In mathematics, the Klein bottle (/ˈklaɪn/) is an example of a nonorientable surface; that is, informally, a onesided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down. More formally, the Klein bottle is a twodimensional manifold on which one cannot define a normal vector at each point that varies continuously over the whole manifold. Other related nonorientable surfaces include the Möbius strip and the real projective plane. While 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 mathematician Felix Klein.^{[1]}
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Transcription
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 selfintersecting Klein bottle.^{[2]}
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 selfintersection; this is thus an immersion of the Klein bottle in the threedimensional 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 nonorientable, as reflected in the onesidedness of the immersion.
The common physical model of a Klein bottle is a similar construction. The Science Museum in London has a collection of handblown glass Klein bottles on display, exhibiting many variations on this topological theme. The bottles date from 1995 and were made for the museum by Alan Bennett.^{[3]}
The Klein bottle, proper, does not selfintersect. Nonetheless, there is a way to visualize the Klein bottle as being contained in four dimensions. By adding a fourth dimension to the threedimensional space, the selfintersection can be eliminated. Gently push a piece of the tube containing the intersection along the fourth dimension, out of the original threedimensional space. A useful analogy is to consider a selfintersecting curve on the plane; selfintersections can be eliminated by lifting one strand off the plane.^{[4]}
Suppose for clarification that we adopt time as that fourth dimension. Consider how the figure could be constructed in xyztspace. 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 everexpanding 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 4figure as defined cannot exist in 3space but is easily understood in 4space.^{[4]}
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 twodimensional 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 threedimensional Euclidean space R^{3}, the Klein bottle cannot. It can be embedded in R^{4}, however.^{[4]}
Continuing this sequence, for example creating a surface 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}.^{[5]}
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. 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 (mirrored) Möbius strips, as described in the following limerick by Leo Moser:^{[6]}
A mathematician named Klein
Thought the Möbius band was divine.
Said he: "If you glue
The edges of two,
You'll get a weird bottle like mine."
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 0cell P, two 1cells C_{1}, C_{2} and one 2cell D. Its Euler characteristic is therefore 1 − 2 + 1 = 0. The boundary homomorphism is given by ∂D = 2C_{1} and ∂C_{1} = ∂C_{2} = 0, yielding the homology groups of the Klein bottle K to be H_{0}(K, Z) = Z, H_{1}(K, Z) = Z×(Z/2Z) and H_{n}(K, Z) = 0 for n > 1.
There is a 21 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 R^{2}.
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^{−1}a⟩.
Six colors suffice to color any map on the surface of a Klein bottle; this is the only exception to the Heawood conjecture, a generalization of the four color theorem, which would require seven.
A Klein bottle is homeomorphic to the connected sum of two projective planes.^{[7]} It is also homeomorphic to a sphere plus two crosscaps.
When embedded in Euclidean space, the Klein bottle is onesided. However, there are other topological 3spaces, and in some of the nonorientable examples a Klein bottle can be embedded such that it is twosided, though due to the nature of the space it remains nonorientable.^{[2]}
Dissection
Dissecting a Klein bottle into halves along its plane of symmetry results in two mirror image Möbius strips, i.e. one with a lefthanded halftwist and the other with a righthanded halftwist (one of these is pictured on the right). Remember that the intersection pictured is not really there.^{[8]}
Simpleclosed curves
One description of the types of simpleclosed curves that may appear on the surface of the Klein bottle is given by the use of the first homology group of the Klein bottle calculated with integer coefficients. This group is isomorphic to Z×Z_{2}. Up to reversal of orientation, the only homology classes which contain simpleclosed curves are as follows: (0,0), (1,0), (1,1), (2,0), (0,1). Up to reversal of the orientation of a simple closed curve, if it lies within one of the two crosscaps that make up the Klein bottle, then it is in homology class (1,0) or (1,1); if it cuts the Klein bottle into two Möbius strips, then it is in homology class (2,0); if it cuts the Klein bottle into an annulus, then it is in homology class (0,1); and if bounds a disk, then it is in homology class (0,0).^{[4]}
Parametrization
The figure 8 immersion
To make the "figure 8" or "bagel" immersion of the Klein bottle, one can start with a Möbius strip and curl it to bring the edge to the midline; since there is only one edge, it will meet itself there, passing through the midline. It has a particularly simple parametrization as a "figure8" torus with a halftwist:^{[4]}
for 0 ≤ θ < 2π, 0 ≤ v < 2π and r > 2.
In this immersion, the selfintersection circle (where sin(v) is zero) is a geometric circle in the xy plane. The positive constant r is the radius of this circle. The parameter θ gives the angle in the xy plane as well as the rotation of the figure 8, and v specifies the position around the 8shaped cross section. With the above parametrization the cross section is a 2:1 Lissajous curve.
