Usage.

Torsion bar suspensions are used on combat vehicles and tanks like the T-72, Leopard 1, Leopard 2, M26 Pershing, M18 Hellcat, and the M1 Abrams (many tanks from World War II used this suspension), and on modern trucks and SUVs from Ford, Chrysler, GM, Mitsubishi, Mazda, Nissan, Isuzu, LuAZ, and Toyota. Manufacturers change the torsion bar or key to adjust the ride height, usually to compensate for engine weight. While the ride height may be adjusted by turning the adjuster bolts on the stock torsion key, rotating the stock key too far can bend the adjusting bolt and place the shock piston outside its standard travel. Over-rotating the torsion bars can also cause the suspension to hit the bump-stop prematurely, causing a harsh ride. Aftermarket forged-metal torsion key kits use relocked adjuster keys to prevent over-rotation, and shock brackets to keep the piston travel in the stock range.

Advantages and disadvantages.

The main advantages of a torsion bar suspension are soft ride due to elasticity of the bar, durability, easy adjustability of ride height, and small profile along the width of the vehicle. It takes up less of the vehicle's interior volume than coil springs.  Torsion bars reached the height of their popularity on mass-production road cars in the middle of the 20th century at the same time that unitary construction was being adopted. At a time when the mechanics of stress and metal fatigue in monocoque body frames was poorly understood, torsion bars were very attractive to vehicle designers as the bars could be mounted to reinforced parts of the central structure, typically the bulkhead. Using MacPherson struts to achieve independent front suspension with coil springs meant providing strong turrets in the frontal structure of the car.  A disadvantage is that torsion bars, unlike coil springs, usually cannot provide a progressive spring rate. In most torsion bar systems, ride height (and therefore many handling features) may be changed by simply adjusting bolts that connect the torsion bars to the steering knuckles. In most cars with this type of suspension, swapping torsion bars for a different spring rate is usually an easy task. Longitudinal torsion bars extend under the passenger compartment, cutting into interior space by raising the floor, while in transverse systems, torsion bar length is limited by vehicle width.

Leveling.

Some vehicles use torsion bars to provide automatic levelling, using a motor to pre-stress the bars to provide greater resistance to load and, in some cases (depending on the speed with which the motors can act), to respond to changes in road conditions. Height adjustable suspension has been used to implement a wheel-change mode where the vehicle is raised on three wheels so that the remaining wheel is lifted off the ground without the aid of a jack.

History.

Before World War II (1934), the front wheel drive Citroen Traction Avant had independent front torsion bar suspension and a flexible trailing dead axle, also sprung by torsion bars. The flexibility of the axle beam provided wheel location features like a twist beam axle. Also in the 1930s, prototypes of the first Volkswagen Beetle incorporated torsion bars—especially their transverse mounting style. Czechoslovakian Tatra's 1948 T600 Tatraplan employed rear torsion bar suspension, the only Tatra to do so.  The system first saw military use in the Swedish Stridsvagn L-60 tank of 1934. It was used extensively in European cars like Renault, Citroën and Volkswagen, as well as by Packard in the 1950s. The Packard used torsion bars at both front and rear, and interconnected the front and rear systems to improve ride quality. Morris Minor and Oxford from the late 1940s onwards used a front torsion bar system very similar to the Citroen, as did the Riley RM models. The revolutionary Jaguar E-Type introduced in 1961 had a torsion bar front suspension very similar to the Citroen and Morris Minor, and an independent coil spring rear suspension using four shock absorbers with concentric springs.  An early application of a torsion bar in an American car was by Hudson Motor Car Company of Detroit who had introduced the innovative front axle flex suspension in 1934 Hudson and Terraplane cars and realized for 1935 that a transverse torsion bar linked to the rear axle was needed as an anti-roll bar to stabilize the cars. The single torsion bar was mounted through the frame sides behind the rear axle and then attached by arms and links to the front side of the spring U-bolt plates. Axle flex was discontinued for the 1936 model year.  Post-war the use of torsion bar front suspension was a defining feature of British Morris cars, starting with the Morris Minor of 1948, its larger Morris Oxford MO counterpart and the upmarket Morris Six MS, plus the Wolseley-badged upmarket variants of the latter two models. The designer of these cars, Alec Issigonis, was inspired by the Traction Avant's suspension, although the Morris cars were rear-wheel drive and used conventional leaf springs for their rear axles. The Minor used lever arm dampers with its torsion bars while the Oxford and the Six used innovative telescopic dampers. The Minor remained in production largely unchanged until 1972 and was replaced by the Morris Marina which also used the torsion bar-lever arm damper system for its front suspension—one of the last new cars worldwide to be introduced with the system and which remained in production until 1984. The Oxford/Six platform was developed through several revised series which used Issigonis' torsion bar system until 1959 when the new Farina Oxford was introduced using front suspension with coil springs, lower wishbones and lever arm dampers.  The most famous American passenger car application was the Chrysler system used beginning with the 1957 model year in cars such as the Imperial Crown series, although Chrysler's "Torsion-Air" suspension was only for the front; the same basic system (longitudinal mounting) was maintained until the 1981 introduction of the K-car. A reengineered torsion bar suspension, introduced with the 1976 Dodge Aspen, introduced transverse-mounted torsion bars (possibly based on the Volkswagen Type 3 passenger car) until production ended in 1989 (with Chrysler's M platform). Some generations of the Dodge Dakota and Durango used torsion bars on the front suspension.  General Motors first used torsion bars on their light-duty pickup trucks in 1960 until it was phased out in 1963 where traditional coil springs are used up front for their 2WD trucks. Its first use in a passenger car was in 1966, starting with the E-platform vehicles (Oldsmobile Toronado, Cadillac Eldorado), 4WD S-10 pickups and Astro vans with optional AWD, and since 1988, full size trucks and SUVs with 4WD (GMT400, GMT800, and GMT900 series).  Porsche used four-wheel torsion bar suspension for their 356 and 911 series from 1948 until 1989 with the introduction of the 964. They are also used in the front suspension of the 914 as well as the rear suspension of the 924, 944, and 968. Honda also used front torsion bars on the third generation Civic and other variants built on the same platform including the Ballade and first generation CRX.

Variations.

The German World War II Panther tank (and some Tigers) had double torsion bars. Needing bars longer than the width of the tank to get the required spring rate and maximum elastic bend angle from available steel alloys, designer Ernst Lehr created a suspension that effectively folded the bars in half. For each wheel, one rod was attached to the suspension arm, while another was mounted to a nearby point on the frame. On the opposite side of the tank, the two rods were attached to each other and fitted into a pivot. Deflection of the suspension arm caused both halves of the double torsion bar to twist. A disadvantage of the torsion bar suspension used in Tiger and Panther tanks (and many other WWII-era tanks and other AFVs) was the inability to incorporate an escape hatch through the bottom of the hull, a common feature of WWII-era tanks, as the torsion bar arrangement would have blocked crew access to such a hatch; however, the absence of leaf, coil or volute springs often left a large expanse of the side of the hull clear to include a side-escape hatch, and it was rare for a tank to be flipped over in such a way that all top-side hatches were unable to open, which is the purpose of ventral hatches.  Many contemporary main battle tanks use torsion bar suspension, including the American M1 Abrams, German Leopard 2, and Chinese MBT-3000, though the newest generation of tanks such as the Russian T-14 Armata utilize an adjustable hydraulic suspension. Due to their small size, tremendous load capacity, and relative ease of service, torsion bar suspension has been ideal for tanks, though it is not without disadvantage. The large travel and high elasticity of the torsion bars results in a "rocking" motion when the tank is moving or coming to a sudden stop. A gun stabilizer must be used to compensate for the rocking motion. Due to the massive weight of a main battle tank, compared to an automobile, there is a much greater risk of breaking a torsion bar on sudden bumps or maneuvers, and if it is not replaced in short order the reduced suspension can affect the maneuverability of the vehicle, and in extreme cases risk immobilizing the vehicle as the reduced capacity of the suspension causes additional torsion bars to break.  Some front-wheel drive automobiles use a related type of torsion beam suspension, usually called a twist-beam rear suspension, in which the rear wheels are carried on trailing arms connected by a laterally mounted torsion beam, as found on the Mitsubishi Debonair. The actual springing medium for these is usually coil springs. The torsion beam functions both as wheel locating arm and as an anti-roll bar to resist lateral motion of the wheels as the body leans in turns. Its advantages are that it is inexpensive to manufacture and install, and engages a minimum amount of interior volume, leaving more space for the carriage of passengers, cargo, and other components. Because the torsion beam acts in the lateral plane, not vertically, the twist beam axle cannot provide ride height adjustment, and it suffers, to some extent, similar car handling limitations as other beam axle suspensions. However these limitations may not be apparent on the road, because of the trend towards firmer, more sporty suspension setups with more limited wheel travel. Twist-beam rear suspensions were pioneered on the Volkswagen Golf in the early 1970s, and remain common on compact cars and minivans.

Other uses.

