In topology, a branch of mathematics, the **Klein bottle** ( /ˈklaɪn/ ) is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, 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. Other related non-orientable objects 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.

- Construction
- Properties
- Dissection
- Simple-closed curves
- Parametrization
- The figure 8 immersion
- 4-D non-intersecting
- 3D pinched torus / 4D Möbius tube
- Bottle shape
- Homotopy classes
- Generalizations
- Klein surface
- See also
- References
- Citations
- Sources
- External links

The Klein bottle was first described in 1882 by the German mathematician Felix Klein.

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 circle of self-intersection – this is an immersion of the Klein bottle in three dimensions.

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 date from 1995 and were made for the museum by Alan Bennett.^{ [1] }

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.

Like the Möbius strip, the Klein bottle is a two-dimensional manifold which is not orientable. Unlike the Möbius strip, the Klein bottle is a *closed* manifold, meaning it is a compact manifold without boundary. While the Möbius strip can be embedded in three-dimensional Euclidean space **R**^{3}, the Klein bottle cannot. It can be embedded in **R**^{4}, however.

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 together, as described in the following limerick by Leo Moser:^{ [2] }

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 0-cell *P*, two 1-cells *C*_{1}, *C*_{2} and one 2-cell *D*. Its Euler characteristic is therefore 1 − 2 + 1 = 0. The boundary homomorphism is given by ∂*D* = 2*C*_{1} and ∂*C*_{1} = ∂*C*_{1} = 0, yielding the homology groups of the Klein bottle *K* to be H_{0}(*K*, **Z**) = **Z**, H_{1}(*K*, **Z**) = **Z**×(**Z**/2**Z**) and H_{n}(*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 **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. It is also homeomorphic to a sphere plus two cross 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.^{ [3] }

Dissecting a Klein bottle into halves along its plane of symmetry results in two mirror image Möbius 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.

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 to **Z**×**Z**_{2}. 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 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).

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 "figure-8" torus with a half-twist:

for 0 ≤ *θ* < 2π, 0 ≤ *v* < 2π and *r* > 2.

In this immersion, the self-intersection 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 8-shaped cross section. With the above parametrization the cross section is a 2:1 Lissajous curve.

A non-intersecting 4-D 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 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 and *ε* sin*v* is a small *v* depended bump in *z-w* space to avoid self intersection. The *v* 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. When *ε=0* the self intersection is a circle in the z-w plane <0, 0, cos*θ*, sin*θ*>.

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.

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).

The parametrization of the 3-dimensional immersion of the bottle itself is much more complicated.

for 0 ≤ *u* < π and 0 ≤ *v* < 2π.

Regular 3D embeddings of the Klein bottle fall into three regular homotopy classes (four if one paints them).^{ [4] } The three are represented by:

- The "traditional" Klein bottle
- Left-handed figure-8 Klein bottle
- Right-handed figure-8 Klein bottle

The traditional Klein bottle embedding is achiral. The figure-8 embedding is chiral (the pinched torus embedding above is not regular as it has pinch points so it's not relevant in this section). The three embeddings above cannot be smoothly transformed into each other in three dimensions. If the traditional Klein bottle is cut lengthwise it deconstructs into two, oppositely chiral Möbius strips.

If a left handed figure-8 Klein bottle is cut it deconstructs into two left handed Möbius strips, and similarly for the right handed figure-8 Klein bottle.

Painting the traditional Klein bottle two colors induces chirality on it, creating four homotopy classes.

The generalization of the Klein bottle to higher genus is given in the article on the fundamental polygon.

In another order of ideas, constructing 3-manifolds, 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 non-orientable version of the **solid torus**, equivalent to

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 so-called dianalytic structure of the space.

In mathematics, the **polar coordinate system** is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point is called the *pole*, and the ray from the pole in the reference direction is the *polar axis*. The distance from the pole is called the *radial coordinate*, *radial distance* or simply *radius*, and the angle is called the *angular coordinate*, *polar angle*, or *azimuth*. The radial coordinate is often denoted by *r* or *ρ*, and the angular coordinate by *φ*, *θ*, or *t*. Angles in polar notation are generally expressed in either degrees or radians.

In mathematics, a **spherical coordinate system** is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the *radial distance* of that point from a fixed origin, its *polar angle* measured from a fixed zenith direction, and the *azimuthal angle* of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the three-dimensional version of the polar coordinate system.

