title: Troll content: from the Proto-Germanic verb *"trulljanan", a derivative of *"trullan". In Norse mythology, "troll", like "thurs", is a term applied to "jötnar" and is mentioned throughout the Old Norse corpus. In Old Norse sources, trolls are said to dwell in isolated mountains, rocks, and caves, sometimes live together (usually as father-and-daughter or mother-and-son), and are rarely described as helpful or friendly. The "Prose Edda" book "Skáldskaparmál" describes an encounter between an unnamed troll woman and the 9th-century skald Bragi Boddason. According to the section, Bragi was driving through "a certain forest" late one evening when a troll woman aggressively asked him


title: Troll content: who he was, in the process describing herself: Bragi responds in turn, describing himself and his abilities as a skillful skald, before the scenario ends. There is much confusion and overlap in the use of Old Norse terms "jötunn", "troll", "þurs", and "risi", which describe various beings. Lotte Motz theorized that these were originally four distinct classes of beings: lords of nature ("jötunn"), mythical magicians ("troll"), hostile monsters ("þurs"), and heroic and courtly beings ("risi"), the last class being the youngest addition. On the other hand, Ármann Jakobson is critical of Motz's interpretation and calls this theory "unsupported by any


title: Troll content: convincing evidence". Ármann highlights that the term is used to denote various beings, such as a jötunn or mountain-dweller, a witch, an abnormally strong or large or ugly person, an evil spirit, a ghost, a "blámaðr", a magical boar, a heathen demi-god, a demon, a brunnmigi, or a berserker. Later in Scandinavian folklore, trolls become defined as a particular type of being. Numerous tales are recorded about trolls in which they are frequently described as being extremely old, very strong, but slow and dim-witted, and are at times described as man-eaters and as turning to stone upon contact with sunlight.


title: Troll content: However, trolls are also attested as looking much the same as human beings, without any particularly hideous appearance about them, but living far away from human habitation and generally having "some form of social organization" — unlike the "rå" and "näck", who are attested as "solitary beings". According to John Lindow, what sets them apart is that they are not Christian, and those who encounter them do not know them. Therefore, trolls were in the end dangerous, regardless of how well they might get along with Christian society, and trolls display a habit of "bergtagning" ('kidnapping'; literally "mountain-taking") and overrunning


title: Troll content: a farm or estate. Lindow states that the etymology of the word "troll" remains uncertain, though he defines trolls in later Swedish folklore as "nature beings" and as "all-purpose otherworldly being[s], equivalent, for example, to fairies in Anglo-Celtic traditions". They "therefore appear in various migratory legends where collective nature-beings are called for". Lindow notes that trolls are sometimes swapped out for cats and "little people" in the folklore record. A Scandinavian folk belief that lightning frightens away trolls and jötnar appears in numerous Scandinavian folktales, and may be a late reflection of the god Thor's role in fighting such beings.


title: Troll content: In connection, the lack of trolls and jötnar in modern Scandinavia is sometimes explained as a result of the "accuracy and efficiency of the lightning strokes". Additionally, the absence of trolls in regions of Scandinavia are described in folklore as being a "consequence of the constant din of the church-bells". This ring caused the trolls to leave for other lands, although not without some resistance; numerous traditions relate how trolls destroyed a church under construction or hurled boulders and stones at completed churches. Large local stones are sometimes described as the product of a troll's toss. Additionally, into the 20th


title: Troll content: century, the origins of particular Scandinavian landmarks, such as particular stones, are ascribed to trolls who may, for example, have turned to stone upon exposure to sunlight. Lindow compares the trolls of the Swedish folk tradition to Grendel, the supernatural mead hall invader in the Old English poem "Beowulf", and notes that "just as the poem "Beowulf" emphasizes not the harrying of Grendel but the cleansing of the hall of Beowulf, so the modern tales stress the moment when the trolls are driven off." Smaller trolls are attested as living in burial mounds and in mountains in Scandinavian folk tradition.


title: Troll content: In Denmark, these creatures are recorded as "troldfolk" ("troll-folk"), "bjergtrolde" ("mountain-trolls"), or "bjergfolk" ("mountain-folk") and in Norway also as "troldfolk" ("troll-folk") and "tusser". Trolls may be described as small, human-like beings or as tall as men depending on the region of origin of the story. In Norwegian tradition, similar tales may be told about the larger trolls and the Huldrefolk ("hidden-folk") yet a distinction is made between the two. The use of the word "trow" in Orkney and Shetland, to mean beings which are very like the Huldrefolk in Norway may suggest a common origin for the terms. The word


title: Troll content: "troll" may have been used by pagan Norse settlers in Orkney and Shetland as a collective term for supernatural beings who should be respected and avoided rather than worshiped. "Troll" could later have become specialized as a description of the larger, more menacing Jötunn-kind whereas "Huldrefolk" may have developed as the term for smaller trolls. John Arnott MacCulloch posited a connection between the Old Norse vættir and trolls, suggesting that both concepts may derive from spirits of the dead. Troll A troll is a class of being in Norse mythology and Scandinavian folklore. In Old Norse sources, beings described as


title: Tea content: to achieve the final product. Almost all tea in bags and most loose tea sold in the West is blended. Such teas may combine others from the same cultivation area or several different ones. The aim is to obtain consistency, better taste, higher price, or some combination of the three. Flavoured and scented teas add new aromas and flavours to the base tea. This can be accomplished through directly adding flavouring agents, such as Ginger or dried Ginger, Cloves, Mintleaves, Elaichi, bergamot (found in Earl Grey), vanilla, and spearmint. Alternatively, because tea easily retains odours, it can be placed in


title: "Taurus (constellation)" content: its discovery by German-born English astronomer William Herschel in 1790. Prior to that time, astronomers had assumed that nebulae were simply unresolved groups of stars. However, Herschel could clearly resolve a star at the center of the nebula that was surrounded by a nebulous cloud of some type. In 1864, English astronomer William Huggins used the spectrum of this nebula to deduce that the nebula is a luminous gas, rather than stars. To the west, the two horns of the bull are formed by Beta (β) Tauri and Zeta (ζ) Tauri; two star systems that are separated by 8°. Beta


title: Taco content: Taco A taco (, , ) is a traditional Mexican dish consisting of a corn or wheat tortilla folded or rolled around a filling. A taco can be made with a variety of fillings, including beef, pork, chicken, seafood, vegetables, and cheese, allowing great versatility and variety. Tacos are generally eaten without utensils, often garnished with salsa, chili pepper, avocado, guacamole, cilantro (coriander), tomatoes, onions, and lettuce. The origins of the taco are not precisely known, and etymologies for the culinary usage of the word are generally theoretical. According to the Real Academia Española, publisher of "Diccionario de la Lengua


title: Taco content: Española", the word "taco" describes a typical Mexican dish of a maize tortilla folded around food. This meaning of the Spanish word "taco" is a Mexican innovation, but in other dialects "taco" is used to mean "wedge; wad, plug; billiard cue; blowpipe; ramrod; short, stocky person; [or] short, thick piece of wood." In this non-culinary usage, the word "taco" has cognates in other European languages, including the French word "tache" and the English word "tack (nail)." According to one etymological theory, the culinary meaning of "taco" derives from its "plug" meaning as employed among Mexican silver miners, who used explosive


title: Taco content: charges in plug form consisting of a paper wrapper and gunpowder filling. Indigenous origins for the culinary word "taco" are also proposed. One possibility is that the word derives from the Nahuatl word "tlahco", meaning "half" or "in the middle," in the sense that food would be placed in the middle of a tortilla. Furthermore, dishes analogous to the taco were known to have existed in Pre-Columbian society—for example, the Náhuatl word "tlaxcalli" (a type of corn tortilla). The taco predates the arrival of the Spanish in Mexico. There is anthropological evidence that the indigenous people living in the lake


title: Taco content: region of the Valley of Mexico traditionally ate tacos filled with small fish. Writing at the time of the Spanish conquistadors, Bernal Díaz del Castillo documented the first taco feast enjoyed by Europeans, a meal which Hernán Cortés arranged for his captains in Coyoacán. There are many traditional varieties of tacos: As an accompaniment to tacos, many taco stands will serve whole or sliced red radishes, lime slices, salt, pickled or grilled chilis (hot peppers), and occasionally cucumber slices, or grilled cambray onions. Beginning from the early part of the twentieth century, various types of tacos have become popular in


title: Taco content: the United States and Canada. An early appearance of a description of the taco in the United States in English was in a 1914 cookbook, "California Mexican-Spanish Cookbook", by Bertha Haffner Ginger. The most common type of taco in the US is the hard-shell, U-shaped version, first described in a cookbook, "The good life: New Mexican food", authored by Fabiola Cabeza de Baca Gilbert and published in Santa Fe, New Mexico, in 1949. These types are sold by restaurants and by fast food chains, while kits are readily available in most supermarkets. Even non-Mexican oriented fast food restaurants have sold


title: Taco content: tacos. Mass production of this type of taco was encouraged by the invention of devices to hold the tortillas in the U-shape as they were deep-fried. A patent for such a device was issued to New York restaurateur Juvencio Maldonado in 1950, based on his patent filing of 1947 (U.S. Patent No. 2,506,305). Such tacos are crisp-fried corn tortillas filled with seasoned ground beef, cheese, lettuce, and sometimes tomato, onion, salsa, sour cream, and avocado or guacamole. Traditionally, soft-shelled tacos referred to corn tortillas that were cooked to a softer state than a hard taco - usually by grilling or


title: Taco content: steaming. More recently, the term has come to include flour tortilla based tacos mostly from large manufacturers and restaurant chains. In this context, "soft tacos" are tacos made with wheat flour tortillas and filled with the same ingredients as a hard taco. The breakfast taco, found in Tex-Mex cuisine, is a soft corn or flour tortilla filled with meat, eggs, or cheese, and can also contain other ingredients. Some have claimed that Austin, Texas is the home of the breakfast taco. However, food writer and "OC Weekly" editor Gustavo Arellano responded that such a statement reflects a common trend of


