Inmathematics, theKlein bottle (/ˈklaɪn/) is an example of asurface 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 anon-orientablesurface, atwo-dimensionalmanifold on which one cannot define a consistent direction perpendicular to the surface (normal vector) that variescontinuously over the whole shape.
The Klein bottle is related to other non-orientable surfaces like theMöbius strip, which also has only one side but does have aboundary. 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 mathematicianFelix Klein.[1]
The following square is afundamental 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.[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 self-intersection; this is thus animmersion of the Klein bottle in thethree-dimensional space.
This immersion is useful for visualizing many properties of the Klein bottle. For example, the Klein bottle has noboundary, where the surface stops abruptly, and it isnon-orientable, as reflected in the one-sidedness of the immersion.
The common physical model of a Klein bottle is a similar construction. TheScience 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.[3]
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.[4]
Suppose for clarification that we adopt time as that fourth dimension. Consider how the figure could be constructed inxyzt-space. The accompanying illustration ("Time evolution...") shows one useful evolution of the figure. Att = 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 theCheshire 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.[4]
More formally, the Klein bottle is thequotient space described as thesquare [0,1] × [0,1] with sides identified by the relations(0,y) ~ (1,y) for0 ≤y ≤ 1 and(x, 0) ~ (1 −x, 1) for0 ≤x ≤ 1.
Like theMöbius strip, the Klein bottle is a two-dimensionalmanifold which is notorientable. Unlike the Möbius strip, it is aclosed manifold, meaning it is acompact manifold without boundary. While the Möbius strip can be embedded in three-dimensionalEuclidean spaceR3, the Klein bottle cannot. It can be embedded inR4, however.[4]
Continuing this sequence, for example creating a 3-manifold which cannot be embedded inR4 but can be inR5, is possible; in this case, connecting two ends of aspherinder 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 inR4.[5]
The Klein bottle can be seen as afiber bundle over thecircleS1, with fibreS1, as follows: one takes the square (modulo the edge identifying equivalence relation) from above to beE, the total space, while the base spaceB is given by the unit interval iny, modulo1~0. The projection π:E→B is then given byπ([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 followinglimerick byLeo Moser:[6]
A mathematician namedKlein
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 aCW complex structure with one 0-cellP, two 1-cellsC1,C2 and one 2-cellD. ItsEuler characteristic is therefore1 − 2 + 1 = 0. The boundary homomorphism is given by∂D = 2C1 and∂C1 = ∂C2 = 0, yielding thehomology groups of the Klein bottleK to beH0(K,Z) =Z,H1(K,Z) =Z×(Z/2Z) andHn(K,Z) = 0 forn > 1.
There is a 2-1covering map from thetorus to the Klein bottle, because two copies of thefundamental region of the Klein bottle, one being placed next to the mirror image of the other, yield a fundamental region of the torus. Theuniversal cover of both the torus and the Klein bottle is the planeR2.
Thefundamental group of the Klein bottle can be determined as thegroup of deck transformations of the universal cover and has thepresentation⟨a,b |ab =b−1a⟩. It follows that it is isomorphic to, the only nontrivial semidirect product of the additive group of integers with itself.
Six colors suffice to color any map on the surface of a Klein bottle; this is the only exception to theHeawood conjecture, a generalization of thefour color theorem, which would require seven.
A Klein bottle is homeomorphic to theconnected sum of twoprojective planes.[7] It is also homeomorphic to a sphere plus twocross-caps.
When embedded in Euclidean space, the Klein bottle is one-sided. However, there are other topological 3-spaces, and in some of the non-orientable examples a Klein bottle can be embedded such that it is two-sided, though due to the nature of the space it remains non-orientable.[2]
Dissecting a Klein bottle into halves along itsplane of symmetry results in two mirror imageMöbius strips, i.e. one with a left-handed half-twist and the other with a right-handed half-twist (one of these is pictured on the right). Remember that the intersection pictured is not really there.[8]
One description of the types of simple-closed 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 toZ×Z2. Up to reversal of orientation, the only homology classes which contain simple-closed 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 cross-caps 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]
To make the "figure 8" or "bagel"immersion of the Klein bottle, one can start with aMöbius 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 "figure-8" torus with a half-twist:[4]
for 0 ≤θ < 2π, 0 ≤v < 2π andr > 2.
In this immersion, the self-intersection circle (where sin(v) is zero) is a geometriccircle in thexy plane. The positive constantr is the radius of this circle. The parameterθ gives the angle in thexy plane as well as the rotation of the figure 8, andv specifies the position around the 8-shaped cross section. With the above parametrization the cross section is a 2:1Lissajous curve.
A non-intersecting 4-D parametrization can be modeled after that of theflat torus:
whereR andP are constants that determine aspect ratio,θ andv are similar to as defined above.v determines the position around the figure-8 as well as the position in the x-y plane.θ determines the rotational angle of the figure-8 as well and the position around the z-w plane.ε is any small constant andε sinv is a smallv dependent bump inz-w space to avoid self intersection. Thev bump causes the self intersecting 2-D/planar figure-8 to spread out into a 3-D stylized "potato chip" or saddle shape in the x-y-w and x-y-z space viewed edge on. Whenε=0 the self intersection is a circle in the z-w plane <0, 0, cosθ, sinθ>.[4]
The pinched torus is perhaps the simplest parametrization of the Klein bottle in both three and four dimensions. It can be viewed as a variant of a torus that, in three dimensions, flattens and passes through itself on one side. Unfortunately, in three dimensions this parametrization has twopinch points, which makes it undesirable for some applications. In four dimensions thez amplitude rotates into thew 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 closedspherinder (solidspheritorus).
The following parametrization of the usual 3-dimensional immersion of the bottle itself is much more complicated.
for 0 ≤u < π and 0 ≤v < 2π.[4]
Regular 3D immersions of the Klein bottle fall into threeregular homotopy classes.[9] The three are represented by:
The traditional Klein bottle immersion isachiral. The figure-8 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 figure-8 Klein bottle can be cut into two Möbius strips of thesame chirality, and cannot be regularly deformed into its mirror image.[4]
The generalization of the Klein bottle to highergenus is given in the article on thefundamental polygon.[10]
In another order of ideas, constructing3-manifolds, it is known that asolid Klein bottle ishomeomorphic to theCartesian product of aMöbius strip and a closed interval. Thesolid Klein bottle is the non-orientable version of thesolid torus, equivalent to
