![]() More formally, the Klein bottle is the quotient space described as the square × with sides identified by the relations (0, y) ~ (1, y) for 0 ≤ y ≤ 1 and ( x, 0) ~ (1 − x, 1) for 0 ≤ x ≤ 1. The 4-figure as defined cannot exist in 3-space but is easily understood in 4-space. By the time the growth front gets to where the bud had been, there is nothing there to intersect and the growth completes without piercing existing structure. 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. At t = 0 the wall sprouts from a bud somewhere near the "intersection" point. The accompanying illustration ("Time evolution.") shows one useful evolution of the figure. Consider how the figure could be constructed in xyzt-space. Suppose for clarification that we adopt time as that fourth dimension. Time evolution of a Klein figure in xyzt-space This creates a curve of self-intersection this is thus an immersion of the Klein bottle in the three-dimensional space. To glue the ends of the cylinder together so that the arrows on the circles match, one would pass one end through the side of the cylinder. To construct the Klein bottle, glue the red arrows of the square together (left and right sides), resulting in a cylinder. 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. The idea is to 'glue' together the corresponding red and blue edges with the arrows matching, as in the diagrams below. The following square is a fundamental polygon of the Klein bottle. The Klein bottle was first described in 1882 by the mathematician Felix Klein. For comparison, a sphere is an orientable surface with no boundary. While a Möbius strip is a surface with boundary, a Klein bottle has no boundary. Other related non-orientable surfaces include the Möbius strip and the real projective plane. More formally, the Klein bottle is a two-dimensional manifold on which one cannot define a normal vector at each point that varies continuously over the whole manifold. ![]() In mathematics, the Klein bottle ( / ˈ k l aɪ n/) is an example of a non-orientable surface that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down. A two-dimensional representation of the Klein bottle immersed in three-dimensional space
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