title: Taco content: "whitewashed" foodways reporting, noting that predominantly Hispanic San Antonio, Texas "never had to brag about its breakfast taco love—folks there just call it 'breakfast'. Indian tacos, or "Navajo tacos", are made using frybread instead of tortillas. They are commonly eaten at pow-wows, festivals, and other gatherings by and for indigenous people in the United States and Canada. Since at least 1978, a variation called the "puffy taco" has been popular. "Henry's Puffy Tacos", opened by Henry Lopez in San Antonio, Texas, claims to have invented the variation, in which uncooked corn tortillas (flattened balls of masa dough) are quickly fried


title: Taco content: in hot oil until they expand and become "puffy". Fillings are similar to hard-shell versions. Restaurants offering this style of taco have since appeared in other Texas cities, as well as in California, where Henry's brother, Arturo Lopez, opened "Arturo's Puffy Taco" in Whittier, not long after Henry's opened. Henry's continues to thrive, managed by the family's second generation. Kits are available at grocery and convenience stores and usually consist of taco shells (corn tortillas already fried in a U-shape), seasoning mix and taco sauce. Commercial vendors for the home market also market soft taco kits with tortillas instead of


title: Taco content: taco shells. The tacodilla contains melted cheese in between the two folded tortillas, thus resembling a quesadilla. In the United States, the "National Taco Day" is celebrated annually on October 4. Taco A taco (, , ) is a traditional Mexican dish consisting of a corn or wheat tortilla folded or rolled around a filling. A taco can be made with a variety of fillings, including beef, pork, chicken, seafood, vegetables, and cheese, allowing great versatility and variety. Tacos are generally eaten without utensils, often garnished with salsa, chili pepper, avocado, guacamole, cilantro (coriander), tomatoes, onions, and lettuce. The origins


title: "The Hound of the Baskervilles" content: (d.1861) of Fowelscombe) and Brook Hall, in the parish of Buckfastleigh, about two miles east of Hayford, the actual home of Richard Cabell (d.1677), husband of Elizabeth Fowell. It has also been claimed that Baskerville Hall is based on a property in Mid Wales, built in 1839 by one Thomas Mynors Baskerville. The house was formerly named Clyro Court and was renamed Baskerville Hall towards the end of the last century. Arthur Conan Doyle was apparently a family friend who often stayed there and may have been aware of a local legend of the hound of the Baskervilles. Still other


title: Tax content: money is fungible. Furthermore, it often happens that taxes or excises initially levied to fund some specific government programs are then later diverted to the government general fund. In some cases, such taxes are collected in fundamentally inefficient ways, for example, though highway tolls. Since governments also resolve commercial disputes, especially in countries with common law, similar arguments are sometimes used to justify a sales tax or value added tax. Some (libertarians, for example) portray most or all forms of taxes as immoral due to their involuntary (and therefore eventually coercive/violent) nature. The most extreme anti-tax view, anarcho-capitalism, holds that


title: TARDIS content: that the Daleks, created and led by Davros, would have no problem breaching the TARDIS defences. "They're experts at fighting TARDISes, they can do anything. Right now, that wooden door is just wood." One of the main TARDIS defenses is a force field. The field is seen in use in "The Runaway Bride" (2006), when the Tenth Doctor and the Bride, Donna Noble, are trying to escape the Empress of the Racnoss and in "The Beast Below" (2010), when the Doctor is showing Amy Pond the wonders of the universe. In "Journey to the Centre of the TARDIS" (2013), the


title: "The X-Files" content: vaccine is revealed in the latter parts of season five, as well as the 1998 film. Counter to the alien colonization effort, another faction of aliens, the faceless rebels, are working to stop alien colonization. Eventually, in the season six episodes "Two Fathers"/"One Son", the rebels manage to destroy the Syndicate. The colonists, now without human liaisons, dispatch the "Super Soldiers": beings that resemble humans, but are biologically alien. In the latter parts of season eight, and the whole of season nine, the Super Soldiers manage to replace key individuals in the government, forcing Mulder and Scully to go into


title: "Twin Peaks" content: the filming of Sarah Palmer having a vision, the camera operator told Lynch that the shot was ruined because "Frank [Silva] was reflected in the mirror." Lynch comments, "Things like this happen and make you start dreaming. And one thing leads to another, and if you let it, a whole other thing opens up." Lynch used the panning shot of Silva in the bedroom, and the shot featuring Silva's reflection, in the closing scenes of the European version of the pilot episode. Silva's reflection in the mirror can also be glimpsed during the scene of Sarah's vision at the end


title: Tetris content: a factor of formula_2 for any constant formula_3. It is NP-hard to approximate the last problem within a factor of formula_4 for any constant formula_3. To prove NP-completeness, it was shown that there is a polynomial reduction between the 3-partition problem, which is also NP-Complete, and the "Tetris" problem. Atari's arcade version of "Tetris", and its NES port by subsidiary Tengen have original pieces composed by Brad Fuller, as well as traditional Russian tunes also arranged by Fuller. According to research from Dr. Richard Haier, "et al." prolonged "Tetris" activity can also lead to more efficient brain activity during play.


title: "Trigonometric functions" content: where just translated reciprocal functions are summed up, such that the poles of the cotangent function and the reciprocal functions match: This identity can be proven with the Herglotz trick. Combining the th with the th term lead to absolutely convergent series: It can be shown from the series definitions that the sine and cosine functions are respectively the imaginary and real parts of the exponential function of a purely imaginary argument. That is, if is real, we have and The latter identity, although primarily established for real , remains valid for every complex , and is called Euler's formula.


title: Thylacine content: 2017 using the DNA extracted from an ethanol-preserved pouch young specimen provided by Museums Victoria. Researchers used the genome to study aspects of the thylacine's evolution and natural history, including the genetic basis of its convergence with canids, clarifying its evolutionary relationships with other marsupials and examining changes in its population size over time. Also in 2017 a reference library of 159 micrographic images of thylacine hair was jointly produced by CSIRO and Where Light Meets Dark, using scanning electron microscopy, metal-coated scanning electron microscopy, confocal laser scanning microscopy and optical light microscopy. In 2018 Rehberg published a study into


title: Torque content: axis of the rotation that the "force vector" (starting from rest) would initiate. The resulting "torque vector" direction is determined by the right-hand rule. The unbalanced torque on a body along axis of rotation determines the rate of change of the body's angular momentum, where L is the angular momentum vector and "t" is time. If multiple torques are acting on the body, it is instead the net torque which determines the rate of change of the angular momentum: For the motion of a point particle, where is the moment of inertia and ω is the angular velocity. It follows


title: "The Marriage of Figaro" content: Count's estate, a preemptive attempt by Figaro to commit the Count to a formal gesture symbolizing his promise that Susanna would enter into the marriage unsullied. The Count evades Figaro's plan by postponing the gesture. The Count says that he forgives Cherubino, but he dispatches him to his own regiment in Seville for army duty, effective immediately. Figaro gives Cherubino mocking advice about his new, harsh, military life from which all luxury, and especially women, will be totally excluded (aria: "Non più andrai" – "No more gallivanting"). "A handsome room with an alcove, a dressing room on the left, a


title: "The Sims (video game)" content: one of eight broad categories: seating, surfaces, decorative, electronics, appliances, plumbing, lighting and miscellaneous. The original inspiration for "The Sims" was Christopher Alexander's 1977 book on architecture and urban design, "A Pattern Language". Game designer Will Wright was inspired by the book's focus on functionality in architecture, as Alexander based his design principles on structural usability rather than aesthetic values. Wright wanted to create a simulation game about enabling human behavior and interaction through design. Scott McCloud's 1993 book "Understanding Comics" became a big influence on the design of "The Sims" later on, as it advocates a certain type of


title: "Theodore Roosevelt" content: a vein and traveled to his lungs. He was 60 years old. Upon receiving word of his death, his son Archibald telegraphed his siblings: "The old lion is dead." Woodrow Wilson's vice president, Thomas R. Marshall, said that "Death had to take Roosevelt sleeping, for if he had been awake, there would have been a fight." Following a private farewell service in the North Room at Sagamore Hill, a simple funeral was held at Christ Episcopal Church in Oyster Bay. Vice President Thomas R. Marshall, Charles Evans Hughes, Warren Harding, Henry Cabot Lodge, and William Howard Taft were among the


title: "Telecommunications in Thailand" content: NBTC Act) and the Telecommunications Business Act 2001 (the Telecommunications Business Act). Additionally, any operator wishing to issue telephone numbers shall obtain a separate licence from the NBTC, subject to a Telecommunications Numbering Plan issued by the NBTC. In 2015 the NBTC arranged two rounds of auctions for 1800 MHz and 900Mhz. In November, Advance Info Service and True Corporation won the 1800 MHz license in the auction which takes nearly 33 hours to complete. In December, True Corporation and Jasmine International won the 900 MHz auction. The significance of this auction is the establishment of Jasmine International in the


title: Tonga content: in 1793; the first London missionaries in 1797; and the Wesleyan Methodist Reverend Walter Lawry in 1822. The US Exploring Expedition visited in 1840. In 1845, the ambitious young warrior, strategist, and orator Tāufaʻāhau united Tonga into a kingdom. He held the chiefly title of Tuʻi Kanokupolu, but had been baptised by Methodist missionaries with the name "Siaosi" ("George") in 1831. In 1875, with the help of missionary Shirley Waldemar Baker, he declared Tonga a constitutional monarchy; formally adopted the western royal style; emancipated the "serfs"; enshrined a code of law, land tenure, and freedom of the press; and limited