Torsion bars were sometimes used instead of conventional coil valve springs in some older motorcycles, such as the Honda CB450, and also on the Panhard Dyna X and Panhard Dyna Z cars of the 1950s. They were also used in the door mechanism of the DMC DeLorean automobile.

A torsion bar suspension, also known as a torsion spring suspension, is any vehicle suspension that uses a torsion bar as its main weight-bearing spring. One end of a long metal bar is attached firmly to the vehicle chassis; the opposite end terminates in a lever, the torsion key, mounted perpendicular to the bar, that is attached to a suspension arm, a spindle, or the axle. Vertical motion of the wheel causes the bar to twist around its axis and is resisted by the bar's torsion resistance. The effective spring rate of the bar is determined by its length, cross section, shape, material, and manufacturing process.

Eine Drehstabfeder (auch Torsionsstab oder Drehstab) ist eine stabförmige Feder. Beim Verdrehen des Stabes um seine Längsachse entstehen in dessen Querschnitten Torsionsspannungen genannte Scherspannungen, die mit dem von außen angebrachten Torsionsmoment im Gleichgewicht sind.


Family.

Publius belonged to the gens Mucia, a noble Plebeian family of Rome, of which the Scaevola were the main branch. Several Scaevola appear in Roman magistracies before the appearance of Publius Mucius, including a certain Publius Mucius Scaevola who served as a tribune of the plebs in 486 BC and a Publius Mutius Scaevola—who, while not of the same branch, clearly belongs to the Scaevola clan—who held the Tribunate of the Soldiers in the same year, suggesting the Scaevola family was an entrenched Republican family of senatorial class from at least 486 BC. In legend the Scaevolas draw their name from a Gaius Mucius Scaevola of 508 BC who supposedly attempted to assassinate the Etruscan king of Clusium, Lars Porsena and upon killing his secretary due to a mix up in interpreting Etruscan dress, thrust his arm into a brazier and declared that 300 young men like him would come for Porsena, leading to the king's withdrawal out of fear for his personal safety.  Publius Mucius Scaevola had a father of the same name, who was consul in 175 BC with Marcus Aemilius Lepidus. Scaevola served as Pontifex Maximus following his younger brother Publius Licinius Crassus Dives Mucianus’s death. Mucianus was elected consul alongside with high priest Lucius Valerius Flaccus in the year 131 BC, after previously serving as Pontifex Maximus in 132 BC.  Scaevola was also the father of Quintus Mucius Scaevola who was a consul in 95 BC. He was responsible for the "Lex Licinia Mucia", which sent Italian residents of Rome falsely claiming citizenship back to their own towns.

Early political career.

Tribune of the Plebs 141 BC.

Publius Mucius Scaevola served as tribune of the plebs in 141 BC. The consuls of this year were Cnaeus Servilius Caepio and Quintus Pompeius. Not much is known of Scaevola’s actions during his year as tribune. Most significant in the historical record is his carrying of a plebiscite which placed Hostilius Tubulus on trial for accepting bribes during his year as praetor in 142 BC.

Praetor 136 BC.

Scaevola was elected as praetor in 136 BC. Lucius Furius Philus and Sextus Atilius Serranus were consuls during this year. During his year as a praetor Scaevola argued vehemently against the citizenship rights of Mancinus, who had demonstrated cowardice the previous year during a campaign in the Numantine War and had subsequently been surrendered to the Numantines as punishment, but had been rejected.

Consulship 133 BC.

Rome during Scaevola’s Consulship.

Publius was elected consul at a pivotal point in Roman history. The Punic Wars had ended only 13 years beforehand and Rome now controlled many new territories around the Mediterranean through military expeditions such as Scipio Aemilianus’ conquest of Numantia in modern-day Spain, as well as the annexation of Asia Minor. Upon the death of Attallus III of Pergamum in 133 BC Rome was granted the land belonging to Attallus in his will. However, an Attalid pretender called Eumenes III tried to retain the lands. In 130 BC he defeated Scaevola’s brother and then-consul Publius Licinius Crassus Dives Mucianus, killing him. Later, Scaevola’s son Quintus Mucius Scaevola engaged in more fighting in the region, though not of clear nature, receiving a proconsular command in 121 BC. The rapid expansionism throughout this period caused internal unrest that had triggered the First Servile War in Sicily two years earlier in 135 BC. Publius Mucius Scaevola’s fellow consul Lucius Calpernius Piso led armies in 133 BC against the slave revolt on the island of Sicily. This revolt was led by a man named Eunus, a semi-prophetic military leader who managed to win several small scale engagements against larger Roman forces. Piso managed to suppress this long-running revolt, though Scaevola had little involvement in this campaign and therefore shared little of the popularity and prestige it incurred. At some point during this period both Piso and Scaevola were absent from the city of Rome, meaning the Senate was presided over by the urban praetor.

Tiberius Gracchus.

Lex Sempronia Agraria.

Before Tiberius Gracchus drew up the highly significant "Lex Agraria", he consulted with the preeminent citizens of the day, including Publius Mucius Scaevola, who was consul at the time. Plutarch describes Publius Mucius as a jurist, foremost in virtue and reputation. Following his consultation with Publius Mucius, among others, Tiberius Gracchus drew up a law that was more lenient towards those occupying public lands than expected.

Jury (Gracchus vs. Scipio).

When conflict between Tiberius Gracchus and his opponents came to a head, Publius Mucius Scaevola publicly refused to support Scipio Nasica’s attempt to depose Tiberius Gracchus. But following Tiberius’ death he tried to restore stability to the Senate, by retrospectively approving the violent events he had previously refused to support. Twentieth century historians have debated whether this about-face demonstrates a man of integrity standing above the factional squabbles in order to secure stability in precarious circumstances, or the propensity to change his factional allegiance at the drop of a hat. For example, in 1965 Erich S. Gruen depicted Publius Mucius Scaevola as a man constantly changing factional allegiance – describing him as a successful but unprincipled politician. In contrast, Wiseman, Bernstein and Badian all argued in the early 1970s for a more favourable assessment of Scaevola as a man of independence and integrity.

Later life.

Pontifex Maximus 130-115 BC.

Publius Mucius Scaevola became Pontifex Maximus in 130 BC, after his brother, Publius Licinius Crassus Dives Mucianus, was killed in battle while fighting in the kingdom of Pergamum. His most notable contribution during this period was the publication of the final "Annales Maximi". The "Annales Maximi" were annals maintained by the Pontifex Maximus, dating back to 400 BC. The Pontifex Maximus, the highest-ranking priest in the Roman Republic, was responsible for recording the names of the magistrates of each year, as well as significant events. The annals ceased being written in the 130s BC, and Publius Mucius Scaevola reportedly published the complete record in his capacity as Pontifex Maximus.

Death.

According to historian David Stockton, the exact year of Publius Mucius Scaevola’s death is unknown. Later juristic writers indicate he was still alive in 121 BC, but he must have died some time before 114 BC, when a different chief pontiff presided over the first trial of the Vestal Virgins.

Publius Mucius Scaevola was a prominent Roman politician and jurist who was consul in 133 BC. In his earlier political career he was tribune of the plebs in 141 BC and praetor in 136 BC. He also held the position of Pontifex Maximus for sixteen years after his consulship and died circa 115 BC.

Publius Mucius Scaevola (* 180 v. Chr.; † 115 v. Chr.) war ein römischer republikanischer Politiker und Jurist. Der Sohn des gleichnamigen Konsuls des Jahres 175 v. Chr. wurde 141 v. Chr. zum Volkstribunen gewählt, 136 zum Prätor. 133 v. Chr. war er gemeinsam mit Lucius Calpurnius Piso Frugi Konsul, als der er die Agrarreformen des Tiberius Gracchus unterstützte, während Piso sie ablehnte. Nach Pomponius gilt Publius Mucius als einer der maßgeblichen Autoren zum römischen Zivilrecht (Ius civile) und schrieb ein zehnbändiges juristisches Werk, das durch Cicero und spätere Juristen teilweise überliefert ist. Er gilt mit seinem Gelehrtenkreis als Mitbegründer einer regelrechten Rechtswissenschaft, die allerdings erst sein Sohn Quintus Mucius Scaevola systematisierte. Im Jahre 131 v. Chr. wurde er als Nachfolger seines verstorbenen Bruders Publius Licinius Crassus Dives Mucianus Pontifex Maximus. Sein Enkel Manius Acilius Glabrio war 67 v. Chr. Konsul.


Background.

On behalf of Charles XII, who had occupied much of the Polish-Lithuanian Commonwealth during the Great Northern War, Stanisław Leszczyński was crowned king of Poland on 4 October 1705. A faction of the commonwealth, organized in the Sandomierz Confederation, remained loyal to Saxon elector Augustus the Strong, Polish king since 1697 and allied against Charles XII with Russian tsar Peter the Great.  The resulting civil war in Poland (1704-1706) did not go well for August. His attempt to regain control in Poland-Lithuania was thwarted by Charles XII in the Battle of Grodno and by Carl Gustav Rehnskiöld in the Battle of Fraustadt, both in the first months of 1706. As a consequence of Fraustadt, the Saxon electorate was virtually undefended, and when Charles XII combined his forces with Rehskiöld and moved through Silesia to occupy it, he met no resistance.