4D nonintersecting
A nonintersecting 4D parametrization can be modeled after that of the flat torus:
where R and P are constants that determine aspect ratio, θ and v are similar to as defined above. v determines the position around the figure8 as well as the position in the xy plane. θ determines the rotational angle of the figure8 as well and the position around the zw plane. ε is any small constant and ε sinv is a small v depended bump in zw space to avoid self intersection. The v bump causes the self intersecting 2D/planar figure8 to spread out into a 3D stylized "potato chip" or saddle shape in the xyw and xyz space viewed edge on. When ε=0 the self intersection is a circle in the zw plane <0, 0, cosθ, sinθ>.^{[4]}
3D pinched torus / 4D Möbius tube
The pinched torus is perhaps the simplest parametrization of the klein bottle in both three and four dimensions. It's a torus that, in three dimensions, flattens and passes through itself on one side. Unfortunately, in three dimensions this parametrization has two pinch points, which makes it undesirable for some applications. In four dimensions the z amplitude rotates into the w amplitude and there are no self intersections or pinch points.^{[4]}
One can view this as a tube or cylinder that wraps around, as in a torus, but its circular cross section flips over in four dimensions, presenting its "backside" as it reconnects, just as a Möbius strip cross section rotates before it reconnects. The 3D orthogonal projection of this is the pinched torus shown above. Just as a Möbius strip is a subset of a solid torus, the Möbius tube is a subset of a toroidally closed spherinder (solid spheritorus).
Bottle shape
The parametrization of the 3dimensional immersion of the bottle itself is much more complicated.
for 0 ≤ u < π and 0 ≤ v < 2π.^{[4]}
Homotopy classes
Regular 3D immersions of the Klein bottle fall into three regular homotopy classes.^{[9]} The three are represented by:
 the "traditional" Klein bottle;
 the lefthanded figure8 Klein bottle;
 the righthanded figure8 Klein bottle.
The traditional Klein bottle immersion is achiral. The figure8 immersion is chiral. (The pinched torus immersion above is not regular, as it has pinch points, so it is not relevant to this section.)
If the traditional Klein bottle is cut in its plane of symmetry it breaks into two Möbius strips of opposite chirality. A figure8 Klein bottle can be cut into two Möbius strips of the same chirality, and cannot be regularly deformed into its mirror image.^{[4]}
Painting the traditional Klein bottle in two colors can induce chirality on it, splitting its homotopy class in two.^{[citation needed]}
Generalizations
The generalization of the Klein bottle to higher genus is given in the article on the fundamental polygon.^{[10]}
In another order of ideas, constructing 3manifolds, it is known that a solid Klein bottle is homeomorphic to the Cartesian product of a Möbius strip and a closed interval. The solid Klein bottle is the nonorientable version of the solid torus, equivalent to
Klein surface
A Klein surface is, as for Riemann surfaces, a surface with an atlas allowing the transition maps to be composed using complex conjugation. One can obtain the socalled dianalytic structure of the space and has only one side .^{[11]}
See also
References
Citations
 ^ Stillwell 1993, p. 65, 1.2.3 The Klein Bottle.
 ^ ^{a} ^{b} Weeks, Jeffrey (2020). The Shape of Space, 3rd Edn. CRC Press. ISBN 9781138061217.
 ^ "Strange Surfaces: New Ideas". Science Museum London. Archived from the original on 20061128.
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} ^{h} ^{i} Alling & Greenleaf 1969.
 ^ Marc ten Bosch  https://marctenbosch.com/news/2021/12/4dtoysversion17kleinbottles/
 ^ David Darling (11 August 2004). The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes. John Wiley & Sons. p. 176. ISBN 9780471270478.
 ^ Shick, Paul (2007). Topology: PointSet and Geometric. WileyInterscience. pp. 191–192. ISBN 9780470096055.
 ^ Cutting a Klein Bottle in Half – Numberphile on YouTube
 ^ Séquin, Carlo H (1 June 2013). "On the number of Klein bottle types". Journal of Mathematics and the Arts. 7 (2): 51–63. CiteSeerX 10.1.1.637.4811. doi:10.1080/17513472.2013.795883. S2CID 16444067.
 ^ Day, Adam (17 February 2014). "Quantum gravity on a Klein bottle". CQG+.
 ^ Bitetto, Dr Marco (20200214). Hyperspatial Dynamics. Dr. Marco A. V. Bitetto.
Sources
 This article incorporates material from Klein bottle on PlanetMath, which is licensed under the Creative Commons Attribution/ShareAlike License.
 Weisstein, Eric W. "Klein Bottle". MathWorld.
 Alling, Norman; Greenleaf, Newcomb (1969). "Klein surfaces and real algebraic function fields". Bulletin of the American Mathematical Society. 75 (4): 627–888. doi:10.1090/S000299041969123323. MR 0251213. PE euclid.bams/1183530665. (A classical on the theory of Klein surfaces)
 Stillwell, John (1993). Classical Topology and Combinatorial Group Theory (2nd ed.). SpringerVerlag. ISBN 0387979700.
External links
 Imaging Maths  The Klein Bottle
 The biggest Klein bottle in all the world
 Klein Bottle animation: produced for a topology seminar at the Leibniz University Hannover.
 Klein Bottle animation from 2010 including a car ride through the bottle and the original description by Felix Klein: produced at the Free University Berlin.
 Klein Bottle, XScreenSaver "hack". A screensaver for X 11 and OS X featuring an animated Klein Bottle.