title: "Thirty Years' War" content: control of political and administrative affairs, in particular in Northern Catalonia, and a firm military focus on the neighbouring Spanish kingdoms of Valencia and Aragon, in line with Richelieu's war against Spain, gradually undermined Catalan enthusiasm for the French. In parallel, in December 1640, the Portuguese rose up against Spanish rule and once again Richelieu supplied aid to the insurgents.. The ensuing conflict with Spain brought Portugal into the Thirty Years' War as, at least, a peripheral player. From 1641 to 1668, the period during which the two nations were at war, Spain sought to isolate Portugal militarily and diplomatically,


title: "Table tennis" content: Table tennis Table tennis, also known as ping-pong, is a sport in which two or four players hit a lightweight ball back and forth across a table using small rackets. The game takes place on a hard table divided by a net. Except for the initial serve, the rules are generally as follows: players must allow a ball played toward them to bounce one time on their side of the table, and must return it so that it bounces on the opposite side at least once. A point is scored when a player fails to return the ball within the


title: "Table tennis" content: rules. Play is fast and demands quick reactions. Spinning the ball alters its trajectory and limits an opponent's options, giving the hitter a great advantage. Table tennis is governed by the worldwide organization International Table Tennis Federation (ITTF), founded in 1926. ITTF currently includes 226 member associations. The table tennis official rules are specified in the ITTF handbook. Table tennis has been an Olympic sport since 1988, with several event categories. From 1988 until 2004, these were men's singles, women's singles, men's doubles and women's doubles. Since 2008, a team event has been played instead of the doubles. The sport


title: "Table tennis" content: originated in Victorian England, where it was played among the upper-class as an after-dinner parlour game. It has been suggested that makeshift versions of the game were developed by British military officers in India in around 1860s or 1870s, who brought it back with them. A row of books stood up along the center of the table as a net, two more books served as rackets and were used to continuously hit a golf-ball. The name "ping-pong" was in wide use before British manufacturer J. Jaques & Son Ltd trademarked it in 1901. The name "ping-pong" then came to describe


title: "Table tennis" content: the game played using the rather expensive Jaques's equipment, with other manufacturers calling it table tennis. A similar situation arose in the United States, where Jaques sold the rights to the "ping-pong" name to Parker Brothers. Parker Brothers then enforced its trademark for the term in the 1920s making the various associations change their names to "table tennis" instead of the more common, but trademarked, term. The next major innovation was by James W. Gibb, a British enthusiast of table tennis, who discovered novelty celluloid balls on a trip to the US in 1901 and found them to be ideal


title: "Table tennis" content: for the game. This was followed by E.C. Goode who, in 1901, invented the modern version of the racket by fixing a sheet of pimpled, or stippled, rubber to the wooden blade. Table tennis was growing in popularity by 1901 to the extent that tournaments were being organized, books being written on the subject, and an unofficial world championship was held in 1902. In 1921, the Table Tennis Association was founded, and in 1926 renamed the English Table Tennis Association. The International Table Tennis Federation (ITTF) followed in 1926. London hosted the first official World Championships in 1926. In 1933,


title: "Table tennis" content: the United States Table Tennis Association, now called USA Table Tennis, was formed. In the 1930s, Edgar Snow commented in "Red Star Over China" that the Communist forces in the Chinese Civil War had a "passion for the English game of table tennis" which he found "bizarre". On the other hand, the popularity of the sport waned in 1930s Soviet Union, partly because of the promotion of team and military sports, and partly because of a theory that the game had adverse health effects. In the 1950s, paddles that used a rubber sheet combined with an underlying sponge layer changed


title: "Table tennis" content: the game dramatically, introducing greater spin and speed. These were introduced to Britain by sports goods manufacturer S.W. Hancock Ltd. The use of speed glue increased the spin and speed even further, resulting in changes to the equipment to "slow the game down". Table tennis was introduced as an Olympic sport at the Olympics in 1988. After the 2000 Olympics in Sydney, the ITTF instituted several rule changes that were aimed at making table tennis more viable as a televised spectator sport. First, the older balls were officially replaced by balls in October 2000. This increased the ball's air resistance


title: "Table tennis" content: and effectively slowed down the game. By that time, players had begun increasing the thickness of the fast sponge layer on their paddles, which made the game excessively fast and difficult to watch on television. A few months later, the ITTF changed from a 21-point to an 11-point scoring system (and the serve rotation was reduced from five points to two), effective in September 2001. This was intended to make games more fast-paced and exciting. The ITTF also changed the rules on service to prevent a player from hiding the ball during service, in order to increase the average length


title: "Table tennis" content: of rallies and to reduce the server's advantage, effective in 2002. For the opponent to have time to realize a serve is taking place, the ball must be tossed a minimum of 16 cm in the air. The ITTF states that all events after July 2014 are played with a new poly material ball. The international rules specify that the game is played with a sphere having a mass of and a diameter of . The rules say that the ball shall bounce up when dropped from a height of onto a standard steel block thereby having a coefficient of


title: "Table tennis" content: restitution of 0.89 to 0.92. balls are now made of a polymer instead of celluloid as of 2015, colored white or orange, with a matte finish. The choice of ball color is made according to the table color and its surroundings. For example, a white ball is easier to see on a green or blue table than it is on a grey table. Manufacturers often indicate the quality of the ball with a star rating system, usually from one to three, three being the highest grade. As this system is not standard across manufacturers, the only way a ball may


title: "Table tennis" content: be used in official competition is upon ITTF approval (the ITTF approval can be seen printed on the ball). The 40 mm ball was introduced after the 2000 Summer Olympics. However, this created some controversy at the time as the Chinese National Team argued that this was merely to give non-Chinese players a better chance of winning since the new type of ball has a slower speed (a 40 mm table tennis ball is slower and spins less than the original 38 mm one, and at that time, most Chinese players were playing with fast attack and smashes). China won


title: "Table tennis" content: all four Olympic gold medals, three silvers and one bronze in 2000. The table is long, wide, and high with any continuous material so long as the table yields a uniform bounce of about when a standard ball is dropped onto it from a height of , or about . The table or playing surface is uniformly dark coloured and matte, divided into two halves by a net at in height. The ITTF approves only wooden tables or their derivates. Concrete tables with a steel net or a solid concrete partition are sometimes available in outside public spaces, such as


title: "Table tennis" content: parks. Players are equipped with a laminated wooden racket covered with rubber on one or two sides depending on the grip of the player. The ITTF uses the term "racket", though "bat" is common in Britain, and "paddle" in the U.S. and Canada. The wooden portion of the racket, often referred to as the "blade", commonly features anywhere between one and seven plies of wood, though cork, glass fiber, carbon fiber, aluminum fiber, and Kevlar are sometimes used. According to the ITTF regulations, at least 85% of the blade by thickness shall be of natural wood. Common wood types include


title: "Table tennis" content: balsa, limba, and cypress or "hinoki", which is popular in Japan. The average size of the blade is about long and wide, although the official restrictions only focus on the flatness and rigidity of the blade itself, these dimensions are optimal for most play styles. Table tennis regulations allow different surfaces on each side of the racket. Various types of surfaces provide various levels of spin or speed, and in some cases they nullify spin. For example, a player may have a rubber that provides much spin on one side of their racket, and one that provides no spin on


title: "Table tennis" content: the other. By flipping the racket in play, different types of returns are possible. To help a player distinguish between the rubber used by his opposing player, international rules specify that one side must be red while the other side must be black. The player has the right to inspect their opponent's racket before a match to see the type of rubber used and what colour it is. Despite high speed play and rapid exchanges, a player can see clearly what side of the racket was used to hit the ball. Current rules state that, unless damaged in play, the


title: "Table tennis" content: racket cannot be exchanged for another racket at any time during a match. According to ITTF rule 2.13.1, the first service is decided by lot, normally a coin toss. It is also common for one player (or the umpire/scorer) to hide the ball in one or the other hand, usually hidden under the table, allowing the other player to guess which hand the ball is in. The correct or incorrect guess gives the "winner" the option to choose to serve, receive, or to choose which side of the table to use. (A common but non-sanctioned method is for the players


title: "Table tennis" content: to play the ball back and forth three times and then play out the point. This is commonly referred to as "serve to play", "rally to serve", "play for serve", or "volley for serve".) In game play, the player serving the ball commences a play. The server first stands with the ball held on the open palm of the hand not carrying the paddle, called the freehand, and tosses the ball directly upward without spin, at least high. The server strikes the ball with the racket on the ball's descent so that it touches first his court and then touches


title: "Table tennis" content: directly the receiver's court without touching the net assembly. In casual games, many players do not toss the ball upward; however, this is technically illegal and can give the serving player an unfair advantage. The ball must remain behind the endline and above the upper surface of the table, known as the playing surface, at all times during the service. The server cannot use his/her body or clothing to obstruct sight of the ball; the opponent and the umpire must have a clear view of the ball at all times. If the umpire is doubtful of the legality of a


title: "Table tennis" content: service they may first interrupt play and give a warning to the server. If the serve is a clear failure or is doubted again by the umpire after the warning, the receiver scores a point. If the service is "good", then the receiver must make a "good" return by hitting the ball back before it bounces a second time on receiver's side of the table so that the ball passes the net and touches the opponent's court, either directly or after touching the net assembly. Thereafter, the server and receiver must alternately make a return until the rally is over.