Terms.

Augustus the Strong made peace with the Swedish Empire. He renounced his claims to the Polish crown, accepted Stanisław Leszczyński as the Polish king and had to congratulate him.  Augustus' alliance with Russia, formalized in the Treaty of Preobrazhenskoye and the Treaty of Narva, was declared void. All Russians under Augustus' command were to be handed over to the Swedish as prisoners. Johann Patkul was declared a criminal and likewise taken in Swedish custody.  The treaty was concluded in secrecy.

Consequences.

The diplomat and politician Johann Patkul was executed.  When the treaty was revealed to him, Peter the Great was disappointed. His diplomats, though aware of the possibility of a separate peace between Saxony and Sweden and actually negotiating a separate peace for Russia since 1703, had been unable to intervene. Furthermore, Peter had regarded Augustus not just as an ally, but as a close friend. With Augustus resigned, Peter unsuccessfully offered the yet to be conquered Polish crown to Hungarian rebel prince Francis II Rákóczi, to British general John Churchill, 1st Duke of Marlborough, to Polish Jakub Sobieski, to Savoyan commander in Habsburg service Eugene of Savoy, and to others.  In the Polish-Lithuanian Commonwealth, the treaty had improved Stanisław Leszczyński's position. He subsequently gained the loyalty of part of the lesser gentry.  By the Treaty of Thorn (1709), Augustus the Strong was restored as Polish king and renewed the alliance with Russia which was made possible by Peter the Great's victory over Charles XII in the Battle of Poltava 27 June 1709.

The Treaty of Altranstädt was concluded between Charles XII of Sweden and Augustus the Strong of Saxony and Poland-Lithuania, on 13 October 1706, during the Great Northern War. Augustus had to renounce his claims to the Polish throne and his alliance with Russia.

Der Altranstädter Friede wurde während des Großen Nordischen Krieges am 24. September 1706 im Schloss zu Altranstädt zwischen dem schwedischen König Karl XII. und dem Kurfürsten von Sachsen August II. geschlossen.


Club career.

Early career.

Vonlanthen played as a junior for BSC Young Boys in Switzerland. In the 2001–02 season, he made his debut as a 16-year-old in the Swiss Super League starting eight games and appearing once as a substitute. In the summer of 2003, he was transferred to the Dutch Eredivisie club PSV Eindhoven. On 21 June 2004, Vonlanthen became the youngest goalscorer when he scored for Switzerland vs France in Coimbra, Portugal at the age of 18 years and 141 days old. He made a good initial impression and helped PSV qualify for the UEFA Champions League. After a good first season, he began to lose form and as a consequence was loaned out to Italian side Brescia Calcio for the last six months of the 2004–05 season. He was again loaned out to NAC Breda for the 2005–06 season.

Red Bull Salzburg.

At the start of the 2006–07 season, Vonlanthen transferred to Red Bull Salzburg in the Austrian Bundesliga.  On 13 July 2009, FC Zürich signed the Swiss forward on a season long loan deal. The move was not made permanent, and Vonlanthen returned to Salzburg for the start of the 2010–11 season.

Initial retirement and return.

On 30 May 2012, Vonlanthen announced his retirement at age 26. According to him, he 'could not face the prospect of undergoing a knee operation'.  However, on 13 June 2013, Vonlanthen came out of retirement to join Grasshoppers, signing a one-year deal with an option for a further two years.  On 27 December 2013, Vonlanthen was loaned out to Swiss second division side Schaffhausen, due to a lack of playtime at Grasshoppers.

International career.

On 6 June 2004, Vonlanthen made his senior debut for the Swiss National Team against Liechtenstein. He came on as a substitute in the 81st minute to replace Alexander Frei. Switzerland went on to beat Liechtenstein 1–0.  At Euro 2004, Vonlanthen became the second youngest player to play in the tournament when he came on as a substitute against England. On 21 June 2004, Vonlanthen became the youngest scorer ever in the European Championships when he equalised against France, beating Wayne Rooney's record – which had only been set four days before – by three months.  Since the European Championships in Portugal, Vonlanthen has managed to hold a regular position in the national team and as well as playing frequently in the under 21s side. He was part of the squad that very narrowly lost out to Spain on a place in the 2009 European Championships. He had scored the equaliser in the first leg in a 2–1 win but the Spanish came back to win 3–1 in the second leg and 4–3 on aggregate.  Vonlanthen was called up to the Swiss squad to play in 2006 World Cup, but was unable to participate due to a hamstring injury. He was also called up for Euro 2008.

Johan Vonlanthen Benavídez (born 1 February 1986) is a Swiss former professional footballer who played as a winger. His stepfather is a Swiss national, giving him the right to play for that country.

Johan Vonlanthen Benavídez (* 1. Februar 1986 in Santa Marta, Kolumbien) ist ein ehemaliger Schweizer Fussballspieler kolumbianischer Herkunft. Von 2016 bis 2018 spielte er für den Schweizer Verein FC Wil in der zweitklassigen Challenge League.


Background.

By 1840, European powers had established small trading posts along the coast, but they seldom moved inland, preferring to stay near the sea and mainly just used the continent for trade. Large parts of the continent were essentially uninhabitable for Europeans because of the high mortality rates from diseases such as malaria. In the middle decades of the 19th century, European explorers had mapped areas of East Africa and Central Africa.  Even as late

Causes.

Africa and global markets.

Sub-Saharan Africa, one of the last regions of the world largely untouched by "informal imperialism", was also attractive to Europe's ruling elites for economic, political and social reasons. During a time when Britain's balance of trade showed a growing deficit, with shrinking and increasingly protectionist continental markets due to the Long Depression (1873–96), Africa offered Britain, Germany, France, and other countries an open market that would garner them a trade surplus: a market that bought more from the colonial power than it sold overall.  Surplus capital was often more profitably invested overseas, where cheap materials, limited competition, and abundant raw materials made a greater premium possible. Another inducement for imperialism arose from the demand for raw materials, especially copper, cotton, rubber, palm oil, cocoa, diamonds, tea, and tin, to which European consumers had grown accustomed and upon which European industry had grown dependent.

Strategic rivalry.

The rivalry between Britain, France, Germany, and the other Western European powers accounts for a large part of the colonization.  While tropical Africa was not a large zone of investment, other overseas regions were. The vast interior between Egypt and the gold and diamond-rich southern Africa had strategic value in securing the flow of overseas trade. Britain was under political pressure to secure lucrative markets against encroaching rivals in China and its eastern colonies, particularly India, Malaya, Australia and New Zealand. Thus, it was crucial to secure the key waterway between East and West – the Suez Canal. However, a theory that Britain sought to annex East Africa during the 1880 onwards, out of geostrategic concerns connected to Egypt (especially the Suez Canal), has been challenged by historians such as John Darwin (1997) and Jonas F. Gjersø (2015).  The scramble for African territory also reflected concern for the acquisition of military and naval bases, for strategic purposes and the exercise of power. The growing navies, and new ships driven by steam power, required coaling

Crises prior to World War I.

Colonisation of the Congo.

David Livingstone's explorations, carried on by Henry Morton Stanley, excited imaginations with Stanley's grandiose ideas for colonisation; but these found little support owing to the problems and scale of action required, except from Leopold II of Belgium, who in 1876 had organised the International African Association (the Congo Society). From 1869 to 1874, Stanley was secretly sent by Leopold II to the Congo region, where he made treaties with several African chiefs along the Congo River and by 1882 had sufficient territory to form the basis of the Congo Free State. Leopold II personally owned the colony from 1885 and used it as a source of ivory and rubber.  While Stanley was exploring Congo on behalf of Leopold II of Belgium, the Franco-Italian marine officer Pierre de

Suez Canal.

French diplomat Ferdinand de Lesseps had obtained many concessions from Isma'il Pasha, the Khedive of Egypt and Sudan, in 1854–56, to build the Suez Canal. Some sources estimate the workforce at 30,000, but others estimate that 120,000 workers died over the ten years of construction due to malnutrition, fatigue and disease, especially cholera. Shortly before its completion in 1869, Khedive Isma'il borrowed enormous sums from British and French bankers at high rates of interest. By 1875, he was facing financial difficulties and was forced to sell his block of shares in the Suez Canal. The shares were snapped up by Britain, under its Prime Minister, Benjamin

Berlin Conference (1884–85).

The occupation of Egypt, and the acquisition of the Congo were the first major moves in what came to be a precipitous scramble for African territory. In 1884, Otto von Bismarck convened the 1884–85 Berlin Conference to discuss the African problem. The diplomats put on a humanitarian façade by condemning the slave trade, prohibiting the sale of alcoholic beverages and firearms in certain regions, and by expressing concern for missionary

Britain's administration of Egypt and South Africa.

