Non-Euclidean geometry
In mathematics, non-Euclidean geometry consists of two geometries based on axioms related to those specifying Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises when either the metric requirement is relaxed, or the parallel postulate is replaced with an alternative one. In the latter case one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries; when the metric requirement is relaxed there are affine planes associated with the planar algebras which give rise to kinematic geometries that have been called non-Euclidean geometry. The essential difference between the metric geometries is the nature of parallel lines. Euclid's fifth postulate, the parallel postulate, is equivalent to Playfair's postulate, which states that, within a two-dimensional plane, for any given line ℓ and a point A, not on ℓ, there is one line through A that does not intersect ℓ. In hyperbolic geometry, by contrast, there are infinitely many lines through A not intersecting ℓ, while in elliptic geometry, any line through A intersects ℓ.
Another way to describe the differences between these geometries is to consider two straight lines indefinitely extended in a two-dimensional plane that are both perpendicular to a third line: In Euclidean geometry, the lines remain at a constant distance from each other and are known as parallels. In hyperbolic geometry, they "curve away" from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular. In elliptic geometry, the lines intersect. Euclidean geometry, named after the Greek mathematician Euclid, includes some of the oldest known mathematics, geometries that deviated from this were not accepted as legitimate until the 19th century; the debate that led to the discovery of the non-Euclidean geometries began as soon as Euclid's work Elements was written. In the Elements, Euclid began with a limited number of assumptions and sought to prove all the other results in the work; the most notorious of the postulates is referred to as "Euclid's Fifth Postulate," or the "parallel postulate", which in Euclid's original formulation is: If a straight line falls on two straight lines in such a manner that the interior angles on the same side are together less than two right angles the straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
Other mathematicians have devised simpler forms of this property. Regardless of the form of the postulate, however, it appears to be more complicated than Euclid's other postulates: 1. To draw a straight line from any point to any point. 2. To produce a finite straight line continuously in a straight line. 3. To describe a circle with any centre and distance. 4. That all right angles are equal to one another. For at least a thousand years, geometers were troubled by the disparate complexity of the fifth postulate, believed it could be proved as a theorem from the other four. Many attempted to find a proof by contradiction, including Ibn al-Haytham, Omar Khayyám, Nasīr al-Dīn al-Tūsī, Giovanni Girolamo Saccheri; the theorems of Ibn al-Haytham, Khayyam and al-Tusi on quadrilaterals, including the Lambert quadrilateral and Saccheri quadrilateral, were "the first few theorems of the hyperbolic and the elliptic geometries." These theorems along with their alternative postulates, such as Playfair's axiom, played an important role in the development of non-Euclidean geometry.
These early attempts at challenging the fifth postulate had a considerable influence on its development among European geometers, including Witelo, Levi ben Gerson, John Wallis and Saccheri. All of these early attempts made at trying to formulate non-Euclidean geometry, provided flawed proofs of the parallel postulate, containing assumptions that were equivalent to the parallel postulate; these early attempts did, provide some early properties of the hyperbolic and elliptic geometries. Khayyam, for example, tried to derive it from an equivalent postulate he formulated from "the principles of the Philosopher": "Two convergent straight lines intersect and it is impossible for two convergent straight lines to diverge in the direction in which they converge." Khayyam considered the three cases right and acute that the summit angles of a Saccheri quadrilateral can take and after proving a number of theorems about them, he refuted the obtuse and acute cases based on his postulate and hence derived the classic postulate of Euclid which he didn't realize was equivalent to his own postulate.
