The Klein bottle immersed in three-dimensional space.

In mathematics, the Klein bottle is a certain non-orientable surface, i.e., a surface (a two-dimensional topological space) with no distinction between the "inside" and "outside" surfaces. The Klein bottle was first described in 1882 by the German mathematician Felix Klein. Other related non-orientable objects are the Möbius strip and the real projective plane. It was originally named the Kleinsche Fläche "Klein surface"; however, this was incorrectly interpreted as Kleinsche Flasche "Klein bottle", which ultimately, due to the dominance of the English language in science, led to the adoption of this term in the German language as well.^{[citation needed]} Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... In mathematics, an immersion is a differentiable map between differentiable manifolds whose derivative is everywhere injective. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... The torus is an orientable surface. ... An open surface with X-, Y-, and Z-contours shown. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... Year 1882 (MDCCCLXXXII) was a common year starting on Sunday (link will display the full calendar) of the Gregorian calendar (or a common year starting on Tuesday of the 12-day slower Julian calendar). ... Felix Christian Klein (April 25, 1849, Düsseldorf, Germany – June 22, 1925, Göttingen) was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory. ... A Möbius strip made with a piece of paper and tape. ... The fundamental polygon of the projective plane. ...

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, : In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or gluing together certain points of a given space. ... For other uses, see Square. ...

This square is a fundamental polygon of the Klein bottle. Image File history File links This is a lossless scalable vector image. ... In mathematics, each closed surface in the sense of geometric topology can be constructed from an even-sided oriented polygon, called a fundamental polygon, by pairwise identification of its edges. ...

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. Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ...

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. It has been suggested that this article or section be merged with orientable manifold. ...

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. 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. In mathematics, in particular in topology, a fiber bundle (or fibre bundle) is a space which locally looks like a product of two spaces but may possess a different global structure. ... Circle illustration This article is about the shape and mathematical concept of circle. ... In mathematics, a circle bundle is a fiber bundle where the fiber is the circle , or more precisely, a principal U(1)-bundle with fiber U(1). ...

Like the Möbius strip, the Klein bottle is a two-dimensional differentiable 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 R³, the Klein bottle cannot. It can be embedded in R⁴, however. A Möbius strip made with a piece of paper and tape. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... The torus is an orientable surface. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...

A handblown Klein Bottle made by Mitsugi Ohno

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: Image File history File links Kleinbot2. ... Image File history File links Kleinbot2. ... Mitsugi Ohno (大野 è²¢, June 28, 1926 - October 22, 1999) was a Japanese glassblower who worked at the University of Tokyo (1947 - 1960) and Kansas State University (1961 - 1996). ... Look up anon, anonymity, anonymous in Wiktionary, the free dictionary. ... A limerick is a five-line poem with a strict meter, popularized by Edward Lear. ...

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."

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 in graph theory was an expected formula to give the correct upper bound for the number of colors which are sufficient for graph coloring on a surface of a given genus. ... Example of a four-colored map 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 counties of a state, the regions may be colored using no more than four colors in such...

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.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.
Wikipedia does not have an article with this exact name. ... Wikipedia does not have an article with this exact name. ... In 3-dimensional geometry, a plane of symmetry is a 2-dimensional flat dividing surface placed such that things on one side are symmetrical (mirror image) to things on the other side. ... A Möbius strip made with a piece of paper and tape. ...

Parametrization

The "figure 8" immersion of the Klein bottle.

The "figure 8" immersion of the Klein bottle has a particularly simple parametrization: Wikipedia does not have an article with this exact name. ... Wikipedia does not have an article with this exact name. ... In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ...

In this immersion, the self-intersection circle is a geometric circle in the XY plane. 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. Circle illustration This article is about the shape and mathematical concept of circle. ...

Generalizations

The generalization of the Klein bottle to higher genus is given in the article on the fundamental polygon. In mathematics, the genus has few different meanings Topology The genus of a connected, oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. ... In mathematics, each closed surface in the sense of geometric topology can be constructed from an even-sided oriented polygon, called a fundamental polygon, by pairwise identification of its edges. ...