Britain's administration of Egypt and the Cape Colony contributed to a preoccupation over securing the source of the Nile River. Egypt was taken over by the British in 1882 leaving the Ottoman Empire in a nominal role until 1914, when London made it a protectorate. Egypt was never an actual British colony. Sudan, Nigeria, Kenya, and Uganda were subjugated in the 1890s and early 20th century; and in the south, the Cape Colony (first acquired in 1795) provided a base for the subjugation of neighboring African states and the Dutch Afrikaner settlers who had left the Cape to avoid the British and then

Moroccan Crisis.

Although the 1884–85 Berlin Conference had set the rules for the Scramble for Africa, it had not weakened the rival imperialists. The 1898 Fashoda Incident, which had seen France and the British Empire on the brink of war, ultimately led to the signature of the "Entente Cordiale" of 1904, which countered the influence of the European powers of the Triple Alliance. As a result, the new German Empire decided to test the solidity of such influence, using the contested territory of Morocco as a battlefield.  Thus, Kaiser Wilhelm II visited Tangiers on 31 March 1905 and made a speech in favor of Moroccan independence, challenging French influence in Morocco. France's influence in Morocco had been reaffirmed by Britain and Spain in 1904. The Kaiser's speech bolstered French nationalism, and with British support the French foreign minister, Théophile Delcassé, took a defiant line. The crisis peaked in mid-June 1905, when Delcassé was forced out of the ministry by the more conciliation-minded premier Maurice Rouvier. But by July 1905 Germany was becoming isolated and the French agreed to a conference to solve the crisis. Both France and Germany continued to posture up until the conference,

Dervish resistance.

Following the Berlin Conference at the end of the 19th century, the British, Italians, and Ethiopians sought to claim lands owned by the Somalis such as the Warsangali Sultanate, the Ajuran Sultanate and the Gobroon Dynasty.  The Dervish movement was a state established by Mohammed Abdullah Hassan, a Somali religious leader who gathered Muslim soldiers from across the Horn of Africa and united them into a loyal army known as the "Dervishes". This Dervish

Herero Wars and the Maji-Maji Rebellion.

Between 1904 and 1908, Germany's colonies in German South-West Africa and German East Africa were rocked by separate, contemporaneous native revolts against their rule. In both territories the threat to German rule was quickly defeated once large-scale reinforcements from Germany arrived, with the Herero rebels in German South-West Africa being defeated at the Battle of Waterberg and the Maji-Maji rebels in German East Africa being steadily crushed by

Colonial encounter.

Colonial consciousness and exhibitions.

Colonial lobby.

In its earlier stages, imperialism was generally the act of individual explorers as well as some adventurous merchantmen. The colonial powers were a long way from approving without any dissent the expensive adventures carried out abroad. Various important political leaders, such as Gladstone, opposed colonisation in its first years. However, during his second premiership between 1880 and 1885 he could not resist the colonial lobby in his cabinet, and thus did not execute his electoral promise to disengage from Egypt. Although Gladstone was personally opposed to imperialism, the social tensions caused by the Long Depression pushed him to favor jingoism: the imperialists had become the "parasites of patriotism" (John A. Hobson). In France, then Radical politician Georges Clemenceau also adamantly opposed himself to it: he thought colonisation was a diversion from the "blue line of the Vosges" mountains, that is revanchism and the patriotic

Colonial propaganda and jingoism.

Countering disease.

From the beginning of the 20th century, the elimination or control of disease in tropical countries became a driving force for all colonial powers. The sleeping sickness epidemic in Africa was arrested due to mobile teams systematically screening millions of people at risk. In the 1880s cattle brought from British Asia to feed Italian soldiers invading Erithrea turned

Slavery abolition.

The continuing anti-slavery movement in Western Europe became a reason and an excuse for the conquest and colonization of Africa. It was the central theme of the Brussels Anti-Slavery Conference 1889–90. During the Scramble for Africa, an early but secondary focus of all colonial regimes was the suppression of slavery and the slave trade. In French West Africa, following conquest and abolition by the French, over a million

Colonialism on the eve of World War I.

During the New Imperialism period, by the end of the 19th century, Europe added almost – one-fifth of the land area of the globe – to its overseas colonial possessions. Europe's formal holdings now included the entire African continent except Ethiopia, Liberia, and Saguia el-Hamra, the latter of which would be integrated into Spanish Sahara. Between 1885 and 1914, Britain took nearly 30% of Africa's population under its control; 15% for France, 11% for Portugal, 9% for

African colonies listed by colonising power.

Germany.

After the First World War, Germany's possessions were partitioned among Britain (which took a sliver of

Italy.

During the Interwar period, Italian Ethiopia formed together with Italian Eritrea and

United Kingdom.

The British were primarily interested in maintaining secure communication lines to India, which led to initial interest in Egypt and South Africa. Once these two

Independent states.

Liberia was the only nation in Africa that was a colony and a protectorate of the United States. Liberia was founded, colonised, established and controlled by the American Colonization Society, a private organisation established in order to relocate freed African-American and Caribbean slaves from the United States and the Caribbean

Connections to Modern Day Events.

Anti-neoliberal scholars connect the old scramble to a new scramble for Africa, coinciding with the emergence of a "Afro-Neo-Liberal" capitalist movement in Post-Colonial Africa. When African nations began to gain independence during the Post World War II Era, their post colonial economic structures remained undiversified and linear. In most cases, the

Petro-states.

In the era of globalization, many African countries have emerged as petro-states (for example Sudan, Cameroon, Nigeria, Angola). These are nations with an economic and political partnership between transnational oil companies and the ruling elite class in oil-rich African nations. Numerous countries have entered into a neo-imperial relationship with Africa during this time period. Mary Gilmartin notes that “material and symbolic

The Scramble for Africa, also called the Partition of Africa or the Conquest of Africa, was the invasion, occupation, division, and colonisation of African territory by European powers during a short period known to historians as the New Imperialism (between 1881 and 1914). In 1870, only 10 percent of Africa was under formal European control; by 1914 this had increased to almost 90 percent of the continent, with only Ethiopia (Abyssinia), the Dervish state (a portion of present-day Somalia) and Liberia remaining independent. The European colonialists had several motives: a desire for valuable natural resources, the quest for national prestige, rivalry between European powers, and religious missionary zeal. Internal African native politics also played a role.

Als Wettlauf um Afrika wird die Kolonialisierung des afrikanischen Kontinentes in der Hochphase des Imperialismus zwischen 1880 und dem Ersten Weltkrieg bezeichnet.


History.

Diessenhofen is first mentioned in 757 as "Deozincova". In 2000, the village of Willisdorf was incorporated into the municipality. The earliest traces of a settlement are Stone and Bronze Age scattered objects found in the shallow valleys of the district and on the banks of the Rhine. A hoard of coins from the Roman era (251-270), and the remains of three towers of the Danube-Iller-Rhein limes (4th century) show Roman settlements in the area. The reference in a deed of the Abbey of St. Gall from 757 mention an Alamanni village, which was probably on the plateau south of the church. In 1178 Count Hartmann III of Kyburg, raised the village to town and probably appointed a Ministerialis (unfree knights in the service of a feudal overlord) family as the Stewards of Diessenhofen. By 1245, at the latest, it was the center of a bailiwick and a castle was built in town. Compared to Schaffhausen and Stein am Rhein, Diessenhofen was a modest market town. Hartmann granted the town a measure of freedom and self-government. He used the castle and bridge over the Rhine to secure and defend his possessions on both sides of the Rhine. After the extinction of the Kyburgs and the transition of sovereignty to the Habsburgs in 1264, the city developed as one of the cornerstones of the region. The Stewards, who sat at Unterhof Castle, temporarily united both the Vogt and "Schultheiss" offices into a single person. By 1320 the citizens chose a Town Council of 8-12 members, and during the 15th Century this evolved into a 24-28 member Grand Town Council. In 1349, the Duke of Austria wrestled the Vogt office away from the Stewards and granted it to another Ministerialis. These two developments made citizenship rights in Diessenhofen increasingly important.  The loss of Habsburg influence and the decline of the Habsburg supported Stewards gradually transformed the city between 1415 and 1442. Diessenhofen was captured in 1460 during the conquest of Thurgau by the Swiss Confederation. The town was besieged for ten days before it was captured, but, much like Frauenfeld, it retained certain privileges in the new Thurgau. These included both the high and low courts and the recently acquired customs, tax and "Vogt" rights, as well as the castle. After 1574 they also acquired the rights over the left bank of the Rhine and the possessions of Paradise Monastery. During the 16th Century, the town gained the low court rights over most of the modern Diessenhofen District. Diessenhofen's court decisions did not have to be approved by the Governor in Frauenfeld, but went directly to the Confederation Council, in contrast to the rest of Thurgau. The city only had to pay homage to the Governor every two years.  The patronage of the church of St. Dionysius is mentioned in 1468. The advowson right in the 12th Century was included in the possession of the Count of Thurgau. By 1230 it was held by the Kyburgs. In 1264, the city government through the Habsburgs had the right. Practically, the citizens exercised the right after 1383 and it was legally confirmed in 1415. When the Protestant Reformation entered the town in 1524, many citizens converted to the new religion. The Mass was abolished in 1529 by the Protestant pastor, who also ordered the confiscation of Catholic Church property. Diessenhofen supported the city of Zurich in the Second war of Kappel in 1531, which ended in a Protestant defeat. After the defeat, the Catholic cantons reintroduced the Mass in 1532. The church remained a shared church until the construction of the Catholic Church in 1966–67. In 1349 the Jewish community of Diessenhofen was driven out of town. In 1401 a Jewish man was successfully prosecuted for ritual murder, which led to further persecution in Winterthur and Schaffhausen. During the 13th Century, two Nunneries were built between Diessenhofen and Schaffhausen, St. Katharinental and Paradise, which remained in operation until their dissolution in the 19th Century. Since the Early Middle Ages the district has been known as the granary of the Thurgau due to its extensive and productive grain fields. Additionally, already in the 9th Century widespread viticulture is mentioned in the Upper Rhine region. Until the 19th Century, the city was largely self-supplied for food and served as a collection point for food heading from the farms to the surrounding cities (especially Schaffhausen and Zurich). In the 12th Century are a weekly markets in town. Starting in 1387 two yearly markets opened and by the 19th Century that it had increased to eight yearly markets. The local industry covered only the simplest needs of the city and its limited market area. The limited industry and area provided too little money to support the formation of trade-oriented guilds. Diessenhofen was never dominated by a single industry or a special craft. It never was involved in the international Lake Constance trade industry. The town's location on the Rhine favored the development of fisheries, and it benefited from the salt trade. Bridge and transit tariffs provided the majority of the town's income until the abolition of internal tariffs in 1848.

