Klein bottle

The Klein bottle immersed in three-dimensional space. In Mathematics, an immersion is a Differentiable map between Differentiable manifolds whose derivative is everywhere Injective.

In mathematics, the Klein bottle is a certain non-orientable surface, i. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and A surface S in the Euclidean space R 3 is orientable if a two-dimensional figure (for example) cannot be moved around the surface and back In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. e. , a surface (a two-dimensional manifold) with no distinct "inner" and "outer" sides. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be Other related non-orientable objects include the Möbius strip and the real projective plane. This article is about the mathematical object See Mobius Band (music group for the music group Construction Consider a Sphere, and let the Great circles of the sphere be "lines" and let pairs of Antipodal points be "points" Whereas a Möbius strip is a two dimensional surface with boundary, a Klein bottle has no boundary. (For comparison, a sphere is an orientable surface with no boundary. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe )

The Klein bottle was first described in 1882 by the German mathematician Felix Klein. Year 1882 ( MDCCCLXXXII) was a Common year starting on Sunday (link will display the full calendar of the Gregorian calendar (or a Common Germany, officially the Federal Republic of Germany ( ˈbʊndəsʁepuˌbliːk ˈdɔʏtʃlant is a Country in Central Europe. Felix Christian Klein ( 25 April 1849 &ndash 22 June 1925) was a German Mathematician, known for his work in Group It was originally named the Kleinsche Fläche "Klein surface"; however, this was incorrectly interpreted as Kleinsche Flasche "Klein bottle", which ultimately led to the adoption of this term in the German language as well.

1 Construction
2 Properties
3 Dissection
4 Parametrization
5 Generalizations
6 Klein surface
7 References in popular culture
8 See also
9 References
10 External links

Construction

Start with a square, and then glue together corresponding colored edges, in the following diagram, so that the arrows match. 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:

This square is a fundamental polygon of the Klein bottle. In Topology and related areas of Mathematics, a quotient space (also called an identification space) is intuitively speaking the result of identifying Classification A square (regular Quadrilateral) is a special case of a Rectangle as it has four right angles and equal parallel sides In Mathematics, each closed Surface in the sense of Geometric topology can be constructed from an even-sided oriented Polygon, called a fundamental

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. The Klein bottle, proper, does not self-intersect. Nonetheless, there is a way to visualize the Klein bottle as being contained in four dimensions.

Glue the red arrows of the square together (left and right sides), resulting in a cylinder. To glue the ends together so that the arrows on the circles match, pass one end through the side of the cylinder. Note that this creates a circle of self-intersection. This is an immersion of the Klein bottle in three dimensions. In Mathematics, an immersion is a Differentiable map between Differentiable manifolds whose derivative is everywhere Injective.

By adding a fourth dimension to the three dimensional space, the self-intersection can be eliminated. Gradually push a piece of the tube containing the intersection 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.

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. A surface S in the Euclidean space R 3 is orientable if a two-dimensional figure (for example) cannot be moved around the surface and back

A hand-blown Klein Bottle (emulation)

The common physical model of a Klein bottle is a similar construction. The British Science Museum has on display a collection of hand-blown glass Klein bottles, exhibiting many variations on this topological theme. For science museums in general check out Science museum. The Science Museum on Exhibition Road, South Kensington, London is part The bottles date from 1995 and were made for the museum by Alan Bennett. Year 1995 ( MCMXCV) was a Common year starting on Sunday. Events of 1995 ^[1] Clifford Stoll, author of The Cuckoo's Egg, manufactures Klein bottles and sells them via the Internet at Acme Klein Bottle. Clifford Stoll (or Cliff Stoll) is a US Astronomer, Computer expert and Author. The Cuckoo's Egg Tracking a Spy Through the Maze of Computer Espionage is a 1990 book written by Clifford Stoll. The Internet is a global system of interconnected Computer networks

Properties

The Klein bottle can be seen as a fiber bundle as follows: one takes the square from above to be E, the total space, while the base space B is given by the unit interval in x, and the projection π is given by π(x, y) = x. In Mathematics, in particular in Topology, a fiber bundle (or fibre bundle) is a space which looks locally like a Product space. Since the two endpoints of the unit interval in x are identified, the base space B is actually the circle S¹, and so the Klein bottle is the twisted S¹-bundle (circle bundle) over the circle. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the In Mathematics, a circle bundle is a Fiber bundle where the fiber is the Circle \mathbf{S}^1 or more precisely a principal ''U''(1-bundle

Like the Möbius strip, the Klein bottle is a two-dimensional differentiable manifold which is not orientable. This article is about the mathematical object See Mobius Band (music group for the music group A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be A surface S in the Euclidean space R 3 is orientable if a two-dimensional figure (for example) cannot be moved around the surface and back 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 R³, the Klein bottle cannot. It can be embedded in R⁴, however.

The Klein bottle can be constructed (in a mathematical sense, because it cannot be done without allowing the surface to intersect itself) by joining the edges of two Möbius strips together, as described in the following anonymous limerick:

A mathematician named Klein

Thought the Möbius band was divine. Anonymity is derived from the Greek word ανωνυμία, meaning "without a Name " or "namelessness" A limerick is a five-line Poem with a strict form originally popularized in English by Edward Lear.

Said he: "If you glue

The edges of two,

You'll get a weird bottle like mine. "

It can also be constructed by folding a Möbius strip in half lengthwise and attaching the edge to itself.