Another example is al-Tusi's son, Sadr al-Din, who wrote a book on the subject in 1298, based on al-Tusi's thoughts, which presented another hypothesis equivalent to the parallel postulate. "He revised both the Euclidean system of axioms and postulates and the proofs of many propositions from the Elements." His work was published in Rome in 1594 and was studied by European geometers, including Saccheri who criticised this work as well as that of Wallis. Giordano Vitale, in his book Euclide restituo, used the Saccheri quadrilateral to prove that if three points are equidistant on the base AB and the summ
The Mathematical Intelligencer
The Mathematical Intelligencer is a mathematical journal published by Springer Verlag that aims at a conversational and scholarly tone, rather than the technical and specialist tone more common among academic journals. It was started by mathematicians Bruce Chandler and Harold Edwards Jr. and first appeared in 1979. Marjorie Senechal is the editor-in-chief. Branislav Kisacanin Review of Mathematical Conversations. Mathematical Association of America. Home page for Mathematical Intelligencer at Springer Verlag
Germany
Germany the Federal Republic of Germany, is a country in Central and Western Europe, lying between the Baltic and North Seas to the north, the Alps to the south. It borders Denmark to the north and the Czech Republic to the east and Switzerland to the south, France to the southwest, Luxembourg and the Netherlands to the west. Germany includes 16 constituent states, covers an area of 357,386 square kilometres, has a temperate seasonal climate. With 83 million inhabitants, it is the second most populous state of Europe after Russia, the most populous state lying in Europe, as well as the most populous member state of the European Union. Germany is a decentralized country, its capital and largest metropolis is Berlin, while Frankfurt serves as its financial capital and has the country's busiest airport. Germany's largest urban area is the Ruhr, with its main centres of Essen; the country's other major cities are Hamburg, Cologne, Stuttgart, Düsseldorf, Dresden, Bremen and Nuremberg. Various Germanic tribes have inhabited the northern parts of modern Germany since classical antiquity.
A region named Germania was documented before 100 AD. During the Migration Period, the Germanic tribes expanded southward. Beginning in the 10th century, German territories formed a central part of the Holy Roman Empire. During the 16th century, northern German regions became the centre of the Protestant Reformation. After the collapse of the Holy Roman Empire, the German Confederation was formed in 1815; the German revolutions of 1848–49 resulted in the Frankfurt Parliament establishing major democratic rights. In 1871, Germany became a nation state when most of the German states unified into the Prussian-dominated German Empire. After World War I and the revolution of 1918–19, the Empire was replaced by the parliamentary Weimar Republic; the Nazi seizure of power in 1933 led to the establishment of a dictatorship, the annexation of Austria, World War II, the Holocaust. After the end of World War II in Europe and a period of Allied occupation, Austria was re-established as an independent country and two new German states were founded: West Germany, formed from the American and French occupation zones, East Germany, formed from the Soviet occupation zone.
Following the Revolutions of 1989 that ended communist rule in Central and Eastern Europe, the country was reunified on 3 October 1990. Today, the sovereign state of Germany is a federal parliamentary republic led by a chancellor, it is a great power with a strong economy. As a global leader in several industrial and technological sectors, it is both the world's third-largest exporter and importer of goods; as a developed country with a high standard of living, it upholds a social security and universal health care system, environmental protection, a tuition-free university education. The Federal Republic of Germany was a founding member of the European Economic Community in 1957 and the European Union in 1993, it is part of the Schengen Area and became a co-founder of the Eurozone in 1999. Germany is a member of the United Nations, NATO, the G7, the G20, the OECD. Known for its rich cultural history, Germany has been continuously the home of influential and successful artists, musicians, film people, entrepreneurs, scientists and inventors.
Germany has a large number of World Heritage sites and is among the top tourism destinations in the world. The English word Germany derives from the Latin Germania, which came into use after Julius Caesar adopted it for the peoples east of the Rhine; the German term Deutschland diutisciu land is derived from deutsch, descended from Old High German diutisc "popular" used to distinguish the language of the common people from Latin and its Romance descendants. This in turn descends from Proto-Germanic *þiudiskaz "popular", derived from *þeudō, descended from Proto-Indo-European *tewtéh₂- "people", from which the word Teutons originates; the discovery of the Mauer 1 mandible shows that ancient humans were present in Germany at least 600,000 years ago. The oldest complete hunting weapons found anywhere in the world were discovered in a coal mine in Schöningen between 1994 and 1998 where eight 380,000-year-old wooden javelins of 1.82 to 2.25 m length were unearthed. The Neander Valley was the location where the first non-modern human fossil was discovered.