title: "Table tennis" content: Returning the serve is one of the most difficult parts of the game, as the server's first move is often the least predictable and thus most advantageous shot due to the numerous spin and speed choices at his or her disposal. A Let is a rally of which the result is not scored, and is called in the following circumstances: A let is also called foul service, if the ball hits the server's side of the table, if the ball does not pass further than the edge and if the ball hits the table edge and hits the net. A


title: "Table tennis" content: point is scored by the player for any of several results of the rally: A game shall be won by the player first scoring 11 points unless both players score 10 points, when the game shall be won by the first player subsequently gaining a lead of 2 points. A match shall consist of the best of any odd number of games. In competition play, matches are typically best of five or seven games. Service alternates between opponents every two points (regardless of winner of the rally) until the end of the game, unless both players score ten points or


title: "Table tennis" content: the expedite system is operated, when the sequences of serving and receiving stay the same but each player serves for only one point in turn (Deuce). The player serving first in a game receives first in the next game of the match. After each game, players switch sides of the table. In the last possible game of a match, for example the seventh game in a best of seven matches, players change ends when the first player scores five points, regardless of whose turn it is to serve. Service is subject to change on game point of the match. Upon


title: "Table tennis" content: the possible last point of the match, the player with the lesser score serves. If the sequence of serving and receiving is out of turn or the ends are not changed, points scored in the wrong situation are still calculated and the game shall be resumed with the order at the score that has been reached. In addition to games between individual players, pairs may also play table tennis. Singles and doubles are both played in international competition, including the Olympic Games since 1988 and the Commonwealth Games since 2002. In 2005, the ITTF announced that doubles table tennis only


title: "Table tennis" content: was featured as a part of team events in the 2008 Olympics. In doubles, all the rules of single play are applied except for the following. Service Order of play, serving and receiving If a game is unfinished after 10 minutes' play and fewer than 18 points have been scored, the expedite system is initiated. The umpire interrupts the game, and the game resumes with players serving for one point in turn. If the expedite system is introduced while the ball is not in play, the previous receiver shall serve first. Under the expedite system, the server must win the


title: "Table tennis" content: point before the opponent makes 13 consecutive returns or the point goes to the opponent. The system can also be initiated at any time at the request of both players or pairs. Once introduced, the expedite system remains in force until the end of the match. A rule to shorten the time of a match, it is mainly seen in defensive players' games. Though table tennis players grip their rackets in various ways, their grips can be classified into two major families of styles, "penhold" and "shakehand". The rules of table tennis do not prescribe the manner in which one


title: "Table tennis" content: must grip the racket, and numerous grips are employed. The penhold grip is so-named because one grips the racket similarly to the way one holds a writing instrument. The style of play among penhold players can vary greatly from player to player. The most popular style, usually referred to as the Chinese penhold style, involves curling the middle, ring, and fourth finger on the back of the blade with the three fingers always touching one another. Chinese penholders favour a round racket head, for a more over-the-table style of play. In contrast, another style, sometimes referred to as the Japanese/Korean


title: "Table tennis" content: penhold grip, involves splaying those three fingers out across the back of the racket, usually with all three fingers touching the back of the racket, rather than stacked upon one another. Sometimes a combination of the two styles occurs, wherein the middle, ring and fourth fingers are straight, but still stacked, or where all fingers may be touching the back of the racket, but are also in contact with one another. Japanese and Korean penholders will often use a square-headed racket for an away-from-the-table style of play. Traditionally these square-headed rackets feature a block of cork on top of the


title: "Table tennis" content: handle, as well as a thin layer of cork on the back of the racket, for increased grip and comfort. Penhold styles are popular among players originating from East Asian countries such as China, Japan, South Korea, and Taiwan. Traditionally, penhold players use only one side of the racket to hit the ball during normal play, and the side which is in contact with the last three fingers is generally not used. This configuration is sometimes referred to as "traditional penhold" and is more commonly found in square-headed racket styles. However, the Chinese developed a technique in the 1990s in


title: "Table tennis" content: which a penholder uses both sides of the racket to hit the ball, where the player produces a backhand stroke (most often topspin) known as a reverse penhold backhand by turning the traditional side of the racket to face one's self, and striking the ball with the opposite side of the racket. This stroke has greatly improved and strengthened the penhold style both physically and psychologically, as it eliminates the strategic weakness of the traditional penhold backhand. The shakehand grip is so-named because the racket is grasped as if one is performing a handshake. Though it is sometimes referred to


title: "Table tennis" content: as the "tennis" or "Western" grip, it bears no relation to the Western tennis grip, which was popularized on the West Coast of the United States in which the racket is rotated 90°, and played with the wrist turned so that on impact the knuckles face the target. In table tennis, "Western" refers to Western nations, for this is the grip that players native to Europe and the Americas have almost exclusively employed. The shakehand grip's simplicity and versatility, coupled with the acceptance among top-level Chinese trainers that the European style of play should be emulated and trained against, has


title: "Table tennis" content: established it as a common grip even in China. Many world class European and East Asian players currently use the shakehand grip, and it is generally accepted that shakehands is easier to learn than penholder, allowing a broader range of playing styles both offensive and defensive. The Seemiller grip is named after the American table tennis champion Danny Seemiller, who used it. It is achieved by placing your thumb and index finger on either side of the bottom of the racquet head and holding the handle with the rest of your fingers. Since only one side of the racquet is


title: "Table tennis" content: used to hit the ball, two contrasting rubber types can be applied to the blade, offering the advantage of "twiddling" the racket to fool the opponent. Seemiller paired inverted rubber with anti-spin rubber; many players today combine inverted and long-pipped rubber. The grip is considered exceptional for blocking, especially on the backhand side, and for forehand loops of backspin balls. The Seemiller grip's popularity reached its apex in 1985 when four (Danny Seemiller, Ricky Seemiller, Eric Boggan and Brian Masters) of the United States' five participants in the World Championships used it. Table tennis strokes generally break down into offensive


title: "Table tennis" content: and defensive categories. A direct hit on the ball propelling it forward back to the opponent. This stroke differs from speed drives in other racket sports like tennis because the racket is primarily "perpendicular" to the direction of the stroke and most of the energy applied to the ball results in "speed" rather than "spin", creating a shot that does not arc much, but is fast enough that it can be difficult to return. A speed drive is used mostly for keeping the ball in play, applying pressure on the opponent, and potentially opening up an opportunity for a more


title: "Table tennis" content: powerful attack. Perfected during the 1960s, the loop is essentially the reverse of the speed drive. The racket is much more "parallel" to the direction of the stroke ("closed") and the racket thus "grazes" the ball, resulting in a large amount of topspin. A good loop drive will arc quite a bit, and once striking the opponent's side of the table will jump forward, much like a kick serve in tennis. The counter-hit is usually a counterattack against drives, normally high loop drives. The racket is held closed and near to the ball, which is hit with a short movement


title: "Table tennis" content: "off the bounce" (immediately after hitting the table) so that the ball travels faster to the other side. A well-timed, accurate counter-drive can be as effective as a smash. When a player tries to attack a ball that has not bounced beyond the edge of the table, the player does not have the room to wind up in a "backswing". The ball "may still be attacked", however, and the resulting shot is called a flip because the backswing is compressed into a quick wrist action. A flip is not a single stroke and can resemble either a loop drive or


title: "Table tennis" content: a loop in its characteristics. What identifies the stroke is that the backswing is compressed into a short wrist flick. The offensive trump card is the smash. A player will typically execute a smash when his or her opponent has returned a ball that bounces too high or too close to the net. "Smashing" consists of using a large backswing and rapid acceleration to impart as much speed on the ball as possible. The goal of a smash is to get the ball to move so quickly that the opponent simply cannot return it. Because the ball speed is the


title: "Table tennis" content: main aim of this shot, often the spin on the ball is something other than topspin. Sidespin can be used effectively with a smash to alter the ball's trajectory significantly, although most intermediate players will smash the ball with little or no spin. An offensive table tennis player will think of a rally as a build-up to a winning smash. The push (or "slice" in Asia) is usually used for keeping the point alive and creating offensive opportunities. A push resembles a tennis slice: the racket cuts underneath the ball, imparting backspin and causing the ball to float slowly to


title: "Table tennis" content: the other side of the table. While not obvious, a push can be difficult to attack because the backspin on the ball causes it to drop toward the table upon striking the opponent's racket. In order to attack a push, a player must usually loop the ball back over the net. Often, the best option for beginners is to simply push the ball back again, resulting in pushing rallies. Against good players, it may be the worst option because the opponent will counter with a loop, putting the first player in a defensive position. Another response to pushing is flipping


title: "Table tennis" content: the ball when it is close to the net. Pushing can have advantages in some circumstances, such as when the opponent makes easy mistakes. A chop is the defensive, backspin counterpart to the offensive loop drive. A chop is essentially a bigger, heavier push, taken well back from the table. The racket face points primarily horizontally, perhaps a little bit upward, and the direction of the stroke is straight down. The object of a defensive chop is to match the topspin of the opponent's shot with backspin. A good chop will float nearly horizontally back to the table, in some


title: "Table tennis" content: cases having so much backspin that the ball actually "rises". Such a chop can be "extremely" difficult to return due to its enormous amount of backspin. Some defensive players can also impart no-spin or sidespin variations of the chop. The block is a simple shot, but nonetheless can be devastating against an attacking opponent. A block is executed by simply placing the racket in front of the ball right after the ball bounces; thus, the ball rebounds back toward the opponent with nearly as much energy as it came in with. This requires precision, since the ball's spin, speed, and


title: "Table tennis" content: location all influence the correct angle of a block. It is very possible for an opponent to execute a perfect loop, drive, or smash, only to have the blocked shot come back at him just as fast. Due to the power involved in offensive strokes, often an opponent simply cannot recover quickly enough, and will be unable to return the blocked shot. Blocks almost always produce the same spin as was received, many times topspin. Depending on the spin of the ball, the block may be returned to an unexpected side of the table. This may come to your advantage,


title: "Table tennis" content: as the opponent may not expect this. The defensive lob is possibly the most impressive shot, since it propels the ball about five metres in height, only to land on the opponent's side of the table with great amounts of spin. To execute, a defensive player first backs-off the table 4–6 meters; then, the stroke itself consists of lifting the ball to an enormous height before it falls back to the opponent's side of the table. A lob is inherently a creative shot, and can have nearly any kind of spin. Top-quality players use this to their advantage in order


