Klein bottle
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In
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, a branch of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Klein bottle () is an example of a
non-orientable In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space i ...
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
; it is a
two-dimensional In mathematics, a plane is a Euclidean ( flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as ...
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
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 In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has b ...
. While a Möbius strip is a surface with
boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment * ''Boundaries'' (2016 film), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film *Boundary (cricket), the edge of the pla ...
, a Klein bottle has no boundary. For comparison, a
sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
is an orientable surface with no boundary. The concept of a Klein bottle was first described in 1882 by the German mathematician
Felix Klein Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and grou ...
.


Construction

The following square is a
fundamental polygon In mathematics, a fundamental polygon can be defined for every compact Riemann surface of genus greater than 0. It encodes not only the topology of the surface through its fundamental group but also determines the Riemann surface up to conformal eq ...
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 Immersion may refer to: The arts * "Immersion", a 2012 story by Aliette de Bodard * ''Immersion'', a French comic book series by Léo Quievreux#Immersion, Léo Quievreux * Immersion (album), ''Immersion'' (album), the third album by Australian gro ...
of the Klein bottle in three dimensions. Image:Klein Bottle Folding 1.svg Image:Klein Bottle Folding 2.svg Image:Klein Bottle Folding 3.svg Image:Klein Bottle Folding 4.svg Image:Klein Bottle Folding 5.svg Image:Klein Bottle Folding 6.svg 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 In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space i ...
, 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 The Science Museum is a major museum on Exhibition Road in South Kensington, London. It was founded in 1857 and is one of the city's major tourist attractions, attracting 3.3 million visitors annually in 2019. Like other publicly funded ...
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. 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 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 The Cheshire Cat ( or ) is a fictional cat popularised by Lewis Carroll in ''Alice's Adventures in Wonderland'' and known for its distinctive mischievous grin. While now most often used in ''Alice''-related contexts, the association of a "Ch ...
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 In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
,1× ,1with sides identified by the relations for and for .


Properties

Like the Möbius strip, the Klein bottle is a two-dimensional
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
which is not
orientable In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space i ...
. Unlike the Möbius strip, it is a ''closed'' manifold, meaning it is a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
manifold without boundary. While the Möbius strip can be embedded in three-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
R3, the Klein bottle cannot. It can be embedded in R4, however. Continuing this sequence, for example creating a surface which cannot be embedded in R4 but can be in R5, is possible; in this case, connecting two ends of a
spherinder In four-dimensional geometry, the spherinder, or spherical cylinder or spherical prism, is a geometric object, defined as the Cartesian product of a 3- ball (or solid 2-sphere) of radius ''r''1 and a line segment of length 2''r''2: :D = \ Lik ...
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 In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
over the
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
''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 . 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 Limerick ( ; ga, Luimneach ) is a western city in Ireland situated within County Limerick. It is in the province of Munster and is located in the Mid-West which comprises part of the Southern Region. With a population of 94,192 at the 2016 ...
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 A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cl ...
structure with one 0-cell ''P'', two 1-cells ''C''1, ''C''2 and one 2-cell ''D''. Its Euler characteristic is therefore . The boundary homomorphism is given by and , yielding the
homology groups In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
of the Klein bottle ''K'' to be , and for . There is a 2-1
covering map A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete spa ...
from the
torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not tou ...
to the Klein bottle, because two copies of the
fundamental region Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each o ...
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 A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete spa ...
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 presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Presenta ...
. Six colors suffice to color any map on the surface of a Klein bottle; this is the only exception to the
Heawood conjecture In graph theory, the Heawood conjecture or Ringel–Youngs theorem gives a lower bound for the number of colors that are necessary for graph coloring on a surface of a given genus. For surfaces of genus 0, 1, 2, 3, 4, 5, 6, 7, ..., the required ...
, a generalization of the four color theorem, which would require seven. A Klein bottle is homeomorphic to the
connected sum In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classifi ...
of two
projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that do ...
s. 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.


Dissection

Dissecting a Klein bottle into halves along its
plane of symmetry In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2D the ...
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.


Simple-closed curves

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×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).


