Editing Klein bottle

{{Short description|Non-orientable mathematical surface}}
[[Image:Klein bottle.svg|thumb|upright|right|A two-dimensional representation of the Klein bottle [[Immersion (mathematics)|immersed]] in three-dimensional space]]

In [[mathematics]], the '''Klein bottle''' ({{IPAc-en|ˈ|k|l|aɪ|n}}) is an example of a [[Orientability|non-orientable]] [[Surface (topology)|surface]]; that is, informally, 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, the Klein bottle is a [[two-dimensional]] [[manifold]] on which one cannot define a [[normal vector]] at each point that varies [[continuous function|continuously]] over the whole manifold. Other related non-orientable surfaces include the [[Möbius strip]] and the [[real projective plane]]. While a Möbius strip is a surface with a [[Boundary (topology)|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]].{{sfn|Stillwell|1993|p=65|loc=1.2.3 The Klein Bottle}}

==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.<ref name=":0" />

:[[Image:Klein Bottle Folding 1.svg]]
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 (mathematics)|immersion]] of the Klein bottle in the [[three-dimensional space]].

<gallery |="" align="center">
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
</gallery>

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 [[orientability|non-orientable]], as reflected in the one-sidedness of the immersion.

[[File:Science Museum London 1110529 nevit.jpg|thumb|right|150px|Immersed Klein bottles in the [[Science Museum (London)|Science Museum in London]]]]
[[Image:Acme klein bottle.jpg|thumb|150px|right|A hand-blown Klein bottle]]
The common physical model of a Klein bottle is a similar construction. The [[Science Museum (London)|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.<ref>{{cite web|archive-url=https://web.archive.org/web/20061128155852/http://www.sciencemuseum.org.uk/on-line/surfaces/new.asp|archive-date=2006-11-28 |url=http://www.sciencemuseum.org.uk/on-line/surfaces/new.asp|title=Strange Surfaces: New Ideas |publisher=Science Museum London }}</ref>

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.{{sfn|Alling|Greenleaf|1969}}

[[File:Klein bottle time evolution in xyzt-space.gif|thumb|[[Time evolution]] of a Klein figure in ''xyzt''-space]]
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 {{nowrap|1=''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.{{sfn|Alling|Greenleaf|1969}}

More formally, the Klein bottle is the [[Quotient space (topology)|quotient space]] described as the [[Square (geometry)|square]] [0,1] × [0,1] with sides identified by the relations {{nowrap|(0, ''y'') ~ (1, ''y'')}} for {{nowrap|0 ≤ ''y'' ≤ 1}} and {{nowrap|(''x'', 0) ~ (1 − ''x'', 1)}} for {{nowrap|0 ≤ ''x'' ≤ 1}}.

==Properties==
Like the [[Möbius strip]], the Klein bottle is a two-dimensional [[manifold]] which is not [[orientability|orientable]]. Unlike the Möbius strip, it is a ''closed'' manifold, meaning it is a [[compact space|compact]] manifold without boundary. While the Möbius strip can be embedded in three-dimensional [[Euclidean space]] '''R'''<sup>3</sup>, the Klein bottle cannot. It can be embedded in '''R'''<sup>4</sup>, however.{{sfn|Alling|Greenleaf|1969}}

Continuing this sequence, for example creating a 3-manifold which cannot be embedded in '''R'''<sup>4</sup> but can be in '''R'''<sup>5</sup>, 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'''<sup>4</sup>.<ref>[[Marc ten Bosch]] - https://marctenbosch.com/news/2021/12/4d-toys-version-1-7-klein-bottles/</ref>

The Klein bottle can be seen as a [[fiber bundle]] over the [[circle]] ''S''<sup>1</sup>, with fibre ''S''<sup>1</sup>, 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 {{nowrap|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 (poetry)|limerick]] by [[Leo Moser]]:<ref name="Darling2004">{{cite book|author=David Darling|title=The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes|url=https://books.google.com/books?id=nnpChqstvg0C&q=get+a+weird+bottle+like+mine&pg=PA176|date=11 August 2004|publisher=John Wiley & Sons|isbn=978-0-471-27047-8|page=176}}</ref>

{{poemquote|text=A mathematician named [[Felix Klein|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''<sub>1</sub>, ''C''<sub>2</sub> and one 2-cell ''D''. Its [[Euler characteristic]] is therefore {{nowrap|1 − 2 + 1 {{=}} 0}}. The boundary homomorphism is given by {{nowrap|&part;''D'' {{=}} 2''C''<sub>1</sub>}} and {{nowrap|&part;''C''<sub>1</sub> {{=}} &part;''C''<sub>2</sub> {{=}} 0}}, yielding the [[cellular homology|homology groups]] of the Klein bottle ''K'' to be {{nowrap|H<sub>0</sub>(''K'', '''Z''') {{=}} '''Z'''}}, {{nowrap|H<sub>1</sub>(''K'', '''Z''') {{=}} '''Z'''×('''Z'''/2'''Z''')}} and {{nowrap|H<sub>''n''</sub>(''K'', '''Z''') {{=}} 0}} for {{nowrap|''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'''<sup>2</sup>.