Geography.

Diessenhofen has an area,, of. Of this area, or 41.9% is used for agricultural purposes, while or 36.3% is forested. Of the rest of the land, or 16.0% is settled (buildings or roads), or 5.0% is either rivers or lakes.  Of the built up area, industrial buildings made up 5.3% of the total area while housing and buildings made up 1.8% and transportation infrastructure made up 1.8%. Power and water infrastructure as well as other special developed areas made up 1.3% of the area while parks, green belts and sports fields made up 5.8%. Out of the forested land, 33.9% of the total land area is heavily forested and 2.4% is covered with orchards or small clusters of trees. Of the agricultural land, 40.9% is used for growing crops. All the water in the municipality is flowing water.  The municipality borders on north with Germany and the canton of Schaffhausen. On the east is the district of Steckborn, on the west the municipality of Schlatt, and on the south the municipality of Basadingen-Schlattingen.

Demographics.

Diessenhofen has a population () of, 30.7% of the population are foreign nationals. Over the last 10 years (1997–2007) the population has changed at a rate of 1.5%. Most of the population () speaks German(81.8%), with Italian being second most common ( 5.2%) and Albanian being third ( 3.9%). , the gender distribution of the population was 50.7% male and 49.3% female. The population was made up of 1,088 Swiss men (33.4% of the population), and 564 (17.3%) non-Swiss men. There were 1,171 Swiss women (35.9%), and 437 (13.4%) non-Swiss women.  In there were 23 live births to Swiss citizens and 14 births to non-Swiss citizens, and in same time span there were 17 deaths of Swiss citizens and 3 non-Swiss citizen deaths. Ignoring immigration and emigration, the population of Swiss citizens increased by 6 while the foreign population increased by 11. There were 4 Swiss men who emigrated from Switzerland to another country, 2 Swiss women who emigrated from Switzerland to another country, 34 non-Swiss men who emigrated from Switzerland to another country and 28 non-Swiss women who emigrated from Switzerland to another country. The total Swiss population change in 2008 (from all sources) was an increase of 25 and the non-Swiss population change was an increase of 32 people. This represents a population growth rate of 1.8%.  The age distribution,, in Diessenhofen is; 297 children or 9.0% of the population are between 0 and 9 years old and 384 teenagers or 11.6% are between 10 and 19. Of the adult population, 430 people or 13.0% of the population are between 20 and 29 years old. 460 people or 13.9% are between 30 and 39, 516 people or 15.6% are between 40 and 49, and 472 people or 14.2% are between 50 and 59. The senior population distribution is 418 people or 12.6% of the population are between 60 and 69 years old, 224 people or 6.8% are between 70 and 79, there are 99 people or 3.0% who are between 80 and 89, and there are 14 people or 0.4% who are 90 and older. , there were 1,325 private households in the municipality, and an average of 2.4 persons per household. there were 338 single family homes (or 67.1% of the total) out of a total of 504 inhabited buildings. There were 46 two family buildings (9.1%), 36 three family buildings (7.1%) and 84 multi-family buildings (or 16.7%). There were 732 (or 22.7%) persons who were part of a couple without children, and 1,722 (or 53.4%) who were part of a couple with children. There were 175 (or 5.4%) people who lived in single parent home, while there are 6 persons who were adult children living with one or both parents, 9 persons who lived in a household made up of relatives, 23 who lived in a household made up of unrelated persons, and 107 who are either institutionalized or live in another type of collective housing.  The vacancy rate for the municipality,, was 1.72%., the construction rate of new housing units was 0.6 new units per 1000 residents. there were 1,528 apartments in the municipality. The most common apartment size was the 4 room apartment of which there were 471. There were 88 single room apartments and 162 apartments with six or more rooms. the average price to rent an average apartment in Diessenhofen was 900.07 Swiss francs (CHF) per month (US$720, £410, €580 approx. exchange rate from 2000). The average rate for a one-room apartment was 649.00 CHF (US$520, £290, €420), a two-room apartment was about 628.34 CHF (US$500, £280, €400), a three-room apartment was about 788.98 CHF (US$630, £360, €500) and a six or more room apartment cost an average of 1614.00 CHF (US$1290, £730, €1030). The average apartment price in Diessenhofen was 80.7% of the national average of 1116 CHF.  In the 2007 federal election the most popular party was the SVP which received 43.96% of the vote. The next three most popular parties were the SP (16.98%), the CVP (12.61%) and the FDP (11.63%). In the federal election, a total of 909 votes were cast, and the voter turnout was 47.3%.  The historical population is given in the following table:

Heritage sites of national significance.

Unterhof Castle, City Archives in the Rathaus (Town council house), Former Dominican Monastery of St. Katharinental with its granary, the "zum Goldenen Löwen" House and the "Goldener Leuen" Collection, the "Rheinbrücke" (Rhine Bridge), and the city walls with "Siegelturm" are listed as Swiss heritage site of national significance. Both the town of Diessenhofen and St Katharinental Monastery are part of the Inventory of Swiss Heritage Sites.

Architecture.

The "Altstadt" of Diessenhofen centers on the "Siegelturm" a tower in which seals and official documents are kept. The old covered bridge over the Rhine was rebuilt in 1814. It leads to Gailingen am Hochrhein in Baden-Württemberg in Germany.

Economy.

, Diessenhofen had an unemployment rate of 1.59%., there were 74 people employed in the primary economic sector and about 25 businesses involved in this sector. 668 people are employed in the secondary sector and there are 54 businesses in this sector. 901 people are employed in the tertiary sector, with 143 businesses in this sector.

Religion.

From the, 955 or 29.6% were Roman Catholic, while 1,335 or 41.4% belonged to the Swiss Reformed Church. Of the rest of the population, there were 3 Old Catholics (or about 0.09% of the population) who belonged to the Christian Catholic Church of Switzerland there are 98 individuals (or about 3.04% of the population) who belong to the Orthodox Church, and there are 96 individuals (or about 2.97% of the population) who belong to another Christian church. There were 3 individuals (or about 0.09% of the population) who were Jewish, and 318 (or about 9.85% of the population) who are Islamic. There are 11 individuals (or about 0.34% of the population) who belong to another church (not listed on the census), 269 (or about 8.34% of the population) belong to no church, are agnostic or atheist, and 139 individuals (or about 4.31% of the population) did not answer the question.

Transport.

Diessenhofen sits on the Lake Line between Schaffhausen and Rorschach and has two railway stations: Diessenhofen, at the southern edge of the city center, and St. Katharinental, near the former Dominican monastery, west of the city center.

Education.

