Klein bottle

In mathematics, the Klein bottle /ˈklaɪn/ is an example of a non-orientable surface; it is a two-dimensional manifold 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. Whereas a Möbius strip is a surface with 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 German mathematician Felix Klein. It may have been originally named the Kleinsche Fläche ("Klein surface") and then misinterpreted as Kleinsche Flasche ("Klein bottle"), which ultimately led to the adoption of this term in the German language as well.

Construction

The following square is a fundamental polygon of the Klein bottle. The idea is to 'glue' together the corresponding coloured edges so that the arrows match, 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.

Klein

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Mathematical sciences

Mathematical sciences is a group of areas of study that includes, in addition to mathematics, those academic disciplines that are primarily mathematical in nature but may not be universally considered subfields of mathematics proper. Statistics, for example, is mathematical in its methods but grew out of scientific observations which merged with inverse probability and grew through applications in the social sciences, some areas of physics and biometrics to become its own separate, though closely allied field. Computer science, computational science, population genetics, operations research, cryptology, econometrics, theoretical physics, and actuarial science are other fields that may be considered part of mathematical sciences.

Some institutions offer degrees in mathematical sciences (e.g., the United States Military Academy and University of Khartoum) or applied mathematical sciences (e.g., the University of Rhode Island).

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Clifford Stoll

Clifford Paul "Cliff" Stoll (born June 4, 1950) is an American astronomer, author and teacher. He is best known for his investigation in 1986, while working as a systems administrator at the Lawrence Berkeley National Laboratory, that led to the capture of hacker Markus Hess, and for Stoll's subsequent book, The Cuckoo's Egg, in which he details the investigation.

Stoll has written a total of three books as well as technology articles in the non-specialist press (e.g., in Scientific American on the Curta mechanical calculator and the slide rule).

Early life

Cliff Stoll attended Hutchinson Central Technical High School in Buffalo, New York. He earned a B.S. in Astronomy in 1973 from the University at Buffalo (SUNY) and went on to receive his PhD from University of Arizona in 1980. Cliff was married to Martha Stoll in the late 1980s, but they later divorced.

Career

During the 1960s and 1970s, Stoll was assistant chief engineer at WBFO, a public radio station in his hometown of Buffalo, New York.

Research institute

A research institute is an establishment endowed for doing research. Research institutes may specialize in basic research or may be oriented to applied research. Although the term often implies natural science research, there are also many research institutes in the social sciences as well, especially for sociological and historical research purposes.

Famous research institutes

In the early medieval period, several astronomical observatories were built in the Islamic world. The first of these was the 9th-century Baghdad observatory built during the time of the Abbasid caliph al-Ma'mun, though the most famous were the 13th-century Maragheh observatory, 15th-century Ulugh Beg Observatory.

The earliest research institute in Europe was Tycho Brahe's Uraniborg complex on the island of Hven, a 16th-century astronomical laboratory set up to make highly accurate measurements of the stars. In the United States there are numerous notable research institutes including Bell Labs, The Scripps Research Institute,Beckman Institute, and SRI International. Hughes Aircraft used a research institute structure for its organizational model.