In mathematics and physics, **Laplace's equation** is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as

In mathematics, a **Möbius strip**, **band**, or **loop**, also spelled **Mobius** or **Moebius**, is a surface with only one side and only one boundary curve. The Möbius strip is the simplest non-orientable surface. It can be realized as a ruled surface. Its discovery is attributed independently to the German mathematicians Johann Benedict Listing and August Ferdinand Möbius in 1858, though similar structures can be seen in Roman mosaics c. 200–250 AD. Möbius published his results in his articles "Theorie der elementaren Verwandtschaft" (1863) and "Ueber die Bestimmung des Inhaltes eines Polyëders" (1865).

In mathematics, an ** n-sphere** is a topological space that is homeomorphic to a

The **Roman surface** or **Steiner surface** is a self-intersecting mapping of the real projective plane into three-dimensional space, with an unusually high degree of symmetry. This mapping is not an immersion of the projective plane; however, the figure resulting from removing six singular points is one. Its name arises because it was discovered by Jakob Steiner when he was in Rome in 1844.

In geometry, a **torus** is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.

An **ellipsoid** is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.

In mathematics, a **unit vector** in a normed vector space is a vector of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in .

In mathematics and physical science, **spherical harmonics** are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.

In probability theory, the **Borel–Kolmogorov paradox** is a paradox relating to conditional probability with respect to an event of probability zero. It is named after Émile Borel and Andrey Kolmogorov.

In mathematics, a **parametric equation** defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric object such as a curve or surface, in which case the equations are collectively called a **parametric representation** or **parameterization** of the object.

The **superformula** is a generalization of the superellipse and was proposed by Johan Gielis around 2000. Gielis suggested that the formula can be used to describe many complex shapes and curves that are found in nature. Gielis has filed a patent application related to the synthesis of patterns generated by the superformula.

This is a **table of orthonormalized spherical harmonics** that employ the Condon-Shortley phase up to degree * = 10. Some of these formulas give the "Cartesian" version. This assumes **x*, *y*, *z*, and *r* are related to and through the usual spherical-to-Cartesian coordinate transformation:

In mathematics, **Viviani's curve**, also known as **Viviani's window**, is a figure eight shaped space curve named after the Italian mathematician Vincenzo Viviani. It is the intersection of a sphere with a cylinder that is tangent to the sphere and passes through two poles of the sphere. Before Viviani this curve was studied by Simon de La Loubère and Gilles de Roberval.

In mathematics, a **multiple integral** is a definite integral of a function of several real variables, for instance, *f*(*x*, *y*) or *f*(*x*, *y*, *z*). Integrals of a function of two variables over a region in are called **double integrals**, and integrals of a function of three variables over a region in are called **triple integrals**. For multiple integrals of a single-variable function, see the Cauchy formula for repeated integration.

In geometric topology, the **Clifford torus** is the simplest and most symmetric flat embedding of the cartesian product of two circles *S*^{1}_{a} and *S*^{1}_{b}. It is named after William Kingdon Clifford. It resides in **R**^{4}, as opposed to in **R**^{3}. To see why **R**^{4} is necessary, note that if *S*^{1}_{a} and *S*^{1}_{b} each exists in its own independent embedding space **R**^{2}_{a} and **R**^{2}_{b}, the resulting product space will be **R**^{4} rather than **R**^{3}. The historically popular view that the cartesian product of two circles is an **R**^{3} torus in contrast requires the highly asymmetric application of a rotation operator to the second circle, since that circle will only have one independent axis *z* available to it after the first circle consumes *x* and *y*.

In mathematics, **vector spherical harmonics** (**VSH**) are an extension of the scalar spherical harmonics for use with vector fields. The components of the VSH are complex-valued functions expressed in the spherical coordinate basis vectors.

In fluid dynamics, the **Oseen equations** describe the flow of a viscous and incompressible fluid at small Reynolds numbers, as formulated by Carl Wilhelm Oseen in 1910. Oseen flow is an improved description of these flows, as compared to Stokes flow, with the (partial) inclusion of convective acceleration.

- ↑ "Strange Surfaces: New Ideas". Science Museum London. Archived from the original on 2006-11-28.
- ↑ David Darling (11 August 2004).
*The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes*. John Wiley & Sons. p. 176. ISBN 978-0-471-27047-8. - ↑ Weeks, Jeffrey (2020).
*The Shape of Space, 3rd Edn*. CRC Press. ISBN 978-1138061217. - ↑ 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.

*This article incorporates material from Klein bottle on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*- Weisstein, Eric W. "Klein Bottle".
*MathWorld*. - A classical on the theory of
**Klein surfaces**is 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/S0002-9904-1969-12332-3 . MR 0251213. ‹See Tfd› PE euclid.bams/1183530665.

Wikimedia Commons has media related to Klein bottle . |

- Imaging Maths - The Klein Bottle
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- 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.

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