The Neanderthal 1 fossils are known to be 40,000 years old. Evidence of modern humans dated, has been found in caves in the Swabian Jura near Ulm; the finds included 42,000-year-old bird bone and mammoth ivory flutes which are the oldest musical instruments found, the 40,000-year-old Ice Age Lion Man, the oldest uncontested figurative art discovered, the 35,000-year-old Venus of Hohle Fels, the oldest uncontested human figurative art discovered. The Nebra sky disk is a bronze artefact created during the European Bronze Age attributed to a site near Nebra, Saxony-Anhalt, it is part of UNESCO's Memory of the World Programme. The Germanic tribes are thought to date from the Pre-Roman Iron Age. From southern Scandinavia and north Germany, they expanded south and west from the 1st century BC, coming into contact with the Celtic tribes of Gaul as well
Münster
Münster is an independent city in North Rhine-Westphalia, Germany. It is in the northern part of the state and is considered to be the cultural centre of the Westphalia region, it is capital of the local government region Münsterland. Münster was the location of the Anabaptist rebellion during the Protestant Reformation and the site of the signing of the Treaty of Westphalia ending the Thirty Years' War in 1648. Today it is known as the bicycle capital of Germany. Münster gained the status of a Großstadt with more than 100,000 inhabitants in 1915; as of 2014, there are 300,000 people living in the city, with about 61,500 students, only some of whom are recorded in the official population statistics as having their primary residence in Münster. In 793, Charlemagne sent out Ludger as a missionary to evangelise the Münsterland. In 797, Ludger founded a school that became the Cathedral School. Gymnasium Paulinum traces its history back to this school. Ludger was ordained as the first bishop of Münster.
The first cathedral was completed by 850. The combination of ford and crossroad, market place, episcopal administrative centre and school, established Münster as an important centre. In 1040, Heinrich III became the first king of Germany to visit Münster. In the Middle Ages, the Prince-Bishopric of Münster was a leading member of the Hanseatic League. In 1534, the Anabaptists led by John of Leiden, took power in the Münster Rebellion and founded a democratic proto-socialistic state, they claimed all property, burned all books except the Bible, called it the "New Jerusalem". John of Leiden believed he would lead the elect from Münster to capture the entire world and purify it of evil with the sword in preparation for the Second Coming of Christ and the beginning of the Millennium, they went so far as to require all citizens to be naked as preparation for the Second Coming. However, the town was recaptured in 1535. Part of the signing of the Peace of Westphalia of 1648 was held in Münster; this ended the Eighty Years' War.
It guaranteed the future of the prince-bishop and the diocese. The last outstanding palace of the German baroque period was created according to plans by Johann Conrad Schlaun; the University of Münster was established in 1780. It is now a major European centre for excellence in education and research with large faculties in the arts, theology, sciences and law. There are about 40,000 undergraduate and postgraduate students enrolled. In 1802 Münster was conquered by Prussia during the Napoleonic Wars, it was part of the Grand Duchy of Berg between 1806 and 1811 and the Lippe department of the First French Empire between 1811 and 1813, before returning to Prussian rule. It became the capital of the Prussian province of Westphalia. A century in 1899 the city's harbour started operations when the city was linked to the Dortmund-Ems Canal. In the 1940s The Bishop of Münster, Cardinal Clemens August Graf von Galen, was one of the most prominent critics of the Nazi government. In retaliation for his success, Münster was garrisoned during World War II, five large complexes of barracks are still a feature of the city.
Münster was the headquarters for the 6th Military District of the German Wehrmacht, under the command of Infantry General Gerhard Glokke. Made up of Westphalia and the Rhineland, after the Battle of France it was expanded to include the Eupen - Malmedy district of Belgium; the headquarters controlled military operations in Münster, Essen, Düsseldorf, Bielefeld, Paderborn, Minden, Lingen, Osnabrück, Recklinghausen and Cologne. Münster was the home station for the VI and XXIII Infantry Corps, as well as the XXXIII and LVI Panzerkorps. Münster was the home of the 6th, 16th and 25th Panzer Division. A secondary target of the Oil Campaign of World War II, Münster was bombed on 25 October 1944 by 34 diverted B-24 Liberator bombers, during a mission to a nearby primary target, the Scholven/Buer synthetic oil plant at Gelsenkirchen. About 91% of the Old City and 63% of the entire city was destroyed by Allied air raids; the US 17th Airborne Division, employed in a standard infantry role and not in a parachute capacity, attacked Münster with the British 6th Guards Tank Brigade on 2 April 1945 in a ground assault and fought its way into the contested city centre, cleared in urban combat on the following day.
