klein Bottle Folding 3.svg Image:Klein Bottle Folding 4.svg Image:Klein Bottle Folding 5.svg Image:Klein Bottle Folding 6.svg 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. The common physical model of a Klein bottle is a similar construction. The Science Museum in London has on display a collection of hand-blown glass Klein bottles, exhibiting many variations on this topological theme. The bottles date from 1995 and were made for the museum by Alan Bennett. The Klein bottle, proper, does not self-intersect. Nonetheless, there is a way to visualize the Klein bottle as being contained in four dimensions. By adding a fourth dimension to the three-dimensional space, the self-intersection can be eliminated. Gently push a piece of the tube containing the intersection along the fourth dimension, out of the original three-dimensional space. A useful analogy is to consider a self-intersecting curve on the plane; self-intersections can be eliminated by lifting one strand off the plane. Suppose for clarification that we adopt time as that fourth dimension. Consider how the figure could be constructed in ''xyzt''-space. The accompanying illustration ("Time evolution...") shows one useful evolution of the figure. At ''t''=0 the wall sprouts from a bud somewhere near the "intersection" point. 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. The 4-figure as defined cannot exist in 3-space but is easily understood in 4-space. More formally, the Klein bottle is the quotient space described as the square () × () with sides identified by the relations for and for . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Klein bottle」の詳細全文を読む スポンサード リンク
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