The entire Swiss population is generally well educated. In Diessenhofen about 66.5% of the population (between age 25-64) have completed either non-mandatory upper secondary education or additional higher education (either university or a "Fachhochschule").  Diessenhofen is home to the Diessenhofen primary and secondary school district.In the 2008/2009 school year there are 691 students at either the primary or secondary levels. There are 131 children in the kindergarten, and the average class size is 18.71 kindergartners. Of the children in kindergarten, 63 or 48.1% are female, 32 or 24.4% are not Swiss citizens and 30 or 22.9% do not speak German natively. The lower and upper primary levels begin at about age 5-6 and lasts for 6 years. There are 200 children in who are at the lower primary level and 236 children in the upper primary level. The average class size in the primary school is 20.1 students. At the lower primary level, there are 93 children or 46.5% of the total population who are female, 52 or 26.0% are not Swiss citizens and 48 or 24.0% do not speak German natively. In the upper primary level, there are 100 or 42.4% who are female, 59 or 25.0% are not Swiss citizens and 63 or 26.7% do not speak German natively. At the secondary level, students are divided according to performance.  The secondary level begins at about age 12 and usually lasts 3 years. There are 220 teenagers who are in the advanced school, of which 125 or 56.8% are female, 45 or 20.5% are not Swiss citizens and 41 or 18.6% do not speak German natively. There are 19 teenagers who are in the standard school, of which 8 or 42.1% are female, 5 or 26.3% are not Swiss citizens and 5 or 26.3% do not speak German natively. Finally, there are 16 teenagers who are in special or remedial classes, of which 6 or 37.5% are female, 8 or 50.0% are not Swiss citizens and 8 or 50.0% do not speak German natively. The average class size for all classes at the secondary level is 19.92 students.

Diessenhofen is a village and a municipality in Frauenfeld District in the canton of Thurgau in Switzerland. The village is situated on the south shore of the High Rhine just opposite the German town of Gailingen am Hochrhein.

Diessenhofen ist eine Kleinstadt und eine Gemeinde im Bezirk Frauenfeld des Kantons Thurgau in der Schweiz. Die seit 2000 bestehende politische Gemeinde umfasst die ehemalige Munizipalgemeinde Diessenhofen mit deren Ortsgemeinden Diessenhofen und Willisdorf. Von 1798 bis 2010 war Diessenhofen Hauptort des damaligen gleichnamigen Bezirks.


Background.

Nils Dacke and his peasants were dissatisfied with the policies of the Swedish king Gustav Vasa. In his effort to modernize Sweden and gain more power, the king had instituted a more efficient system for tax collection. The heavy tax burden angered many peasants.  Gustav Vasa had also broken relations with Rome and promoted Lutheranism instead of Catholicism in order to confiscate the property of the church (including land), effected by the laws of the Reduction of Gustav I of Sweden. In 1541 the king's men had confiscated many of the belongings of the churches in Småland, such as the church silver and even the church bells, to finance the army. Dacke criticized the new church order and promoted the old faith. He was also supported by many local priests.  The rebellion was one of many rebellions during the rule of Gustav Vasa. In contrast with other contemporary rebellions in Sweden, this one was led by peasants, and not supported by the local nobility.

Uprising.

The uprising began in summer 1542 when the king's bailiffs were attacked and killed when they came to collect taxes. Gustav Vasa responded by sending a military force led by his own father-in-law. He was defeated by Dacke's constantly growing army of peasants. Other attempts to defeat Dacke militarily also failed. Dacke and his army reached as far north as Mjölby at the fringes of the Östergötland plains and enjoyed widespread support around Sommen and Ydre.  Local aristocrat Måns Johansson Natt och Dag sided with the king despite having a troublesome relation to him. He was put in charge of an army to suppress the rebellion.  Next, the Swedish government stopped all supplies of provisions and other necessities to the region. This weakened the rebellion considerably. Defaming propaganda about Dacke was also spread by the government, labeling him a traitor and a heretic.  In March 1543 Gustav Vasa ordered his army of Swedish recruits and German landsknecht mercenaries to attack Småland. This time larger forces were deployed, and Dacke's forces were attacked from two directions – from Östergötland and Västergötland. The uprising was defeated, and Dacke was wounded but managed to flee.

Aftermath.

The king's revenge on the instigators of the rebellion was hard. The leaders that were caught were executed together with the priests who had supported Dacke. Peasants who had supported the rebellion were deported to Finland, where they had to serve in the army, and the counties where the rebellion had taken place had to pay a large fine to the king.  Dacke himself was caught and killed in August 1543 when trying to escape the country. According to legend, his body was taken to Kalmar, where his head was publicly displayed wearing a crown of copper, as a warning to others.  Unrest in Ydre continued well after the death of Dacke and ended only after Gustav Vasa sent a force of 400 men to pacify the hundred.  The rebellion was the most serious threat to the rule of Gustav Vasa, but after having defeated it he managed to consolidate his power, concentrating more and more power in the hands of the monarch.

Idiom.

In the Swedish language, the idiom "[something] hasn't happened since the Dacke War" is used to mean "[something] hasn't happened for a long time". This expression is especially common in the southern parts of Sweden but is also used elsewhere.

The Dacke War () was a peasant uprising led by Nils Dacke in Småland, Sweden, in 1542 against the rule of Gustav Vasa. Dacke and his followers were dissatisfied with the heavy tax burden, the introduction of Lutheranism, and the confiscation of Church property (the confiscation and taxes were introduced to pay for the Swedish War of Liberation that had brought Gustav Vasa to power). In 1543 the uprising was defeated, and Nils Dacke was killed.

Der Dacke-Aufstand (schwedisch "Dackefejden") war ein Bauernaufstand in der schwedischen historischen Provinz Småland in den Jahren 1542 und 1543. Er war der größte genuine Bauernaufstand in Skandinavien und drohte auf seinem Höhepunkt, den regierenden König Gustav Wasa zu stürzen.


Properties.

Relation to Euclidean geometry.

Hyperbolic geometry is more closely 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, Euclidean 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 also has many consequences: concepts that are equivalent in Euclidean geometry are not equivalent in hyperbolic geometry; new concepts need to be introduced. Further, because of the angle of parallelism, hyperbolic geometry has an absolute scale, a relation between distance and angle measurements.

Lines.

Single lines in hyperbolic geometry have exactly the same properties as single straight lines in Euclidean geometry. For example, two points uniquely define a line, and line segments can be infinitely extended.  Two intersecting lines have the same properties as two intersecting lines in Euclidean geometry. For example, two distinct lines can intersect in no more than one point, intersecting lines form equal opposite angles, and adjacent angles of intersecting lines are supplementary.  When a third line is introduced, then there can be properties of intersecting lines that differ from intersecting lines in Euclidean geometry. For example, given two intersecting lines there are infinitely many lines that do not intersect either of the given lines.  These properties are all independent of the model used, even if the lines may look radically different.

Non-intersecting / parallel lines.

Non-intersecting lines in hyperbolic geometry also have properties that differ from non-intersecting lines in Euclidean geometry:  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: Some geometers simply use "parallel" lines instead of "limiting parallel" lines, with "ultraparallel" lines being just "non-intersecting".  These limiting parallels make an angle "θ" with "PB"; this angle depends only on the Gaussian curvature of the plane and the distance "PB" and is called the angle of parallelism.  For ultraparallel lines, the ultraparallel theorem states that there is a unique line in the hyperbolic plane that is perpendicular to each pair of ultraparallel lines.

Circles and disks.

In hyperbolic geometry, the circumference of a circle of radius "r" is greater than formula_1.  Let formula_2, where formula_3 is the Gaussian curvature of the plane. In hyperbolic geometry, formula_4 is negative, so the square root is of a positive number.  Then the circumference of a circle of radius "r" is equal to:  And the area of the enclosed disk is:  Therefore, in hyperbolic geometry the ratio of a circle's circumference to its radius is always strictly greater than formula_7, though it can be made arbitrarily close by selecting a small enough circle.  If the Gaussian curvature of the plane is −1 then the geodesic curvature of a circle of radius "r" is: formula_8

Hypercycles and horocycles.

In hyperbolic geometry, there is no line that remains equidistant from another. Instead, the points that all have the same orthogonal distance from a given line lie on a curve called a hypercycle.  Another special curve is the horocycle, a curve whose normal radii (perpendicular lines) are all limiting parallel to each other (all converge asymptotically in one direction to the same ideal point, the centre of the horocycle).  Through every pair of points there are two horocycles. The centres of the horocycles are the ideal points of the perpendicular bisector of the line-segment between them.  Given any three distinct points, they all lie on either a line, hypercycle, horocycle, or circle.  The length of the line-segment is the shortest length between two points. The arc-length of a hypercycle connecting two points is longer than that of the line segment and shorter than that of a horocycle, connecting the same two points. The arclength of both horocycles connecting two points are equal. The arc-length of a circle between two points is larger than the arc-length of a horocycle connecting two points.  If the Gaussian curvature of the plane is −1 then the geodesic curvature of a horocycle is 1 and of a hypercycle is between 0 and 1.

Triangles.

Unlike Euclidean triangles, where the angles always add up to π radians (180°, a straight angle), in hyperbolic geometry the sum of the angles of a hyperbolic triangle is always strictly less than π radians (180°, a straight angle). The difference is referred to as the defect.  The area of a hyperbolic triangle is given by its defect in radians multiplied by "R". As a consequence, all hyperbolic triangles have an area that is less than or equal to "R"π. The area of a hyperbolic ideal triangle in which all three angles are 0° is equal to this maximum.  As in Euclidean geometry, each hyperbolic triangle has an incircle. In hyperbolic geometry, if all three of its vertices lie on a horocycle or hypercycle, then the triangle has no circumscribed circle.  As in spherical and elliptical geometry, in hyperbolic geometry if two triangles are similar, they must be congruent.