From 1946 to 1998, there was a Latvian secondary school in Münster, in 1947, one of the largest of about 93 Latvian libraries in the West was established in Münster. In the 1950s the Old City was rebuilt to match its pre-war state, though many of the surrounding buildings were replaced with cheaper modern structures, it was for several decades a garrison town for the British forces stationed in West Germany. In 2004, Münster won an honourable distinction: the LivCom-Award for the most livable city in the world with a population
Siegen
Siegen is a city in Germany, in the south Westphalian part of North Rhine-Westphalia. It is located in the district of Siegen-Wittgenstein in the Arnsberg region; the university town is the district seat, is ranked as a "higher centre" in the South Westphalian urban agglomeration. In 1975, municipal reforms and amalgamations lifted Siegen's population above the 100,000 mark; the city of Siegen lies in the basin of the upper reaches of the river Sieg. From there, lateral valleys branch off in many directions; the heights of the surrounding mountains, wherever they are not settled, are covered in coppice. To the north lies the Sauerland, to the northwest the Rothaargebirge and to the southwest the Westerwald; the nearest cities to Siegen, taking into account average travelling distances, are Hagen to the north 83 km, Frankfurt am Main to the southeast 125 km, Koblenz to the southwest 105 km and Cologne to the west 93 km. As the crow flies the distances to these places are, however, 65 km, 95 km, 65 km and 75 km.
The city lies on the German-Dutch holiday road called the Orange Route, joining towns and regions associated with the House of Orange. The city's total land area is 115 km2, its greatest east-west span is about 12 km, its greatest north-south span is about 12 km. City limits are 48 km long. Siegen lies at a median elevation of 290 m above sea level; the city's greatest elevation is the peak of the Pfannenberg at 499 m above sea level at southern city limits. Siegen's lowest point is 215 m above sea level at Niederschelden at southwestern city limits, which there forms the state boundary with Rhineland-Palatinate. 60% of the city's land is wooded, making Siegen one of Germany's greenest cities. The city area is divided into six zones, called Bezirke in German and comparable to boroughs in some cities, which themselves are further divided into various communities; each "borough" has a borough board consisting of 15 voting and 15 non-voting members who are appointed by city council with regard to each party's share of the vote in the municipal elections in the borough in question.
The borough boards decide on matters particular to their respective boroughs. These matters are laid down in Siegen's city charter. District I: Birlenbach, Langenholdinghausen, Dillnhütten, Buchen, Obersetzen District II: District III: Kaan-Marienborn, parts of Alt-Siegen, Bürbach, Breitenbach, Feuersbach District IV: Alt-Siegen District V: Seelbach and parts of Alt-Siegen District VI: Oberschelden, Niederschelden, EisernAlong with the boroughs and communities into which the city is divided, as mandated by law, there are further subdivisions within the communities, each with its own name, but none with distinctly clear borders, they are called Quartiere, which can be rendered as "quarters" or "neighbourhoods". Examples of these include the Unterstadt, the Oberstadt, Hammerhütte, Charlottental, Haardter Berg and the Alte Dreisbach; some neighbourhoods straddle community boundaries, like Sieghütte, parts of which can be found in both Siegen-Mitte and Weidenau. Moreover, some neighbourhoods overlap each other.
Unlike the boroughs or communities, the Quartiere have no administrative importance. They do, serve some function as to their inhabitants' identity, but more than that, they are useful for finding one's way with a city map and using in bus route names and on public notices and traffic signs. Many of the Hüttentalstraße city Autobahn's exits are named after the Quartiere that they serve; the communities of Weidenau, Birlenbach, Langenholdinghausen, Sohlbach, Dillnhütten, Niedersetzen and Meiswinkel formed from 1 July 1966 to 31 December 1974 the town of Hüttental. The communities of Eiserfeld, Gosenbach and Oberschelden formed the town of Eiserfeld between those same two dates; the city of Siegen borders in the north on the town of Kreuztal and the community of Wenden, in the east on the town of Netphen, in the southeast on the community of Wilnsdorf, in the south on the community of Neunkirchen, in the west on the community of Mudersbach and in the northwest on the town of Freudenberg. The name Siegen comes from the Celtic river name Sieg.