title: "Table tennis" content: to control the spin of the ball. For instance, though the opponent may "smash" the ball hard and fast, a good defensive lob could be more difficult to return due to the unpredictability and heavy amounts of the spin on the ball. Thus, though backed off the table by tens of feet and running to reach the ball, a good defensive player can still win the point using good lobs. However, at the professional level, lobbers will lose the point most of the time, so the lob is not used unless it is really necessary. Adding spin onto the ball


title: "Table tennis" content: causes major changes in table tennis gameplay. Although nearly every stroke or serve creates some kind of spin, understanding the individual types of spin allows players to defend against and use different spins effectively. Backspin is where the bottom half of the ball is rotating away from the player, and is imparted by striking the base of the ball with a downward movement. At the professional level, backspin is usually used defensively in order to keep the ball low. Backspin is commonly employed in service because it is harder to produce an offensive return, though at the professional level most


title: "Table tennis" content: people serve sidespin with either backspin or topspin. Due to the initial lift of the ball, there is a limit on how much speed with which one can hit the ball without missing the opponent's side of the table. However, backspin also makes it harder for the opponent to return the ball with great speed because of the required angular precision of the return. Alterations are frequently made to regulations regarding equipment in an effort to maintain a balance between defensive and offensive spin choices. It is actually possible to smash with backspin offensively, but only on high balls that


title: "Table tennis" content: are close to the net. The topspin stroke has a smaller influence on the first part of the ball-curve. Like the backspin stroke, however, the axis of spin remains roughly perpendicular to the trajectory of the ball thus allowing for the Magnus effect to dictate the subsequent curvature. After the apex of the curve, the ball dips downwards as it approaches the opposing side, before bouncing. On the bounce, the topspin will accelerate the ball, much in the same way that a wheel which is already spinning would accelerate upon making contact with the ground. When the opponent attempts to


title: "Table tennis" content: return the ball, the topspin causes the ball to jump upwards and the opponent is forced to compensate for the topspin by adjusting the angle of his or her racket. This is known as "closing the racket". The speed limitation of the topspin stroke is minor compared to the backspin stroke. This stroke is the predominant technique used in professional competition because it gives the opponent less time to respond. In table tennis topspin is regarded as an offensive technique due to increased ball speed, lower bio-mechanical efficiency and the pressure that it puts on the opponent by reducing reaction


title: "Table tennis" content: time. (It is possible to play defensive topspin-lobs from far behind the table, but only highly skilled players use this stroke with any tactical efficiency.) Topspin is the least common type of spin to be found in service at the professional level, simply because it is much easier to attack a top-spin ball that is not moving at high speed. This type of spin is predominantly employed during service, wherein the contact angle of the racket can be more easily varied. Unlike the two aforementioned techniques, sidespin causes the ball to spin on an axis which is vertical, rather than


title: "Table tennis" content: horizontal. The axis of rotation is still roughly perpendicular to the trajectory of the ball. In this circumstance, the Magnus effect will still dictate the curvature of the ball to some degree. Another difference is that unlike backspin and topspin, sidespin will have relatively very little effect on the bounce of the ball, much in the same way that a spinning top would not travel left or right if its axis of rotation were exactly vertical. This makes sidespin a useful weapon in service, because it is less easily recognized when bouncing, and the ball "loses" less spin on the


title: "Table tennis" content: bounce. Sidespin can also be employed in offensive rally strokes, often from a greater distance, as an adjunct to topspin or backspin. This stroke is sometimes referred to as a "hook". The hook can even be used in some extreme cases to circumvent the net when away from the table. Players employ this type of spin almost exclusively when serving, but at the professional level, it is also used from time to time in the lob. Unlike any of the techniques mentioned above, corkspin (or "drill-spin") has the axis of spin relatively parallel to the ball's trajectory, so that the


title: "Table tennis" content: Magnus effect has little or no effect on the trajectory of a cork-spun ball: upon bouncing, the ball will dart right or left (according to the direction of the spin), severely complicating the return. In theory this type of spin produces the most obnoxious effects, but it is less strategically practical than sidespin or backspin, because of the limitations that it imposes upon the opponent during their return. Aside from the initial direction change when bouncing, unless it goes out of reach, the opponent can counter with "either" topspin or backspin. A backspin stroke is similar in the fact that


title: "Table tennis" content: the corkspin stroke has a lower maximum velocity, simply due to the contact angle of the racket when producing the stroke. To impart a spin on the ball which is parallel to its trajectory, the racket must be swung more or less perpendicular to the trajectory of the ball, greatly limiting the forward momentum that the racket transfers to the ball. Corkspin is almost always mixed with another variety of spin, since alone, it is not only less effective but also harder to produce. Competitive table tennis is popular in East Asia and Europe, and has been gaining attention in


title: "Table tennis" content: the United States. The most important international competitions are the World Table Tennis Championships, the Table Tennis World Cup, the Olympics and the ITTF World Tour. Continental competitions include the following: Chinese players have won 60% of the men's World Championships since 1959; in the women's competition for the Corbillin Cup, Chinese players have won all but three of the World Championships since 1971. Other strong teams come from East Asia and Europe, including countries such as Austria, Belarus, Germany, Hong Kong, Portugal, Japan, South Korea, Singapore, Sweden, and Taiwan. There are also professional competitions at the clubs level; the


title: "Table tennis" content: respective leagues of Austria, Belgium, China (specifically, the China Table Tennis Super League), France, Germany and Russia are examples of the highest level. There are also some important international club teams competitions such as the European Champions League and its former competitor, the European Club Cup, where the top club teams from European countries compete. An official hall of fame exists at the ITTF Museum. A "Grand Slam" is earned by a player who wins singles crowns at the Olympic Games, World Championships, and World Cup. Jan-Ove Waldner of Sweden first completed the grand slam at 1992 Olympic Games. Deng


title: "Table tennis" content: Yaping of China is the first female recorded at the inaugural Women's World Cup in 1996. The following table presents an exhaustive list of all players to have completed a grand slam. Jean-Philippe Gatien (France) and Wang Hao (China) won both the World Championships and the World Cup, but lost in the gold medal matches at the Olympics. Jörgen Persson (Sweden) also won the titles except the Olympic Games. Persson is one of the three table tennis players to have competed at seven Olympic Games. Ma Lin (China) won both the Olympic gold and the World Cup, but lost (three


title: "Table tennis" content: times, in 1999, 2005, and 2007) in the finals of the World Championships. Founded in 1926, the International Table Tennis Federation (ITTF) is the worldwide governing body for table tennis, which maintains an international ranking system in addition to organizing events like the World Table Tennis Championships. In 2007, the governance for table tennis for persons with a disability was transferred from the International Paralympic Committee to the ITTF. On many continents, there is a governing body responsible for table tennis on that continent. For example, the European Table Tennis Union (ETTU) is the governing body responsible for table tennis


title: "Table tennis" content: in Europe. There are also national bodies and other local authorities responsible for the sport, such as USA Table Tennis (USATT), which is the national governing body for table tennis in the United States. Table tennis Table tennis, also known as ping-pong, is a sport in which two or four players hit a lightweight ball back and forth across a table using small rackets. The game takes place on a hard table divided by a net. Except for the initial serve, the rules are generally as follows: players must allow a ball played toward them to bounce one time on


title: Tetrahedron content: Tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces. The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex, and may thus also be called a 3-simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point.


title: Tetrahedron content: In the case of a tetrahedron the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid". Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. It has two such nets. For any tetrahedron there exists a sphere (called the circumsphere) on which all four vertices lie, and another sphere (the insphere) tangent to the tetrahedron's faces. A regular tetrahedron is one in which all four faces are equilateral triangles. It is one of the five regular Platonic solids, which


title: Tetrahedron content: have been known since antiquity. In a regular tetrahedron, all faces are the same size and shape (congruent) and all edges are the same length. Regular tetrahedra alone do not tessellate (fill space), but if alternated with regular octahedra in the ratio of two tetrahedra to one octahedron, they form the alternated cubic honeycomb, which is a tessellation. The regular tetrahedron is self-dual, which means that its dual is another regular tetrahedron. The compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula. The following Cartesian coordinates define the four vertices of a tetrahedron with edge


title: Tetrahedron content: length 2, centered at the origin, and two level edges: Expressed symmetrically as 4 points on the unit sphere, centroid at the origin, with lower face level, the vertices are: v1 = ( sqrt(8/9), 0, -1/3 ) v2 = ( -sqrt(2/9), sqrt(2/3), -1/3 ) v3 = ( -sqrt(2/9), -sqrt(2/3), -1/3 ) v4 = ( 0, 0, 1 ) with the edge length of sqrt(8/3). Still another set of coordinates are based on an alternated cube or demicube with edge length 2. This form has Coxeter diagram and Schläfli symbol h{4,3}. The tetrahedron in this case has edge length 2. Inverting


title: Tetrahedron content: these coordinates generates the dual tetrahedron, and the pair together form the stellated octahedron, whose vertices are those of the original cube. For a regular tetrahedron of edge length "a": With respect to the base plane the slope of a face (2) is twice that of an edge (), corresponding to the fact that the "horizontal" distance covered from the base to the apex along an edge is twice that along the median of a face. In other words, if "C" is the centroid of the base, the distance from "C" to a vertex of the base is twice that


title: Tetrahedron content: from "C" to the midpoint of an edge of the base. This follows from the fact that the medians of a triangle intersect at its centroid, and this point divides each of them in two segments, one of which is twice as long as the other (see ). For a regular tetrahedron with side length "a", radius "R" of its circumscribing sphere, and distances "d" from an arbitrary point in 3-space to its four vertices, we have The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron (see above, and also ,


title: Tetrahedron content: showing one of the two tetrahedra in the cube). The symmetries of a regular tetrahedron correspond to half of those of a cube: those that map the tetrahedra to themselves, and not to each other. The tetrahedron is the only Platonic solid that is not mapped to itself by point inversion. The regular tetrahedron has 24 isometries, forming the symmetry group T, [3,3], (*332), isomorphic to the symmetric group, "S". They can be categorized as follows: The regular "tetrahedron" has two special orthogonal projections, one centered on a vertex or equivalently on a face, and one centered on an edge.