Regular apeirogon.

A special polygon in hyperbolic geometry is the regular apeirogon, a uniform polygon with an infinite number of sides.  In Euclidean geometry, the only way to construct such a polygon is to make the side lengths tend to zero and the apeirogon is indistinguishable from a circle, or make the interior angles tend to 180 degrees and the apeirogon approaches a straight line.  However, in hyperbolic geometry, a regular apeirogon has sides of any length (i.e., it remains a polygon).  The side and angle bisectors will, depending on the side length and the angle between the sides, be limiting or diverging parallel (see lines above). If the bisectors are limiting parallel the apeirogon can be inscribed and circumscribed by concentric horocycles.  If the bisectors are diverging parallel then a pseudogon (distinctly different from an apeirogon) can be inscribed in hypercycles (all vertices are the same distance of a line, the axis, also the midpoint of the side segments are all equidistant to the same axis.)

Tessellations.

Like the Euclidean plane it is also possible to tessellate the hyperbolic plane with regular polygons as faces.  There are an infinite number of uniform tilings based on the Schwarz triangles ("p" "q" "r") where 1/"p" + 1/"q" + 1/"r" < 1, where "p", "q", "r" are each orders of reflection symmetry at three points of the fundamental domain triangle, the symmetry group is a hyperbolic triangle group. There are also infinitely many uniform tilings that cannot be generated from Schwarz triangles, some for example requiring quadrilaterals as fundamental domains.

Standardized Gaussian curvature.

Though hyperbolic geometry applies for any surface with a constant negative Gaussian curvature, it is usual to assume a scale in which the curvature "K" is −1.  This results in some formulas becoming simpler. Some examples are:

Cartesian-like coordinate systems.

In hyperbolic geometry, the sum of the angles of a quadrilateral is always less than 360 degrees, and hyperbolic rectangles differ greatly from Euclidean rectangles since there are no equidistant lines, so a proper Euclidean rectangle would need to be enclosed by two lines and two hypercycles. These all complicate coordinate systems.  There are however different coordinate systems for hyperbolic plane geometry. All are based around choosing a point (the origin) on a chosen directed line (the "x"-axis) and after that many choices exist.  The Lobachevski coordinates "x" and "y" are found by dropping a perpendicular onto the "x"-axis. "x" will be the label of the foot of the perpendicular. "y" will be the distance along the perpendicular of the given point from its foot (positive on one side and negative on the other).  Another coordinate system measures the distance from the point to the horocycle through the origin centered around formula_9 and the length along this horocycle.  Other coordinate systems use the Klein model or the Poincare disk model described below, and take the Euclidean coordinates as hyperbolic.

Distance.

Construct a Cartesian-like coordinate system as follows. Choose a line (the "x"-axis) in the hyperbolic plane (with a standardized curvature of −1) and label the points on it by their distance from an origin ("x"=0) point on the "x"-axis (positive on one side and negative on the other). For any point in the plane, one can define coordinates "x" and "y" by dropping a perpendicular onto the "x"-axis. "x" will be the label of the foot of the perpendicular. "y" will be the distance along the perpendicular of the given point from its foot (positive on one side and negative on the other). Then the distance between two such points will be  This formula can be derived from the formulas about hyperbolic triangles.  The corresponding metric tensor is: formula_11.  In this coordinate system, straight lines are either perpendicular to the "x"-axis (with equation "x" = a constant) or described by equations of the form where "A" and "B" are real parameters which characterize the straight line.

History.

Since the publication of Euclid's Elements circa 300 BCE, many geometers made attempts to prove the parallel postulate. Some tried to prove it by assuming its negation and trying to derive a contradiction. Foremost among these were Proclus, Ibn al-Haytham (Alhacen), Omar Khayyám, Nasīr al-Dīn al-Tūsī, Witelo, Gersonides, Alfonso, and later Giovanni Gerolamo Saccheri, John Wallis, Johann Heinrich Lambert, and Legendre. Their attempts were doomed to failure (as we now know, the parallel postulate is not provable from the other postulates), but their efforts led to the discovery of hyperbolic geometry.  The theorems of Alhacen, Khayyam and al-Tūsī on quadrilaterals, including the Ibn al-Haytham–Lambert quadrilateral and Khayyam–Saccheri quadrilateral, were the first theorems on hyperbolic geometry. Their works on hyperbolic geometry had a considerable influence on its development among later European geometers, including Witelo, Gersonides, Alfonso, John Wallis and Saccheri.  In the 18th century, Johann Heinrich Lambert introduced the hyperbolic functions and computed the area of a hyperbolic triangle.

19th-century developments.

In the 19th century, hyperbolic geometry was explored extensively by Nikolai Ivanovich Lobachevsky, János Bolyai, Carl Friedrich Gauss and Franz Taurinus. Unlike their predecessors, who just wanted to eliminate the parallel postulate from the axioms of Euclidean geometry, these authors realized they had discovered a new geometry. Gauss wrote in an 1824 letter to Franz Taurinus that he had constructed it, but Gauss did not publish his work. Gauss called it "non-Euclidean geometry" causing several modern authors to continue to consider "non-Euclidean geometry" and "hyperbolic geometry" to be synonyms. Taurinus published results on hyperbolic trigonometry in 1826, argued that hyperbolic geometry is self consistent, but still believed in the special role of Euclidean geometry. The complete system of hyperbolic geometry was published by Lobachevsky in 1829/1830, while Bolyai discovered it independently and published in 1832.  In 1868, Eugenio Beltrami provided models (see below) of hyperbolic geometry, and used this to prove that hyperbolic geometry was consistent if and only if Euclidean geometry was.  The term "hyperbolic geometry" was introduced by Felix Klein in 1871. Klein followed an initiative of Arthur Cayley to use the transformations of projective geometry to produce isometries. The idea used a conic section or quadric to define a region, and used cross ratio to define a metric. The projective transformations that leave the conic section or quadric stable are the isometries. "Klein showed that if the Cayley absolute is a real curve then the part of the projective plane in its interior is isometric to the hyperbolic plane..."  For more history, see article on non-Euclidean geometry, and the references Coxeter and Milnor.

Philosophical consequences.

The discovery of hyperbolic geometry had important philosophical consequences. Before its discovery many philosophers (for example Hobbes and Spinoza) viewed philosophical rigour in terms of the "geometrical method", referring to the method of reasoning used in "Euclid's Elements".  Kant in the "Critique of Pure Reason" came to the conclusion that space (in Euclidean geometry) and time are not discovered by humans as objective features of the world, but are part of an unavoidable systematic framework for organizing our experiences.  It is said that Gauss did not publish anything about hyperbolic geometry out of fear of the "uproar of the Boeotians", which would ruin his status as "princeps mathematicorum" (Latin, "the Prince of Mathematicians"). The "uproar of the Boeotians" came and went, and gave an impetus to great improvements in mathematical rigour, analytical philosophy and logic. Hyperbolic geometry was finally proved consistent and is therefore another valid geometry.

Geometry of the universe (spatial dimensions only).

Because Euclidean, hyperbolic and elliptic geometry are all consistent, the question arises: which is the real geometry of space, and if it is hyperbolic or elliptic, what is its curvature?  Lobachevsky had already tried to measure the curvature of the universe by measuring the parallax of Sirius and treating Sirius as the ideal point of an angle of parallelism. He realised that his measurements were not precise enough to give a definite answer, but he did reach the conclusion that if the geometry of the universe is hyperbolic, then the absolute length is at least one million times the diameter of the earth's orbit (, 10 parsec). Some argue that his measurements were methodologically flawed.  Henri Poincaré, with his sphere-world thought experiment, came to the conclusion that everyday experience does not necessarily rule out other geometries.  The geometrization conjecture gives a complete list of eight possibilities for the fundamental geometry of our space. The problem in determining which one applies is that, to reach a definitive answer, we need to be able to look at extremely large shapes – much larger than anything on Earth or perhaps even in our galaxy.

Geometry of the universe (special relativity).

Special relativity places space and time on equal footing, so that one considers the geometry of a unified spacetime instead of considering space and time separately. Minkowski geometry replaces Galilean geometry (which is the three-dimensional Euclidean space with time of Galilean relativity).  In relativity, rather than considering Euclidean, elliptic and hyperbolic geometries, the appropriate geometries to consider are Minkowski space, de Sitter space and anti-de Sitter space, corresponding to zero, positive and negative curvature respectively.  Hyperbolic geometry enters special relativity through rapidity, which stands in for velocity, and is expressed by a hyperbolic angle. The study of this velocity geometry has been called kinematic geometry. The space of relativistic velocities has a three-dimensional hyperbolic geometry, where the distance function is determined from the relative velocities of "nearby" points (velocities).

