The only exception to the formula is the Klein bottle, which has Euler characteristic 0 ( hence the formula gives p = 7 ) and requires 6 colors, as shown by P. Franklin in 1934 ( Weisstein ).

A two-dimensional representation of the Klein bottle Immersion ( mathematics ) | immersed in three-dimensional space.

In mathematics, the Klein bottle () is a non-orientable surface, informally, a surface ( a two-dimensional manifold ) in which notions of left and right cannot be consistently defined.

Whereas a Möbius strip is a surface with boundary, a Klein bottle has no boundary.

It may have been originally named the Kleinsche Fläche (" Klein surface ") and that this was incorrectly interpreted as Kleinsche Flasche (" Klein bottle "), which ultimately led to the adoption of this term in the German language as well.

More formally, the Klein bottle is the quotient space described as the square × with sides identified by the relations for and for:

This square is a fundamental polygon of the Klein bottle.

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.

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.

Klein bottle in the Science Museum ( London ) | Science Museum in London.

Dissecting the Klein bottle results in Möbius strips.

The common physical model of a Klein bottle is a similar construction.

The Klein bottle can be seen as a fiber bundle over 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.

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 R < sup > 3 </ sup >, the Klein bottle cannot.

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:

Six colors suffice to color any map on the surface of a Klein bottle ; this is the only exception to

Jacob Klein was one student of Husserl who pursued this line of inquiry, seeking to " desedimentize " mathematics and the mathematical sciences.

Hillview was put up for sale and purchased from the State Government by Edwin Klein in 1957.

In this era some German settlers, Joseph Klein and P W Merkell, tried to establish vineyards in the area, but eventually found the land was not suited for this type of agriculture.

Klein died aboard Conrad's steamer and was interred along the Congo, much like Kurtz in the novel.

Isaac Klein ( September 5, 1905 – 1979 ) was a prominent rabbi and halakhic authority within Conservative Judaism.

Klein was born in the small village of Palanka in what was then Hungary and emigrated with his family to the United States in 1921.

Klein served as rabbi at Kadimoh Congregation in Springfield, Massachusetts from 1934 – 1953 ; Temple Emanu-El, Buffalo, New York, 1953 – 1968 ; Temple Shaarey Zedek, Buffalo, ( which was created from the merger of Emanu-El with Temple Beth David in 1968 ), 1968-1972.

Klein was a leader of the right-wing of the Conservative movement.

Karlheinz Stockhausen (; 22 August 1928 – 5 December 2007 ) was a German composer, widely acknowledged by critics as one of the most important ( Barrett 1988, 45 ; Harvey 1975b, 705 ; Hopkins 1972, 33 ; Klein 1968, 117 ) but also controversial ( Power 1990, 30 ) composers of the 20th and early 21st centuries.

Based on the extent to which the application of neoliberal policies has contributed to income disparities and inequality, both Klein and Noam Chomsky have suggested that the primary role of neoliberalism was as an ideological cover for capital accumulation by multinational corporations.

Wattenscheid, Wanne and Gelsenkirchen were the centers of Masurian emigration and Gelsenkirchen-Schalke was even called Klein ( little )- Ortelsburg before 1914.

Joel I. Klein was the lead prosecutor.

This account was challenged in a book by Aaron Klein, who claims that Al-Gashey died of heart failure in the 1970s, and that Safady was either killed by Christian Phalangists in Lebanon in the early 1980s, or, according to a PLO operative friendly with Safady, is still alive ( as of 2005 ).

After the book's release, Klein was heavily criticized by the news magazine The Economist, leading to a broadcast debate with Klein and the magazine's writers, dubbed " No Logo vs.

Inspired by Klein & MBO's " Dirty Talk " and Sylvester's disco classic, " You Make Me Feel ( Mighty Real )", " Blue Monday " became the best-selling independent 12 " single of all time in the UK ; however, ( much to the chagrin of the buying public ) it was not on the track list of Power, Corruption & Lies.

Chargers owner Eugene Klein tried to get the NFL and Bengals to postpone the game but he was turned down.

In mid-1935 Wernher von Braun initiated a fuel pump project at the southwest German firm Klein, Schanzlin & Becker that was experienced in building large fire-fighting pumps .< ref name = Neufeld >

Through 2011, he was one of seven major leaguers to have had at least four 30-homer, 100-RBI seasons in their first five years, along with Chuck Klein, Joe DiMaggio, Ralph Kiner, Mark Teixeira, Albert Pujols, and Ryan Braun.

Historically, the concept of fundamental group first emerged in the theory of Riemann surfaces, in the work of Bernhard Riemann, Henri Poincaré and Felix Klein, where it describes the monodromy properties of complex functions, as well as providing a complete topological classification of closed surfaces.

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.

The dilaton made its first appearance in Kaluza – Klein theory, a five-dimensional theory that combined gravitation and electromagnetism.

The illative case was used extensively in older Lithuanian ; the first Lithuanian grammar book by Daniel Klein, that mentions both illative and į + accusative, calls the usage of the illative " more elegant ".

In 1968 Klein founded Klein Associates, Inc .( now L-3 / Klein ) and continued to work on improvements including the first commercial high frequency ( 500 kHz ) systems and the first dual-frequency side-scan sonars, and the first combined side-scan and sub-bottom profiling sonar.

In 1985, Charles Mazel of Klein Associates produced the first commercial side-scan sonar training videos and the first Side Scan Sonar Training Manual.

The model for hyperbolic geometry was answered by Eugenio Beltrami, in 1868, who first showed that a surface called the pseudosphere has the appropriate curvature to model a portion of hyperbolic space, and in a second paper in the same year, defined the Klein model which models the entirety of hyperbolic space, and used this to show that Euclidean geometry and hyperbolic geometry were equiconsistent, so that hyperbolic geometry was logically consistent if and only if Euclidean geometry was.

Joshua Klein wrote, " or the first half of her recording career, Janet Jackson sounded like an artist with something to prove.

Mason Klein, curator of a Man Ray exhibition at the Jewish Museum entitled Alias Man Ray: The Art of Reinvention, suggests that the artist may have been " the first Jewish avant-garde artist.

Lyman, along with Simon Ferguson ( Wireds first advertising manager ), introduced revolutionary ad campaigns by a diverse group of industry leaders — such as Apple Computer, Intel, Sony, Calvin Klein, and Absolut — to the readers of the first technology publication with a lifestyle slant.

Since he first came to prominence in the mid-1960s, critics have cited Campbell as one of the leading writers in his field: T. E. D. Klein has written that " Campbell reigns supreme in the field today ", while S. T. Joshi stated, " future generations will regard him as the leading horror writer of our generation, every bit the equal of Lovecraft or Blackwood.