The [[fundamental group]] of the Klein bottle can be determined as the [[Deck transformation#Deck transformation group, regular covers|group of deck transformations]] of the universal cover and has the [[presentation of a group|presentation]] {{nowrap|{{angbr|1=''a'', ''b'' {{!}} ''ab'' = ''b''<sup>&minus;1</sup>''a''}}}}. It follows that it is isomorphic to <math>\mathbb{Z} \rtimes \mathbb{Z}</math>, the only nontrivial semidirect product of the additive group of integers <math>\mathbb{Z}</math> with itself.

[[File:Klein bottle colouring.svg|thumb|upright|A 6-colored Klein bottle, the only exception to the Heawood conjecture]]
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 plane]]s.<ref>{{Cite book |last=Shick |first=Paul |title=Topology: Point-Set and Geometric |publisher=Wiley-Interscience |year=2007 |isbn=9780470096055 |pages=191–192}}</ref> It is also homeomorphic to a sphere plus two [[cross-cap]]s.

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.<ref name=":0">{{Cite book | publisher = CRC Press | isbn = 978-1138061217 | last = Weeks | first = Jeffrey | title = The Shape of Space, 3rd Edn. | year = 2020 | url = https://www.crcpress.com/The-Shape-of-Space/Weeks/p/book/9781138061217 }}</ref>

==Dissection==
[[File:KleinBottle-cut.svg|thumb|right|150px|Dissecting the Klein bottle results in two Möbius strips.]]
Dissecting a Klein bottle into halves along its [[plane of symmetry]] results in two mirror image [[Möbius strip]]s, 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.<ref>[https://www.youtube.com/watch?v=I3ZlhxaT_Ko Cutting a Klein Bottle in Half – Numberphile on YouTube]</ref>

==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'''×'''Z'''<sub>2</sub>. 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).{{sfn|Alling|Greenleaf|1969}}

==Parametrization==
[[Image:KleinBottle-Figure8-01.svg|thumb|left|The "figure 8" immersion of the Klein bottle.]]
[[Image:Kleinbagel cross section.png|thumb|left|Klein bagel cross section, showing a figure eight curve (the [[lemniscate of Gerono]]).]]

=== The figure 8 immersion ===
To make the "figure 8" or "bagel" [[Immersion (mathematics)|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:{{sfn|Alling|Greenleaf|1969}}

:<math>\begin{align}
x & = \left(r + \cos\frac{\theta}{2}\sin v - \sin\frac{\theta}{2}\sin 2v\right) \cos \theta\\
y & = \left(r + \cos\frac{\theta}{2}\sin v - \sin\frac{\theta}{2}\sin 2v\right) \sin \theta\\
z & = \sin\frac{\theta}{2}\sin v + \cos\frac{\theta}{2}\sin 2v
\end{align}</math>
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]].

=== 4-D non-intersecting ===
A non-intersecting 4-D parametrization can be modeled after that of the [[Flat torus#Flat torus|flat torus]]:
:<math>\begin{align}
x & = R\left(\cos\frac{\theta}{2}\cos v - \sin\frac{\theta}{2}\sin 2v\right) \\
y & = R\left(\sin\frac{\theta}{2}\cos v + \cos\frac{\theta}{2}\sin 2v\right) \\
z & = P\cos\theta\left(1 + \varepsilon\sin v\right) \\
w & = P\sin\theta\left(1 + {\varepsilon}\sin v\right)
\end{align}</math>

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'' dependent 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''>.{{sfn|Alling|Greenleaf|1969}}

=== 3D pinched torus / 4D Möbius tube ===
[[Image:Pinched Torus Klein bottle.jpg|thumb|left|The pinched torus immersion of the Klein bottle.]]
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 two [[pinch point (mathematics)|pinch point]]s, 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.{{sfn|Alling|Greenleaf|1969}}

:<math>\begin{align}
  x(\theta, \varphi) &= (R + r \cos \theta) \cos{\varphi} \\
  y(\theta, \varphi) &= (R + r \cos \theta) \sin{\varphi} \\
  z(\theta, \varphi) &= r \sin \theta \cos\left(\frac{\varphi}{2}\right) \\
  w(\theta, \varphi) &= r \sin \theta \sin\left(\frac{\varphi}{2}\right)
\end{align}</math>

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 following parametrization of the usual 3-dimensional immersion of the bottle itself is much more complicated. 
[[File:Klein bottle translucent.png|thumb|right|Klein Bottle with slight transparency]]