title: Tetrahedron content: The first corresponds to the A Coxeter plane. The two skew perpendicular opposite edges of a "regular tetrahedron" define a set of parallel planes. When one of these planes intersects the tetrahedron the resulting cross section is a rectangle. When the intersecting plane is near one of the edges the rectangle is long and skinny. When halfway between the two edges the intersection is a square. The aspect ratio of the rectangle reverses as you pass this halfway point. For the midpoint square intersection the resulting boundary line traverses every face of the tetrahedron similarly. If the tetrahedron is bisected


title: Tetrahedron content: on this plane, both halves become wedges. This property also applies for tetragonal disphenoids when applied to the two special edge pairs. The tetrahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane. An isosceles tetrahedron, also called a disphenoid, is a tetrahedron where all four faces are congruent triangles. A space-filling tetrahedron packs with congruent copies of itself to tile space, like the disphenoid tetrahedral honeycomb. In


title: Tetrahedron content: a trirectangular tetrahedron the three face angles at one vertex are right angles. If all three pairs of opposite edges of a tetrahedron are perpendicular, then it is called an orthocentric tetrahedron. When only one pair of opposite edges are perpendicular, it is called a semi-orthocentric tetrahedron. An isodynamic tetrahedron is one in which the cevians that join the vertices to the incenters of the opposite faces are concurrent, and an isogonic tetrahedron has concurrent cevians that join the vertices to the points of contact of the opposite faces with the inscribed sphere of the tetrahedron. The isometries of an


title: Tetrahedron content: irregular (unmarked) tetrahedron depend on the geometry of the tetrahedron, with 7 cases possible. In each case a 3-dimensional point group is formed. Two other isometries (C, [3]), and (S, [2,4]) can exist if the face or edge marking are included. Tetrahedral diagrams are included for each type below, with edges colored by isometric equivalence, and are gray colored for unique edges. The volume of a tetrahedron is given by the pyramid volume formula: where "A" is the area of the base and "h" is the height from the base to the apex. This applies for each of the four


title: Tetrahedron content: choices of the base, so the distances from the apexes to the opposite faces are inversely proportional to the areas of these faces. For a tetrahedron with vertices , , and , the volume is , or any other combination of pairs of vertices that form a simply connected graph. This can be rewritten using a dot product and a cross product, yielding If the origin of the coordinate system is chosen to coincide with vertex d, then d = 0, so where a, b, and c represent three edges that meet at one vertex, and is a scalar triple


title: Tetrahedron content: product. Comparing this formula with that used to compute the volume of a parallelepiped, we conclude that the volume of a tetrahedron is equal to of the volume of any parallelepiped that shares three converging edges with it. The absolute value of the scalar triple product can be represented as the following absolute values of determinants: Hence which gives where "α", "β", "γ" are the plane angles occurring in vertex d. The angle "α", is the angle between the two edges connecting the vertex d to the vertices b and c. The angle "β", does so for the vertices a


title: Tetrahedron content: and c, while "γ", is defined by the position of the vertices a and b. Given the distances between the vertices of a tetrahedron the volume can be computed using the Cayley–Menger determinant: where the subscripts represent the vertices and "d" is the pairwise distance between them – i.e., the length of the edge connecting the two vertices. A negative value of the determinant means that a tetrahedron cannot be constructed with the given distances. This formula, sometimes called Tartaglia's formula, is essentially due to the painter Piero della Francesca in the 15th century, as a three dimensional analogue of


title: Tetrahedron content: the 1st century Heron's formula for the area of a triangle. Denote "a,b,c" be three edges that meet at a point, and "x,y,z" the opposite edges. Let "V" be the volume of the tetrahedron; then where The above formula uses different expressions with the following formula,The above formula uses six lengths of edges, and the following formula uses three lengths of edges and three angles. If "U", "V", "W", "u", "v", "w" are lengths of edges of the tetrahedron (first three form a triangle; "u" opposite to "U" and so on), then where A plane that divides two opposite edges


title: Tetrahedron content: of a tetrahedron in a given ratio also divides the volume of the tetrahedron in the same ratio. Thus any plane containing a bimedian (connector of opposite edges' midpoints) of a tetrahedron bisects the volume of the tetrahedron. For tetrahedra in hyperbolic space or in three-dimensional elliptic geometry, the dihedral angles of the tetrahedron determine its shape and hence its volume. In these cases, the volume is given by the Murakami–Yano formula. However, in Euclidean space, scaling a tetrahedron changes its volume but not its dihedral angles, so no such formula can exist. Any two opposite edges of a tetrahedron


title: Tetrahedron content: lie on two skew lines, and the distance between the edges is defined as the distance between the two skew lines. Let "d" be the distance between the skew lines formed by opposite edges a and as calculated here. Then another volume formula is given by The tetrahedron has many properties analogous to those of a triangle, including an insphere, circumsphere, medial tetrahedron, and exspheres. It has respective centers such as incenter, circumcenter, excenters, Spieker center and points such as a centroid. However, there is generally no orthocenter in the sense of intersecting altitudes. Gaspard Monge found a center that


title: Tetrahedron content: exists in every tetrahedron, now known as the Monge point: the point where the six midplanes of a tetrahedron intersect. A midplane is defined as a plane that is orthogonal to an edge joining any two vertices that also contains the centroid of an opposite edge formed by joining the other two vertices. If the tetrahedron's altitudes do intersect, then the Monge point and the orthocenter coincide to give the class of orthocentric tetrahedron. An orthogonal line dropped from the Monge point to any face meets that face at the midpoint of the line segment between that face's orthocenter and


title: Tetrahedron content: the foot of the altitude dropped from the opposite vertex. A line segment joining a vertex of a tetrahedron with the centroid of the opposite face is called a "median" and a line segment joining the midpoints of two opposite edges is called a "bimedian" of the tetrahedron. Hence there are four medians and three bimedians in a tetrahedron. These seven line segments are all concurrent at a point called the "centroid" of the tetrahedron. In addition the four medians are divided in a 3:1 ratio by the centroid (see Commandino's theorem). The centroid of a tetrahedron is the midpoint


title: Tetrahedron content: between its Monge point and circumcenter. These points define the "Euler line" of the tetrahedron that is analogous to the Euler line of a triangle. The nine-point circle of the general triangle has an analogue in the circumsphere of a tetrahedron's medial tetrahedron. It is the twelve-point sphere and besides the centroids of the four faces of the reference tetrahedron, it passes through four substitute "Euler points", one third of the way from the Monge point toward each of the four vertices. Finally it passes through the four base points of orthogonal lines dropped from each Euler point to the


title: Tetrahedron content: face not containing the vertex that generated the Euler point. The center "T" of the twelve-point sphere also lies on the Euler line. Unlike its triangular counterpart, this center lies one third of the way from the Monge point "M" towards the circumcenter. Also, an orthogonal line through "T" to a chosen face is coplanar with two other orthogonal lines to the same face. The first is an orthogonal line passing through the corresponding Euler point to the chosen face. The second is an orthogonal line passing through the centroid of the chosen face. This orthogonal line through the twelve-point


title: Tetrahedron content: center lies midway between the Euler point orthogonal line and the centroidal orthogonal line. Furthermore, for any face, the twelve-point center lies at the midpoint of the corresponding Euler point and the orthocenter for that face. The radius of the twelve-point sphere is one third of the circumradius of the reference tetrahedron. There is a relation among the angles made by the faces of a general tetrahedron given by where "α" is the angle between the faces "i" and "j". A tetrahedron is a 3-simplex. Unlike the case of the other Platonic solids, all the vertices of a regular tetrahedron


title: Tetrahedron content: are equidistant from each other (they are the only possible arrangement of four equidistant points in 3-dimensional space). A tetrahedron is a triangular pyramid, and the regular tetrahedron is self-dual. A regular tetrahedron can be embedded inside a cube in two ways such that each vertex is a vertex of the cube, and each edge is a diagonal of one of the cube's faces. For one such embedding, the Cartesian coordinates of the vertices are This yields a tetrahedron with edge-length 2, centered at the origin. For the other tetrahedron (which is dual to the first), reverse all the signs.


title: Tetrahedron content: These two tetrahedra's vertices combined are the vertices of a cube, demonstrating that the regular tetrahedron is the 3-demicube. The volume of this tetrahedron is one-third the volume of the cube. Combining both tetrahedra gives a regular polyhedral compound called the compound of two tetrahedra or stella octangula. The interior of the stella octangula is an octahedron, and correspondingly, a regular octahedron is the result of cutting off, from a regular tetrahedron, four regular tetrahedra of half the linear size (i.e., rectifying the tetrahedron). The above embedding divides the cube into five tetrahedra, one of which is regular. In fact,


title: Tetrahedron content: five is the minimum number of tetrahedra required to compose a cube. Inscribing tetrahedra inside the regular compound of five cubes gives two more regular compounds, containing five and ten tetrahedra. Regular tetrahedra cannot tessellate space by themselves, although this result seems likely enough that Aristotle claimed it was possible. However, two regular tetrahedra can be combined with an octahedron, giving a rhombohedron that can tile space. However, several irregular tetrahedra are known, of which copies can tile space, for instance the disphenoid tetrahedral honeycomb. The complete list remains an open problem. If one relaxes the requirement that the tetrahedra


title: Tetrahedron content: be all the same shape, one can tile space using only tetrahedra in many different ways. For example, one can divide an octahedron into four identical tetrahedra and combine them again with two regular ones. (As a side-note: these two kinds of tetrahedron have the same volume.) The tetrahedron is unique among the uniform polyhedra in possessing no parallel faces. A corollary of the usual law of sines is that in a tetrahedron with vertices "O", "A", "B", "C", we have One may view the two sides of this identity as corresponding to clockwise and counterclockwise orientations of the surface.


