In mathematics, the Klein bottle () is an example of a surface with no distinct inside or outside. In other words, 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. More formally, it is an example of a non-orientable surface, a two-dimensional manifold on which one cannot define a consistent direction perpendicular to the surface (normal vector) that varies continuously over the whole shape. The Klein bottle is related to other non-orientable surfaces like the Möbius strip, which also has only one side but does have a boundary. In contrast, the Klein bottle is boundaryless, like a sphere or torus, though it cannot be embedded in ordinary three-dimensional space without intersecting itself. The Klein bottle was first described in 1882 by the mathematician Felix Klein.
Construction The following square is a fundamental polygon of the Klein bottle. The idea is to 'glue' together the corresponding red and blue edges with the arrows matching, 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, one would pass one end through the side of the cylinder. This creates a curve of self-intersection; this is thus an immersion of the Klein bottle in the three-dimensional space.
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 a collection of hand-blown glass Klein bottles on display, exhibiting many variations on this topological theme. The bottles were made for the museum by Alan Bennett in 1995. 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 is 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 [0,1] × [0,1] with sides identified by the relations (0, y) ~ (1, y) for 0 ≤ y ≤ 1 and (x, 0) ~ (1 − x, 1) for 0 ≤ x ≤ 1.
Properties Like the Möbius strip, the Klein bottle is a two-dimensional manifold which is not orientable. Unlike the Möbius strip, it is a closed manifold, meaning it is a compact manifold without boundary. While the Möbius strip can be embedded in three-dimensional Euclidean space R3, the Klein bottle cannot. It can be embedded in R4, however. Continuing this sequence, for example creating a 3-manifold which cannot be embedded in R4 but can be in R5, 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 R4. The Klein bottle can be seen as a fiber bundle over the circle S1, with fibre S1, 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]. 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 initial construction of the Klein bottle by identifying opposite edges of a square shows that the Klein bottle can be given a CW complex structure with one 0-cell P, two 1-cells C1, C2 and one 2-cell D. Its Euler characteristic is therefore 1 − 2 + 1 = 0. The boundary homomorphism is given by ∂D = 2C1 and ∂C1 = ∂C2 = 0, yielding the homology groups of the Klein bottle K to be H0(K, Z) = Z, H1(K, Z) = Z×(Z/2Z) and Hn(K, Z) = 0 for n > 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. The universal cover of both the torus and the Klein bottle is the plane R2. The fundamental group of the Klein bottle can be determined as the group of deck transformations of the universal cover and has the presentation ⟨a, b | ab = b−1a⟩. It follows that it is isomorphic to Z ⋊ Z {\displaystyle \mathbb {Z} \rtimes \mathbb {Z} } , the only nontrivial semidirect product of the additive group of integers Z {\displaystyle \mathbb {Z} } with itself.
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