Physical realizations of the hyperbolic plane.

The hyperbolic plane is a plane where every point is a saddle point. There exist various pseudospheres in Euclidean space that have a finite area of constant negative Gaussian curvature.  By Hilbert's theorem, it is not possible to isometrically immerse a complete hyperbolic plane (a complete regular surface of constant negative Gaussian curvature) in a three-dimensional Euclidean space.  Other useful models of hyperbolic geometry exist in Euclidean space, in which the metric is not preserved. A particularly well-known paper model based on the pseudosphere is due to William Thurston.  The art of crochet has been used (see ) to demonstrate hyperbolic planes with the first being made by Daina Taimiņa.  In 2000, Keith Henderson demonstrated a quick-to-make paper model dubbed the "hyperbolic soccerball" (more precisely, a truncated order-7 triangular tiling).  Instructions on how to make a hyperbolic quilt, designed by Helaman Ferguson, have been made available by Jeff Weeks.

Models of the hyperbolic plane.

There are different pseudospherical surfaces that have for a large area a constant negative Gaussian curvature, the pseudosphere being the best well known of them.  But it is easier to do hyperbolic geometry on other models.  There are four models commonly used for hyperbolic geometry: the Klein model, the Poincaré disk model, the Poincaré half-plane model, and the Lorentz or hyperboloid model. These models define a hyperbolic plane which satisfies the axioms of a hyperbolic geometry. Despite their names, the first three mentioned above were introduced as models of hyperbolic space by Beltrami, not by Poincaré or Klein. All these models are extendable to more dimensions.

The Beltrami–Klein model.

The Beltrami–Klein model, also known as the projective disk model, Klein disk model and Klein model, is named after Eugenio Beltrami and Felix Klein.  For the two dimensions this model uses the interior of the unit circle for the complete hyperbolic plane, and the chords of this circle are the hyperbolic lines.  For higher dimensions this model uses the interior of the unit ball, and the chords of this "n"-ball are the hyperbolic lines.

The Poincaré disk model.

The Poincaré disk model, also known as the conformal disk model, also employs the interior of the unit circle, but lines are represented by arcs of circles that are orthogonal to the boundary circle, plus diameters of the boundary circle.

The Poincaré half-plane model.

The Poincaré half-plane model takes one-half of the Euclidean plane, bounded by a line "B" of the plane, to be a model of the hyperbolic plane. The line "B" is not included in the model.  The Euclidean plane may be taken to be a plane with the Cartesian coordinate system and the x-axis is taken as line "B" and the half plane is the upper half ("y" > 0 ) of this plane.

The hyperboloid model.

The hyperboloid model or Lorentz model employs a 2-dimensional hyperboloid of revolution (of two sheets, but using one) embedded in 3-dimensional Minkowski space. This model is generally credited to Poincaré, but Reynolds says that Wilhelm Killing used this model in 1885

The hemisphere model.

The hemisphere model is not often used as model by itself, but it functions as a useful tool for visualising transformations between the other models.  The hemisphere model uses the upper half of the unit sphere: formula_14  The hemisphere model is part of a Riemann sphere, and different projections give different models of the hyperbolic plane:  See further: Connection between the models (below)

The Gans model.

In 1966 David Gans proposed a flattened hyperboloid model in the journal American Mathematical Monthly. It is an orthographic projection of the hyperboloid model onto the xy-plane. This model is not as widely used as other models but nevertheless is quite useful in the understanding of hyperbolic geometry.

The band model.

The band model employs a portion of the Euclidean plane between two parallel lines. Distance is preserved along one line through the middle of the band. Assuming the band is given by formula_23, the metric is given by formula_24.

Connection between the models.

All models essentially describe the same structure. The difference between them is that they represent different coordinate charts laid down on the same metric space, namely the hyperbolic plane. The characteristic feature of the hyperbolic plane itself is that it has a constant negative Gaussian curvature, which is indifferent to the coordinate chart used. The geodesics are similarly invariant: that is, geodesics map to geodesics under coordinate transformation. Hyperbolic geometry generally is introduced in terms of the geodesics and their intersections on the hyperbolic plane.  Once we choose a coordinate chart (one of the "models"), we can always embed it in a Euclidean space of same dimension, but the embedding is clearly not isometric (since the curvature of Euclidean space is 0). The hyperbolic space can be represented by infinitely many different charts; but the embeddings in Euclidean space due to these four specific charts show some interesting characteristics.  Since the four models describe the same metric space, each can be transformed into the other.  See, for example:

Isometries of the hyperbolic plane.

Every isometry (transformation or motion) of the hyperbolic plane to itself can be realized as the composition of at most three reflections. In "n"-dimensional hyperbolic space, up to "n"+1 reflections might be required. (These are also true for Euclidean and spherical geometries, but the classification below is different.)  All the isometries of the hyperbolic plane can be classified into these classes:

Hyperbolic geometry in art.

M. C. Escher's famous prints "Circle Limit III" and "Circle Limit IV" illustrate the conformal disc model (Poincaré disk model) quite well. The white lines in "III" are not quite geodesics (they are hypercycles), but are close to them. It is also possible to see quite plainly the negative curvature of the hyperbolic plane, through its effect on the sum of angles in triangles and squares.  For example, in "Circle Limit III" every vertex belongs to three triangles and three squares. In the Euclidean plane, their angles would sum to 450°; i.e., a circle and a quarter. From this, we see that the sum of angles of a triangle in the hyperbolic plane must be smaller than 180°. Another visible property is exponential growth. In "Circle Limit III", for example, one can see that the number of fishes within a distance of "n" from the center rises exponentially. The fishes have an equal hyperbolic area, so the area of a ball of radius "n" must rise exponentially in "n".  The art of crochet has been used to demonstrate hyperbolic planes (pictured above) with the first being made by Daina Taimiņa, whose book "Crocheting Adventures with Hyperbolic Planes" won the 2009 Bookseller/Diagram Prize for Oddest Title of the Year.  HyperRogue is a roguelike game set on various tilings of the hyperbolic plane.

Higher dimensions.

Hyperbolic geometry is not limited to 2 dimensions; a hyperbolic geometry exists for every higher number of dimensions.

Homogeneous structure.

Hyperbolic space of dimension "n" is a special case of a Riemannian symmetric space of noncompact type, as it is isomorphic to the quotient The orthogonal group acts by norm-preserving transformations on Minkowski space R, and it acts transitively on the two-sheet hyperboloid of norm 1 vectors. Timelike lines (i.e., those with positive-norm tangents) through the origin pass through antipodal points in the hyperboloid, so the space of such lines yields a model of hyperbolic "n"-space. The stabilizer of any particular line is isomorphic to the product of the orthogonal groups O("n") and O(1), where O("n") acts on the tangent space of a point in the hyperboloid, and O(1) reflects the line through the origin. Many of the elementary concepts in hyperbolic geometry can be described in linear algebraic terms: geodesic paths are described by intersections with planes through the origin, dihedral angles between hyperplanes can be described by inner products of normal vectors, and hyperbolic reflection groups can be given explicit matrix realizations.  In small dimensions, there are exceptional isomorphisms of Lie groups that yield additional ways to consider symmetries of hyperbolic spaces. For example, in dimension 2, the isomorphisms allow one to interpret the upper half plane model as the quotient and the Poincaré disc model as the quotient. In both cases, the symmetry groups act by fractional linear transformations, since both groups are the orientation-preserving stabilizers in of the respective subspaces of the Riemann sphere. The Cayley transformation not only takes one model of the hyperbolic plane to the other, but realizes the isomorphism of symmetry groups as conjugation in a larger group. In dimension 3, the fractional linear action of on the Riemann sphere is identified with the action on the conformal boundary of hyperbolic 3-space induced by the isomorphism. This allows one to study isometries of hyperbolic 3-space by considering spectral properties of representative complex matrices. For example, parabolic transformations are conjugate to rigid translations in the upper half-space model, and they are exactly those transformations that can be represented by unipotent upper triangular matrices.

In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:

Die hyperbolische Geometrie (auch Lobatschewskische Geometrie oder Lobatschewski-Geometrie genannt) ist ein Beispiel für eine nichteuklidische Geometrie, das man erhält, wenn man zu den Axiomen der absoluten Geometrie anstelle des Parallelenaxioms, das die euklidischen Geometrien kennzeichnet, das diesem widersprechende "hyperbolische Axiom" hinzunimmt. Dieses besagt, dass es zu einer Geraden "g" und einem Punkt "P" (der nicht auf "g" liegt) nicht wie in der euklidischen Geometrie nur genau eine, sondern mindestens zwei Geraden ("h" und "i") gibt, die durch "P" gehen und zu "g" parallel sind. Dass zwei Geraden „parallel“ zueinander sind, bedeutet hier aber lediglich, dass sie in derselben Ebene liegen und keine gemeinsamen Punkte haben, nicht dass sie überall den gleichen Abstand haben ("h" und "i" haben nur einen gemeinsamen Punkt "P").