title: Tetrahedron content: Putting any of the four vertices in the role of "O" yields four such identities, but at most three of them are independent: If the "clockwise" sides of three of them are multiplied and the product is inferred to be equal to the product of the "counterclockwise" sides of the same three identities, and then common factors are cancelled from both sides, the result is the fourth identity. Three angles are the angles of some triangle if and only if their sum is 180° (π radians). What condition on 12 angles is necessary and sufficient for them to be the


title: Tetrahedron content: 12 angles of some tetrahedron? Clearly the sum of the angles of any side of the tetrahedron must be 180°. Since there are four such triangles, there are four such constraints on sums of angles, and the number of degrees of freedom is thereby reduced from 12 to 8. The four relations given by this sine law further reduce the number of degrees of freedom, from 8 down to not 4 but 5, since the fourth constraint is not independent of the first three. Thus the space of all shapes of tetrahedra is 5-dimensional. Let be the points of a


title: Tetrahedron content: tetrahedron. Let Δ be the area of the face opposite vertex "P" and let "θ" be the dihedral angle between the two faces of the tetrahedron adjacent to the edge "PP". The law of cosines for this tetrahedron, which relates the areas of the faces of the tetrahedron to the dihedral angles about a vertex, is given by the following relation: Let "P" be any interior point of a tetrahedron of volume "V" for which the vertices are "A", "B", "C", and "D", and for which the areas of the opposite faces are "F", "F", "F", and "F". Then For


title: Tetrahedron content: vertices "A", "B", "C", and "D", interior point "P", and feet "J", "K", "L", and "M" of the perpendiculars from "P" to the faces, Denoting the inradius of a tetrahedron as "r" and the inradii of its triangular faces as "r" for "i" = 1, 2, 3, 4, we have with equality if and only if the tetrahedron is regular. If "A", "A", "A" and "A" denote the area of each faces, the value of "r" is given by This formula is obtained from dividing the tetrahedron into four tetrahedra whose points are the three points of one of the


title: Tetrahedron content: original faces and the incenter. Since the four subtetrahedra fill the volume, we have formula_28. Denote the circumradius of a tetrahedron as "R". Let "a,b,c" be the length of the three edges that meet at a point, and "A,B,C" the length of the opposite edges. Let "V" be the volume of the tetrahedron. Then The sum of the areas of any three faces is greater than the area of the fourth face. There exist tetrahedra having integer-valued edge lengths, face areas and volume. One example has one edge of 896, the opposite edge of 990 and the other four edges


title: Tetrahedron content: of 1073; two faces have areas of and the other two have areas of , while the volume is . A tetrahedron can have integer volume and consecutive integers as edges, an example being the one with edges 6, 7, 8, 9, 10, and 11 and volume 48. A regular tetrahedron can be seen as a triangular pyramid. A regular tetrahedron can be seen as a degenerate polyhedron, a uniform "digonal antiprism", where base polygons are reduced digons. A regular tetrahedron can be seen as a degenerate polyhedron, a uniform dual "digonal trapezohedron", containing 6 vertices, in two sets of


title: Tetrahedron content: colinear edges. A truncation process applied to the tetrahedron produces a series of uniform polyhedra. Truncating edges down to points produces the octahedron as a rectified tetrahedron. The process completes as a birectification, reducing the original faces down to points, and producing the self-dual tetrahedron once again. This polyhedron is topologically related as a part of sequence of regular polyhedra with Schläfli symbols {3,"n"}, continuing into the hyperbolic plane. The tetrahedron is topologically related to a series of regular polyhedra and tilings with order-3 vertex figures. An interesting polyhedron can be constructed from five intersecting tetrahedra. This compound of five


title: Tetrahedron content: tetrahedra has been known for hundreds of years. It comes up regularly in the world of origami. Joining the twenty vertices would form a regular dodecahedron. There are both left-handed and right-handed forms, which are mirror images of each other. The square hosohedron is another polyhedron with four faces, but it does not have triangular faces. In numerical analysis, complicated three-dimensional shapes are commonly broken down into, or approximated by, a polygonal mesh of irregular tetrahedra in the process of setting up the equations for finite element analysis especially in the numerical solution of partial differential equations. These methods have


title: Tetrahedron content: wide applications in practical applications in computational fluid dynamics, aerodynamics, electromagnetic fields, civil engineering, chemical engineering, naval architecture and engineering, and related fields. The tetrahedron shape is seen in nature in covalently bonded molecules. All sp-hybridized atoms are surrounded by atoms (or lone electron pairs) at the four corners of a tetrahedron. For instance in a methane molecule () or an ammonium ion (), four hydrogen atoms surround a central carbon or nitrogen atom with tetrahedral symmetry. For this reason, one of the leading journals in organic chemistry is called "Tetrahedron". The central angle between any two vertices of a


title: Tetrahedron content: perfect tetrahedron is arccos(−), or approximately 109.47°. Water, , also has a tetrahedral structure, with two hydrogen atoms and two lone pairs of electrons around the central oxygen atoms. Its tetrahedral symmetry is not perfect, however, because the lone pairs repel more than the single O–H bonds. Quaternary phase diagrams in chemistry are represented graphically as tetrahedra. However, quaternary phase diagrams in communication engineering are represented graphically on a two-dimensional plane. If six equal resistors are soldered together to form a tetrahedron, then the resistance measured between any two vertices is half that of one resistor. Since silicon is the


title: Tetrahedron content: most common semiconductor used in solid-state electronics, and silicon has a valence of four, the tetrahedral shape of the four chemical bonds in silicon is a strong influence on how crystals of silicon form and what shapes they assume. The Royal Game of Ur, dating from 2600 BC, was played with a set of tetrahedral dice. Especially in roleplaying, this solid is known as a 4-sided die, one of the more common polyhedral dice, with the number rolled appearing around the bottom or on the top vertex. Some Rubik's Cube-like puzzles are tetrahedral, such as the Pyraminx and Pyramorphix. Tetrahedra


title: Tetrahedron content: are used in color space conversion algorithms specifically for cases in which the luminance axis diagonally segments the color space (e.g. RGB, CMY). The Austrian artist Martina Schettina created a tetrahedron using fluorescent lamps. It was shown at the light art biennale Austria 2010. It is used as album artwork, surrounded by black flames on "The End of All Things to Come" by Mudvayne. Stanley Kubrick originally intended the monolith in "" to be a tetrahedron, according to Marvin Minsky, a cognitive scientist and expert on artificial intelligence who advised Kubrick on the HAL 9000 computer and other aspects of


title: Tetrahedron content: the movie. Kubrick scrapped the idea of using the tetrahedron as a visitor who saw footage of it did not recognize what it was and he did not want anything in the movie regular people did not understand. In Season 6, Episode 15 of "Futurama", named "Möbius Dick", the Planet Express crew pass through an area in space known as the Bermuda Tetrahedron. Many other ships passing through the area have mysteriously disappeared, including that of the first Planet Express crew. In the 2013 film "Oblivion" the large structure in orbit above the Earth is of a tetrahedron design and


title: Tetrahedron content: referred to as the Tet. The tetrahedral hypothesis, originally published by William Lowthian Green to explain the formation of the Earth, was popular through the early 20th century. A tetrahedron having stiff edges is inherently rigid. For this reason it is often used to stiffen frame structures such as spaceframes. At some airfields, a large frame in the shape of a tetrahedron with two sides covered with a thin material is mounted on a rotating pivot and always points into the wind. It is built big enough to be seen from the air and is sometimes illuminated. Its purpose is


title: Tetrahedron content: to serve as a reference to pilots indicating wind direction. The skeleton of the tetrahedron (the vertices and edges) form a graph, with 4 vertices, and 6 edges. It is a special case of the complete graph, K, and wheel graph, W. It is one of 5 Platonic graphs, each a skeleton of its Platonic solid. Tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only


title: "Track cycling" content: Track cycling Track cycling is a bicycle racing sport usually held on specially built banked tracks or velodromes (but many events are held at older velodromes where the track banking is relatively shallow) using track bicycles. Track cycling has been around since at least 1870. When cycling was in its infancy, wooden indoor tracks were laid which resemble those of modern velodromes. These velodromes consisted of two straights and slightly banked turns. One appeal of indoor track racing was that spectators could be easily controlled, and hence an entrance fee could be charged, making track racing a lucrative sport. Early


title: "Track cycling" content: track races attracted crowds of up to 2000 people. Indoor tracks also enabled year-round cycling for the first time. The main early centers for track racing in Britain were Birmingham, Sheffield, Liverpool, Manchester and London. The most noticeable changes in over a century of track cycling have concerned the bikes themselves, engineered to be lighter and more aerodynamic to enable ever-faster times. With the exception of the 1912 Olympics, track cycling has been featured in every modern Olympic Games. Women's track cycling was first included in the modern Olympics in 1988. The sport was moved indoors since 2000 Summer Olympics


title: "Track cycling" content: in Sydney. Along the decades, track lengths have been gradually reduced. Early velodromes varied in length between 130 and 500 metres long. By the 1960s, a standard length of 333.33 metres length were commonly used for international competitions. Since 1990, international velodromes usually have a length of 250 metres. Track cycling is particularly popular in Europe, notably Belgium, France, Germany and the United Kingdom where it is often used as off-season training by road racers (professional six-day events were often entered by two-man teams comprising a leading road racer and a track specialist). The sport also has significant followings in


title: "Track cycling" content: Japan and Australia. In the United States, track racing reached a peak of popularity in the 1930s when six-day races were held in Madison Square Garden in New York. The word "Madison" is still used as the name for this type of race in six-day racing. A group of US velodromes formed the American Track Racing Association (ATRA) and its membership now includes more than half of all velodromes in the US. Track cycling events fit into two broad categories: sprint races and endurance races. Riders will typically fall into one category and not compete in the other. Riders with


title: "Track cycling" content: good all round ability in the junior ranks will decide to focus on one area or another before moving up to the senior ranks. Sprint races are generally between 8 and 10 laps in length and focus on raw sprinting power and race tactics over a small number of laps to defeat opponents. Sprint riders will train specifically to compete in races of this length and will not compete in longer endurance races. The main sprint events are:

 Endurance races are held over much longer distances. While these primarily test the riders endurance abilities, the ability


title: "Track cycling" content: to sprint effectively is also required in the Madison, points race and scratch race. The length of these races varies from 12–16 laps for the individual and team pursuit races, up to 120 laps for a full length Madison race in World Championships or Olympic Games. The main endurance events are: Held every four years as part of the Summer Olympics. There are currently 10 events in the Olympics, fewer than appear in the World Championships. At the 2008 Summer Olympics, seven of these events were for men while only three were for women. For the 2012 Summer Olympics, there


title: "Track cycling" content: were five events for both men and women. For the 2016 Summer Olympics the events remained the same. The UCI Track Cycling World Championships are held every year, usually in March or April at the end of the winter track season. There are currently 20 events in the World Championships, 10 for men and 10 for women. Qualification places are determined by different countries performance during the World Cup Classic series held through the season. The UCI Track Cycling World Cup series consists of four or five meetings, held in different countries throughout the world during the winter track cycling


title: "Track cycling" content: season. These meeting include 17 of the 19 events (excluding the omnium for men and women) that take place in a World Championship over three days. Events won and points scored by the riders throughout this series count towards qualification places individually and for their nation in the World Championships at the end of the season. The overall leader in each event wears the white points leaders jersey at each race, with the overall winner at the end of the season keeping the jersey and wearing it at the World Championships. Riders compete for either national teams or trade teams.


title: "Track cycling" content: As World Championship qualification is at stake, the events attract a top field of riders. However, it is common for top riders not to compete at all the events of the series, with teams often using the events to field younger riders or attempt different line-ups at some events. Top riders can still win the series, or obtain good a placing for qualification points for their country, without competing at every event. The UCI Track Cycling World Ranking is based upon the results in all women's UCI-sanctioned races over a twelve-month period. The ranking includes an individual and a nations


title: "Track cycling" content: ranking and includes the disciplines: individual pursuit, points race, scratch, sprint, time trial, keirin, omnium (since 2010-11), team pursuit, team sprint and madison (men only). Several countries run a series of national level events held as part of series throughout each of those countries and sometimes across country borders. Examples of these are the Revolution track series held in both the UK and Australia, and the ATRA NCS series in the United States. Aerodynamic drag is a significant factor in both road and track racing. Frames are often constructed of moulded carbon fiber, for a lightweight design. More recently, track


title: "Track cycling" content: (and road) bikes have employed airfoil designs on the tubes of the frame to reduce aerodynamic drag. On a dedicated track bicycle there are few components, with no need for brakes or derailleurs (and therefore shifters). In fact the only moving part on a bicycle is the drivetrain, which includes the cranks, pedals, the single chainring, one cog fixed to the hub without a freewheel mechanism and the chain itself. Because of this the need for aerodynamic components is minimised, and the aerodynamics to weight ratio makes aerodynamic componentry, at present, an idea with more cons than pros. Given the


title: "Track cycling" content: importance of aerodynamics, the riders' sitting position becomes extremely important. The riding position is similar to the road racing position, but is ultimately dependent on the frame geometry of the bicycle and the handlebars used. Handlebars on track bikes used for longer events such as the points race are similar to the drop bars found on road bicycles. However, in the sprint event the rider's position is more extreme compared with a road rider. The bars are lower and the saddle is higher and more forward. Bars are often narrower with a deeper drop. Steel bars, as opposed to lighter


title: "Track cycling" content: alloys or carbon fibre, are still used by many sprinters for their higher rigidity and durability. In timed events such as the pursuit and the time trial, riders often use aerobars or 'triathlon bars' similar to those found on road time trial bicycles, allowing the rider to position the arms closer together in front of the body. This results in a more horizontal back and presents the minimum frontal area to reduce drag. Aerobars can be separate bars that are attached to time trial or bull horn bars, or they can be part of a one-piece monocoque design. Use of


title: "Track cycling" content: aerobars is permitted only in pursuit and time trial events. Formats of track cycle races are also heavily influenced by aerodynamics. If one rider closely follows, they draft or slipstream another, because the leading rider pushes air around themselves; any rider closely following has to push out less air than the lead rider and thus can travel at the same speed while expending less effort. This fact has led to a variety of racing styles that allow skilled riders or teams to exploit this tactical advantage, as well as formats that simply test strength, speed and endurance. During the early


title: "Track cycling" content: 1990s in individual pursuit events, some riders, most notably Graeme Obree, adopted a straight-armed "Superman"-like position with their arms fully extended horizontally, but this position was subsequently outlawed by the Union Cycliste Internationale (UCI), the sport's ruling body. Recumbent bicycles can actually be ridden faster due to their aerodynamic advantage, but are similarly banned from UCI competition. In addition to regular track racing, tracks are also the venue for many speed records. These are over either a fixed distance or for a fixed period of time. The most famous of these is the hour record, which involves simply riding as


title: "Track cycling" content: far as possible in one hour. The history of the hour record is replete with exploits by some of the greatest names in cycling from both road and track racing (including, among others, Major Taylor, Henri Desgrange, Fausto Coppi, Jacques Anquetil, Eddy Merckx, Francesco Moser, Miguel Indurain and Tony Rominger). Originally, attempts were made at velodromes with reputations for being fast (such as the Velodromo Vigorelli in Milan). More recently, attempts have moved to high-altitude locations, such as Mexico City, where the thinner air results in lower aerodynamic drag, which more than offsets the added difficulty of breathing. Innovations in


title: "Track cycling" content: equipment and the rider's position on the bike have also led to dramatic improvements in the hour record, but have also been a source of controversy (see Graeme Obree). Track cycling Track cycling is a bicycle racing sport usually held on specially built banked tracks or velodromes (but many events are held at older velodromes where the track banking is relatively shallow) using track bicycles. Track cycling has been around since at least 1870. When cycling was in its infancy, wooden indoor tracks were laid which resemble those of modern velodromes. These velodromes consisted of two straights and slightly banked


title: Tetracycline content: the fact that tetracycline binds to the small ribosomal subunit of both prokaryotes and eukaryotes (30S and 40S, respectively). This is because bacteria actively pump tetracycline into their cytoplasm, even against a concentration gradient, whereas mammalian cells do not. This accounts for the relatively small off-site effect of tetracycline on human cells. The tetracyclines, a large family of antibiotics, were discovered as natural products by Benjamin Minge Duggar in 1945 and first prescribed in 1948. Benjamin Duggar, working under Yellapragada Subbarow at Lederle Laboratories, discovered the first tetracycline antibiotic, chlortetracycline (Aureomycin), in 1945. In 1950, Harvard University professor R.B. Woodward


title: Triangle content: Triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices "A", "B", and "C" is denoted formula_1. In Euclidean geometry any three points, when non-collinear, determine a unique triangle and simultaneously, a unique plane (i.e. a two-dimensional Euclidean space). In other words, there is only one plane that contains that triangle, and every triangle is contained in some plane. If the entire geometry is only the Euclidean plane, there is only one plane and all triangles are contained in it; however, in higher-dimensional Euclidean spaces,


title: Triangle content: this is no longer true. This article is about triangles in Euclidean geometry, and in particular, the Euclidean plane, except where otherwise noted. Triangles can be classified according to the lengths of their sides: Hatch marks, also called tick marks, are used in diagrams of triangles and other geometric figures to identify sides of equal lengths. A side can be marked with a pattern of "ticks", short line segments in the form of tally marks; two sides have equal lengths if they are both marked with the same pattern. In a triangle, the pattern is usually no more than 3


title: Triangle content: ticks. An equilateral triangle has the same pattern on all 3 sides, an isosceles triangle has the same pattern on just 2 sides, and a scalene triangle has different patterns on all sides since no sides are equal. Similarly, patterns of 1, 2, or 3 concentric arcs inside the angles are used to indicate equal angles. An equilateral triangle has the same pattern on all 3 angles, an isosceles triangle has the same pattern on just 2 angles, and a scalene triangle has different patterns on all angles since no angles are equal. Triangles can also be classified according to


title: Triangle content: their internal angles, measured here in degrees. A triangle that has two angles with the same measure also has two sides with the same length, and therefore it is an isosceles triangle. It follows that in a triangle where all angles have the same measure, all three sides have the same length, and such a triangle is therefore equilateral. Triangles are assumed to be two-dimensional plane figures, unless the context provides otherwise (see Non-planar triangles, below). In rigorous treatments, a triangle is therefore called a "2-simplex" (see also Polytope). Elementary facts about triangles were presented by Euclid in books 1–4


title: Triangle content: of his "Elements", around 300 BC. The sum of the measures of the interior angles of a triangle in Euclidean space is always 180 degrees. This fact is equivalent to Euclid's parallel postulate. This allows determination of the measure of the third angle of any triangle given the measure of two angles. An "exterior angle" of a triangle is an angle that is a linear pair (and hence supplementary) to an interior angle. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two interior angles that are not adjacent to


title: Triangle content: it; this is the exterior angle theorem. The sum of the measures of the three exterior angles (one for each vertex) of any triangle is 360 degrees. Two triangles are said to be "similar" if every angle of one triangle has the same measure as the corresponding angle in the other triangle. The corresponding sides of similar triangles have lengths that are in the same proportion, and this property is also sufficient to establish similarity. Some basic theorems about similar triangles are: Two triangles that are congruent have exactly the same size and shape: all pairs of corresponding interior angles