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. The Heawood conjecture or Ringel–Youngs theorem in Graph theory gives an Upper bound for the number of colors which are Sufficient for The four color theorem (also known as the four color map theorem) states that given any plane separated into regions such as a political map of the states of a country

A Klein bottle is equivalent to a sphere plus two cross caps. In Mathematics, a cross-cap is a two-dimensional surface that is topologically equivalent to a Möbius strip.

Dissection

Dissecting the Klein bottle results in Möbius strips.

Dissecting a Klein bottle into halves along its plane of symmetry results in two mirror image Möbius strips, i. Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is Symmetry with respect This article is about the mathematical object See Mobius Band (music group for the music group 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 isn't really there. In fact, it is also possible to cut the Klein bottle into a single Möbius strip.

Parametrization

The "figure 8" immersion of the Klein bottle.

The "figure 8" immersion of the Klein bottle has a particularly simple parametrization:

$\begin{array}{rcl} x & = & \left(r + \cos\frac{u}{2}\sin v - \sin\frac{u}{2}\sin 2v\right) \cos u\\ y & = & \left(r + \cos\frac{u}{2}\sin v - \sin\frac{u}{2}\sin 2v\right) \sin u\\ z & = & \sin\frac{u}{2}\sin v + \cos\frac{u}{2}\sin 2v \end{array}$

In this immersion, the self-intersection circle is a geometric circle in the xy plane. In Mathematics, an immersion is a Differentiable map between Differentiable manifolds whose derivative is everywhere Injective. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the The positive constant r is the radius of this circle. The parameter u gives the angle in the xy plane, and v specifies the position around the 8-shaped cross section.

The parametrization of the 3-dimensional immersion of the bottle itself is much more complicated. Here is a simplified version:

$\begin{align} x & = \frac{ \sqrt{2} f(u) \cos u \cos v (3\cos^{2}u - 1) - 2\cos 2u}{80\pi^{3}g(u)}-\frac{3\cos u -3}{4}\\ y & = -\frac{f(u)\sin v}{60\pi^{3}}\\ z & = -\frac{\sqrt{2}f(u)\sin u \,\cos v}{15\pi^{3}g(u)}+\frac{\sin u \cos^{2} u + \sin u}{4}-\frac{\sin u\,\cos u}{2} \end{align}$

where

$f(u) = 20u^{3}-65\pi u^{2}+50\pi^{2}u-16\pi^{3}\,$

$g(u) = \sqrt{8\cos^{2}2u-\cos 2u (24\cos^{3}u-8\cos u + 15) + 6\cos^{4}u (1 - 3\sin^{2}u)+17}$

for 0 ≤ u < 2π and 0 ≤ v < 2π.

In this parametrization, u follows the length of the bottle's body while v goes around its circumference.

Generalizations

The generalization of the Klein bottle to higher genus is given in the article on the fundamental polygon. In Mathematics, genus has a few different but closely related meanings Topology Orientable surface In Mathematics, each closed Surface in the sense of Geometric topology can be constructed from an even-sided oriented Polygon, called a fundamental

Klein surface

A Klein surface is, as for Riemann surfaces, a surface with an atlas allowing that the transition functions can be composed with complex conjugation one can obtains the so called dianalytic structure. In Mathematics, particularly in Complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional In Mathematics, a transition function has several different meanings In Topology, a transition function is a Homeomorphism In Mathematics, the complex conjugate of a Complex number is given by changing the sign of the Imaginary part.

References in popular culture

The TV series Futurama has a brand of beer, Klein's Beer, sold in a Klein bottle. Futurama is an Emmy Award -winning animated American sitcom created by Matt Groening, and developed by Groening and
Russell Hoban's 2001 novel Amaryllis Night and Day makes extensive use of the Klein bottle as a metaphor. Russell Conwell Hoban (born February 4, 1925) is an American writer of Fantasy, Science fiction, mainstream Fiction, The display of bottles at London's Science Museum, and Alan Bennett himself, also feature in the book.
In the book Visitors From Oz, the characters construct a Klein bottle to travel from Oz to Earth. Wikipedia_talkFeatured_lists#Proposed_change_to_all_featured_lists for an explanation of this and other inclusion tags below -->The Oz books form a
In the Infocom game Trinity, a giant Klein bottle figures prominently, and is used to help solve one of the puzzles. Infocom was a Software company, based in Cambridge Massachusetts, that produced numerous works of Interactive fiction.
The Magic: The Gathering card Elkin Bottle shows a 3D representation of a Klein bottle. This article is somewhat over-complete Please do not add significant new content without first discussing it on the talk page The name "Elkin" is an anagram of the word "Klein".
In the animated television show The Real Ghostbusters, Ray Stantz mentions Klein bottles at 01:47 in the episode "Janine Melnitz, Ghostbuster," indicating that they are used as a method of building additional capacity to the containment unit, increasing the amount of ghosts they can store. This article is about the animated spin-off of the 1984 film Ghostbusters. ^[2]

References

Eric W. Weisstein, Klein Bottle at MathWorld. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic In Wormhole theory a non-orientable wormhole is a wormhole connection that appears to reverse the chirality of anything passed through it In Geometry, Boy's surface is an immersion of the Real projective plane in 3-dimensional space found by Werner Boy in 1901 (he discovered it This article is about the mathematical object See Mobius Band (music group for the music group Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
A classical on the theory of Klein surfaces is [1] of Alling-Greenleaf

External links

This article incorporates material from Klein bottle on PlanetMath, which is licensed under the GFDL. PlanetMath is a free, collaborative online Mathematics Encyclopedia.

Dictionary

-noun

(topology) A two-dimensional enclosed surface which doesn't have a distinct "inside" and "outside" (in a similar way that a Möbius strip has only one side).

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