It is, unclear whether there is any relation between this name and the Celtic-Germanic Sicambri people, who in pre-Christian times lived in parts of North Rhine-Westphalia. The first documentary mention of the place called Sigena dates from 1079; the city's history is markedly shaped by mining. Bearing witness to this longtime industry are the many mines. In 1224, Siegen is mentioned as a newly built town whose ownership was shared by the Count of Nassau, Heinrich the Rich, Engelbert II of Berg, Archbishop of Cologne after the latter transferred one half of the ownership to the former. Moreover, there is proof that the Oberes Schloss was standing at this time. On 19 October 1303, the town was granted Soest town rights; the town remained under the two overlords' joint ownership until 1 February 1381, only
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: For any given line R and point P not on R, in the plane containing both line R and point P there are at least two distinct lines through P that do not intersect R. Hyperbolic plane geometry is the geometry of saddle surfaces and pseudospherical surfaces, surfaces with a constant negative Gaussian curvature. A modern use of hyperbolic geometry is in the theory of special relativity Minkowski spacetime and gyrovector space; when geometers first realised they were working with something other than the standard Euclidean geometry they described their geometry under many different names. In the former Soviet Union, it is called Lobachevskian geometry, named after one of its discoverers, the Russian geometer Nikolai Lobachevsky; this page is about the 2-dimensional hyperbolic geometry and the differences and similarities between Euclidean and hyperbolic geometry. Hyperbolic geometry can be extended to three and more dimensions.
Hyperbolic geometry is more related to Euclidean geometry than it seems: the only axiomatic difference is the parallel postulate. When the parallel postulate is removed from Euclidean geometry the resulting geometry is absolute geometry. There are two kinds of absolute geometry and hyperbolic. All theorems of absolute geometry, including the first 28 propositions of book one of Euclid's Elements, are valid in Euclidean and hyperbolic geometry. Propositions 27 and 28 of Book One of Euclid's Elements prove the existence of parallel/non-intersecting lines; this difference has many consequences: concepts that are equivalent in Euclidean geometry are not equivalent in hyperbolic geometry. Further, because of the angle of parallelism, hyperbolic geometry has an absolute scale, a relation between distance and angle measurements. Single lines in hyperbolic geometry have the same properties as single straight lines in Euclidean geometry. For example, two points uniquely define a line, lines can be infinitely extended.
Two intersecting lines have the same properties as two intersecting lines in Euclidean geometry. For example, two lines can intersect in no more than one point, intersecting lines have equal opposite angles, adjacent angles of intersecting lines are supplementary; when we add a third line there are properties of intersecting lines that differ from intersecting lines in Euclidean geometry. For example, given 2 intersecting lines there are infinitely many lines that do not intersect either of the given lines; these properties all are independent of the model used if the lines may look radically different. Non-intersecting lines in hyperbolic geometry have properties that differ from non-intersecting lines in Euclidean geometry: For any line R and any point P which does not lie on R, in the plane containing line R and point P there are at least two distinct lines through P that do not intersect R; this implies that there are through P an infinite number of coplanar lines that do not intersect R.
These non-intersecting lines are divided into two classes: Two of the lines are limiting parallels: there is one in the direction of each of the ideal points at the "ends" of R, asymptotically approaching R, always getting closer to R, but never meeting it. All other non-intersecting lines have a point of minimum distance and diverge from both sides of that point, are called ultraparallel, diverging parallel or sometimes non-intersecting; some geometers use parallel lines instead of limiting parallel lines, with ultraparallel lines being just non-intersecting. These limiting parallels make an angle θ with PB. For ultraparallel lines, the ultraparallel theorem states that there is a unique line in the hyperbolic plane, perpendicular to each pair of ultraparallel lines. In hyperbolic geometry, the circumference of a circle of radius r is greater than 2 π r. Let R = 1 − K, where K is the Gaussian curvature of the plane. In hyperbolic geometry, K is negative, so the square root is of a positive number.
The circumference of a circle of radius r is equal to: 2 π R sinh r R. And the area of the enclosed disk is: 4 π R 2 sinh 2 r 2 R = 2 π R 2. Therefore, in hyperbolic geometry the ratio of a circle's circumference to its radius is always greater than 2 π, though
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