![]() ![]() The Klein bottle can be constructed (in a four dimensional space, because in three dimensional space it cannot be done without allowing the surface to intersect itself) by joining the edges of two Möbius strips, as described in the following limerick by Leo Moser: The projection π: E→ B is then given by π() =. The Klein bottle can be seen as a fiber bundle over the circle S 1, with fibre S 1, 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. Ĭontinuing this sequence, for example creating a 3-manifold which cannot be embedded in R 4 but can be in R 5, is possible in this case, connecting two ends of a spherinder to each other in the same manner as the two ends of a cylinder for a Klein bottle, creates a figure, referred to as a "spherinder Klein bottle", that cannot fully be embedded in R 4. While the Möbius strip can be embedded in three-dimensional Euclidean space R 3, the Klein bottle cannot. ![]() Unlike the Möbius strip, it is a closed manifold, meaning it is a compact manifold without boundary. Like the Möbius strip, the Klein bottle is a two-dimensional manifold which is not orientable. 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 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. Gently push a piece of the tube containing the intersection along the fourth dimension, out of the original three-dimensional space. By adding a fourth dimension to the three-dimensional space, the self-intersection can be eliminated. Nonetheless, there is a way to visualize the Klein bottle as being contained in four dimensions. The Klein bottle, proper, does not self-intersect. The bottles date from 1995 and were made for the museum by Alan Bennett. The Science Museum in London has a collection of hand-blown glass Klein bottles on display, exhibiting many variations on this topological theme. The common physical model of a Klein bottle is a similar construction. Immersed Klein bottles in the Science Museum in London A hand-blown 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. This immersion is useful for visualizing many properties of the Klein bottle. 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 a 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. Non-orientable mathematical surface A two-dimensional representation of the Klein bottle immersed in three-dimensional space ![]()
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