:<math>\begin{align}
  x(u, v) = -&\frac{2}{15}\cos u \left(3\cos{v} - 30\sin{u} + 90\cos^4{u}\sin{u}\right. - \\
             &\left.60\cos^6{u}\sin{u} + 5\cos{u}\cos{v}\sin{u}\right) \\[3pt]

  y(u, v) = -&\frac{1}{15}\sin u \left(3\cos{v} - 3\cos^2{u}\cos{v} - 48\cos^4{u}\cos{v} + 48\cos^6{u}\cos{v}\right. -\\
             &60\sin{u} + 5\cos{u}\cos{v}\sin{u} - 5\cos^3{u}\cos{v}\sin{u} -\\
             &\left.80\cos^5{u}\cos{v}\sin{u} + 80\cos^7{u}\cos{v}\sin{u}\right) \\[3pt]

  z(u, v) =  &\frac{2}{15} \left(3 + 5\cos{u}\sin{u}\right) \sin{v}
\end{align}</math>
for 0 ≤ ''u'' < π and 0 ≤ ''v'' < 2π.{{sfn|Alling|Greenleaf|1969}}

== Homotopy classes ==
Regular 3D immersions of the Klein bottle fall into three [[regular homotopy]] classes.<ref>{{cite journal|last1=Séquin|first1=Carlo H|title=On the number of Klein bottle types|journal=Journal of Mathematics and the Arts|date=1 June 2013|volume=7|issue=2|pages=51–63|doi=10.1080/17513472.2013.795883|citeseerx=10.1.1.637.4811|s2cid=16444067}}</ref> 
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 [[chirality|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.{{sfn|Alling|Greenleaf|1969}}

==Generalizations==
The generalization of the Klein bottle to higher [[genus (mathematics)|genus]] is given in the article on the [[fundamental polygon]].<ref>{{Cite news |last=Day |first=Adam |date=17 February 2014 |title=Quantum gravity on a Klein bottle |url=https://cqgplus.com/2014/02/17/quantum-gravity-on-a-klein-bottle/ |website=CQG+}}</ref>

In another order of ideas, constructing [[3-manifold]]s, 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 <math>D^2 \times S^1.</math>

==Klein surface==
A '''Klein surface''' is, as for [[Riemann surface]]s, a surface with an atlas allowing the [[transition map]]s to be composed using [[complex conjugation]]. One can obtain the so-called [[dianalytic structure]] of the space, and it has only one side.<ref>{{Cite book |last=Bitetto |first=Dr Marco |url=https://books.google.com/books?id=K4DQDwAAQBAJ&dq=A+Klein+surface+is%2C+as+for+Riemann+surfaces%2C+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&pg=PA222 |title=Hyperspatial Dynamics |date=2020-02-14 |publisher=Dr. Marco A. V. Bitetto |language=en}}</ref>

==See also==
* [[Algebraic topology]]
* [[Alice universe]]
* [[Systoles of surfaces#Klein bottle|Bavard's Klein bottle systolic inequality]]
* [[Boy's surface]]

== References ==
=== Citations ===
{{Reflist}}

=== Sources ===
{{refbegin}}
* {{PlanetMath attribution|id=4249|title=Klein bottle}}
* {{MathWorld|urlname=KleinBottle|title=Klein Bottle}}
* {{cite journal |title = Klein surfaces and real algebraic function fields |first1 = Norman |last1 = Alling |first2 = Newcomb |last2 = Greenleaf |journal = [[Bulletin of the American Mathematical Society]] |id = {{Project Euclid|euclid.bams/1183530665}} |mr=0251213 |volume = 75 |number= 4 |year=1969 |pages=627–888 |doi=10.1090/S0002-9904-1969-12332-3  |doi-access = free }} (A classical on the theory of Klein surfaces) 
* {{cite book | title = Classical Topology and Combinatorial Group Theory | edition = 2nd | author-link = John Stillwell | author-last = Stillwell | author-first = John | publisher = [[Springer-Verlag]] | isbn = 0-387-97970-0 | year = 1993}}
{{refend}}

==External links==
{{Commons category|Klein bottle}}
* [https://plus.maths.org/content/os/issue26/features/mathart/index Imaging Maths - The Klein Bottle]
* [http://www.kleinbottle.com/meter_tall_klein_bottle.html The biggest Klein bottle in all the world]
* [https://www.youtube.com/watch?v=E8rifKlq5hc Klein Bottle animation: produced for a topology seminar at the Leibniz University Hannover.]
* [https://www.youtube.com/watch?v=sRTKSzAOBr4&fmt=22 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.]
* [https://archive.today/20130713133627/https://github.com/danfuzz/xscreensaver/blob/master/hacks/glx/klein.man Klein Bottle], [[XScreenSaver]] "hack". A screensaver for [[X Window System|X 11]] and [[OS X]] featuring an animated Klein Bottle.

{{Compact topological surfaces}}
{{Manifolds}}

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