Parametrization


The figure 8 immersion

To make the "figure 8" or "bagel"
immersion Immersion may refer to: The arts * "Immersion", a 2012 story by Aliette de Bodard * ''Immersion'', a French comic book series by Léo Quievreux#Immersion, Léo Quievreux * Immersion (album), ''Immersion'' (album), the third album by Australian gro ...
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: :\begin x & = \left(r + \cos\frac\sin v - \sin\frac\sin 2v\right) \cos \theta\\ y & = \left(r + \cos\frac\sin v - \sin\frac\sin 2v\right) \sin \theta\\ z & = \sin\frac\sin v + \cos\frac\sin 2v \end 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 A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
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 Lissajous curve , also known as Lissajous figure or Bowditch curve , is the graph of a system of parametric equations : x=A\sin(at+\delta),\quad y=B\sin(bt), which describe the superposition of two perpendicular oscillations in x and y dire ...
.


4-D non-intersecting

A non-intersecting 4-D parametrization can be modeled after that of the
flat torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not tou ...
: :\begin x & = R\left(\cos\frac\cos v - \sin\frac\sin 2v\right) \\ y & = R\left(\sin\frac\cos v + \cos\frac\sin 2v\right) \\ z & = P\cos\theta\left(1 + \epsilon\sin v\right) \\ w & = P\sin\theta\left(1 + \sin v\right) \end 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''>.


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. :\begin x(\theta, \varphi) &= (R + r \cos \theta) \cos \\ y(\theta, \varphi) &= (R + r \cos \theta) \sin \\ z(\theta, \varphi) &= r \sin \theta \cos\left(\frac\right) \\ w(\theta, \varphi) &= r \sin \theta \sin\left(\frac\right) \end 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 In four-dimensional geometry, the spherinder, or spherical cylinder or spherical prism, is a geometric object, defined as the Cartesian product of a 3- ball (or solid 2-sphere) of radius ''r''1 and a line segment of length 2''r''2: :D = \ Lik ...
(solid spheritorus).


Bottle shape

The parametrization of the 3-dimensional immersion of the bottle itself is much more complicated. :\begin x(u, v) = -&\frac\cos u \left(3\cos - 30\sin + 90\cos^4\sin\right. - \\ &\left.60\cos^6\sin + 5\cos\cos\sin\right) \\ pt y(u, v) = -&\frac\sin u \left(3\cos - 3\cos^2\cos - 48\cos^4\cos + 48\cos^6\cos\right. -\\ &60\sin + 5\cos\cos\sin - 5\cos^3\cos\sin -\\ &\left.80\cos^5\cos\sin + 80\cos^7\cos\sin\right) \\ pt z(u, v) = &\frac \left(3 + 5\cos\sin\right) \sin \end for 0 ≤ ''u'' < π and 0 ≤ ''v'' < 2π.


Homotopy classes

Regular 3D immersions of the Klein bottle fall into three regular homotopy classes. The three are represented by: * the "traditional" Klein bottle; * the left-handed figure-8 Klein bottle; * the right-handed figure-8 Klein bottle. The traditional Klein bottle immersion is achiral. 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 the ''same'' chirality, and cannot be regularly deformed into its mirror image. Painting the traditional Klein bottle in two colors can induce chirality on it, splitting its homotopy class in two.


Generalizations

The generalization of the Klein bottle to higher
genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...
is given in the article on the
fundamental polygon In mathematics, a fundamental polygon can be defined for every compact Riemann surface of genus greater than 0. It encodes not only the topology of the surface through its fundamental group but also determines the Riemann surface up to conformal eq ...
. 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 D^2 \times S^1.


Klein surface

A Klein surface is, as for
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
s, a surface with an atlas allowing the
transition map In mathematics, particularly topology, one describes a manifold using an atlas. An atlas consists of individual ''charts'' that, roughly speaking, describe individual regions of the manifold. If the manifold is the surface of the Earth, then an a ...
s to be composed using complex conjugation. One can obtain the so-called dianalytic structure of the space.


See also

*
Algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
* Alice universe * Bavard's Klein bottle systolic inequality * Boy's surface


References


Citations


Sources

* * * A classical on the theory of Klein surfaces is *


External links


Imaging Maths - The Klein Bottle



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 XScreenSaver is a free and open-source collection of 240+ screensavers for Unix, macOS, iOS and Android operating systems. It was created by Jamie Zawinski in 1992 and is still maintained by him, with new releases coming out several times ...
"hack". A screensaver for X 11 and OS X featuring an animated Klein Bottle. {{Manifolds Geometric topology Manifolds Surfaces Topological spaces