Trivia

• A mounted Klein bottle is the trophy for the BASIC WonderCup Challenge.
• The TV series Futurama has a brand of beer, Klein's Beer, sold in a Klein bottle.
• The British Science Museum has on display a beautiful collection of hand-blown glass Klein bottles, exhibiting many variations on the same topological theme. The bottles date from 1995 and were made for the museum by Alan Bennett. [1]
• Russell Hoban's 2001 novel Amaryllis Night and Day makes extensive use of the Klein bottle as a metaphor. The display of bottles at London's Science Museum, and Alan Bennett himself, also feature in the book.
• In the computer game Nethack, attempting to dip a potion into itself gives the message, "That is a potion bottle, not a Klein bottle!"
• Joe Strummer and the Mescaleros have a song entitled "Mega Bottle Ride" which describes a journey "into the fourth dimension" via the "Banchoff-Klein Mega Bottle Ride".
• In the Janine Melnitz, Ghostbuster episode of the TV show The Real Ghostbusters, Ray mentions adding another Klein bottle to the Containment Unit.
• In the book Visitors From Oz, the characters construct a Klein bottle to travel from Oz to Earth.
• In the Infocom game Trinity, a giant Klein bottle figures prominently, and is used to help solve one of the puzzles.
• Clifford Stoll, author of The Cuckoo's Egg, manufactures Klein bottles and sells them via the Internet at Acme Klein Bottle.
• The Magic: The Gathering card Elkin Bottle shows a 3D representation of a Klein bottle. The card name is an obvious anagram of Klein.
• American computer scientist Ted Kaehler was visiting Germany and walked into a glass-blower's shop. He asked for "ein Klein Flasche." The glass-blower handed him a small bottle, as "klein" means "small". Kaehler eventually had to draw a Klein bottle on paper before the artisan understood what he was asking for; he was able to make one, although it was not a real Klein bottle.
• The unix fortune cookie program says, "Klein bottle for rent, enquire within."

The WonderCup Challenge gives teams of high school students the opportunity to test their knowledge of basic biology, chemistry, and physics. ... Futurama is an Emmy Award-winning animated American sitcom created by Matt Groening (creator of The Simpsons) and David X. Cohen for the Fox network. ... This is a list of fictional products in the animated television series Futurama. ... The Science Museum on Exhibition Road, South Kensington, London is part of the National Museum of Science and Industry. ... Year 1995 (MCMXCV) was a common year starting on Sunday (link will display full 1995 Gregorian calendar). ... Russell Conwell Hoban (born February 4, 1925) is an American writer of fantasy, science fiction, mainstream fiction, magic realism, poetry, and childrens books. ... This article is about the role-playing game. ... The Mescaleros were the backing band for Joe Strummer for three albums prior to his death in 2002. ... This article is about the animated spin-off of the 1984 film Ghostbusters. ... Dr. Raymond Ray Stantz, PhD is a fictional ghostbuster appearing in the films Ghostbusters and Ghostbusters II (played by Dan Aykroyd) and in the animated television series The Real Ghostbusters (voiced by Frank Welker). ... Ghostbusters equipment is the equipment used by the Ghostbusters in the 1984 film and all subsequent Ghostbusters fiction used to aid in the capture and containment of ghosts. ... The Oz books form a book series that begins with The Wonderful Wizard of Oz, and that relates the history of the Land of Oz. ... Zork universe Zork games Zork Anthology Zork trilogy Zork I   Zork II   Zork III Beyond Zork   Zork Zero Enchanter trilogy Enchanter   Sorcerer   Spellbreaker Other games Wishbringer   Return to Zork Zork: Nemesis   Zork Grand Inquisitor Zork: The Undiscovered Underground Topics in Zork Encyclopedia Frobozzica Characters   Kings   Creatures Timeline   Magic   Calendar Zorkmid... Trinity is an interactive fiction computer game written by Brian Moriarty and published in 1986 by Infocom. ... Clifford Stoll (or Cliff Stoll) is an astronomer, computer systems administrator, and author. ... The Cuckoos Egg is a book written by Clifford Stoll. ... Magic: The Gathering (colloq. ... fortune is a simple program that displays a random message from a database of quotes. ...

See also

• Topology
• Algebraic topology
• Surface
• Alice universe
• Boy's surface

A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... An open surface with X-, Y-, and Z-contours shown. ... In theoretical physics, an Alice universe is a hypothetical universe with no global definition of charge. ... In geometry, Boys surface is an immersion of the real projective plane in 3-dimensional space found by Werner Boy in 1901. ...

References

• Eric W. Weisstein, Klein Bottle at MathWorld.

Dr. Eric W. Weisstein Encyclopedist Dr. Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is a noted encyclopedist in several technical areas of science and mathematics. ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...

External links

Wikimedia Commons has media related to:

Surfaces

• Imaging Maths - The Klein Bottle
• The biggest Klein bottle in all the world