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People around the world are taking a new approach when identifying other identities as well as their own; however, there are still places that follow the traditional process of categorization as stated in the following quote. "The impact of patriarchy and traditional assumptions about gender and families are evident in the lives of Chinese migrant workers (Chow, Tong), sex workers and their clients in South Korea (Shin), and Indian widows (Chauhan), but also Ukrainian migrants (Amelina) and Australian men of the new global middle class (Connell)." This text suggests that there are many more intersections of discrimination for people around the globe than Crenshaw originally accounted for in her definition.
For example, Chandra Mohanty discusses alliances between women throughout the world as intersectionality in a global context. She rejects the western feminist theory, especially when it writes about global women of color and generally associated "third world women". She argues that "third world women" are often thought of as a homogenous entity, when, in fact, their experience of oppression is informed by their geography, history, and culture. When western feminists write about women in the global South in this way, they dismiss the inherent intersecting identities that are present in the dynamic of feminism in the global South. Mohanty questions the performance of intersectionality and relationality of power structures within the US and colonialism and how to work across identities with this history of colonial power structures.
This lack of homogeneity and intersecting identities can be seen through Feminism in India, which goes over how women in India practice feminism within social structures and the continuing effects of colonization that differ from that of Western and other non-Western countries. This is elaborated on by Christine Bose, who discusses a global use of intersectionality which works to remove associations of specific inequalities with specific institutions, while showing that these systems generate intersectional effects. She uses this approach to develop a framework that can analyze gender inequalities across different nations and differentiates this from an approach (the one that Mohanty was referring to) which, one, paints national-level inequalities as the same and, two, differentiates only between the global North and South.
This is manifested through the intersection of global dynamics like economics, migration, or violence, with regional dynamics, like histories of the nation or gendered inequalities in education and property education. There is an issue globally with the way the law interacts with intersectionality, for example, the UK's legislation to protect workers rights has a distinct issue with intersectionality. Under the Equality Act 2010, the things that are listed as 'protected characteristics' are "age, disability, gender reassignment, marriage or civil partnership, pregnancy and maternity, race, religion or belief, sex and sexual orientation". "Section 14 contains a provision to cover direct discrimination on up to two combined grounds—known as combined or dual discrimination.
However, this section has never been brought into effect as the government deemed it too 'complicated and burdensome' for businesses." This demonstrates a systematic neglect of the issues that intersectionality presents, because the UK courts have explicitly decided not to cover intersectional discrimination in their courts. Transnational intersectionality Third World feminists and transnational feminists criticize intersectionality as a concept emanating from WEIRD (Western, educated, industrialized, rich, democratic) societies that unduly universalizes women's experiences. Third world feminists have worked to revise Western conceptualizations of intersectionality that assume all women experience the same type of gender and racial oppression. Shelly Grabe coined the term "transnational intersectionality" to represent a more comprehensive conceptualization of intersectionality.
Grabe wrote, "Transnational intersectionality places importance on the intersections among gender, ethnicity, sexuality, economic exploitation, and other social hierarchies in the context of empire building or imperialist policies characterized by historical and emergent global capitalism." Both Third World and transnational feminists advocate attending to "complex and intersecting oppressions and multiple forms of resistance". Social work In the field of social work, proponents of intersectionality hold that unless service providers take intersectionality into account, they will be of less use for various segments of the population, such as those reporting domestic violence or disabled victims of abuse. According to intersectional theory, the practice of domestic violence counselors in the United States urging all women to report their abusers to police is of little use to women of color due to the history of racially motivated police brutality, and those counselors should adapt their counseling for women of color.
Women with disabilities encounter more frequent domestic abuse with a greater number of abusers. Health care workers and personal care attendants perpetrate abuse in these circumstances, and women with disabilities have fewer options for escaping the abusive situation. There is a "silence" principle concerning the intersectionality of women and disability, which maintains an overall social denial of the prevalence of abuse among the disabled and leads to this abuse being frequently ignored when encountered. A paradox is presented by the overprotection of people with disabilities combined with the expectations of promiscuous behavior of disabled women. This leads to limited autonomy and social isolation of disabled individuals, which place women with disabilities in situations where further or more frequent abuse can occur.
Criticism Methods and ideology According to political theorist Rebecca Reilly-Cooper intersectionality relies heavily on standpoint theory, which has its own set of criticisms. Intersectionality posits that an oppressed person is often the best person to judge their experience of oppression; however, this can create paradoxes when people who are similarly oppressed have different interpretations of similar events. Such paradoxes make it very difficult to synthesize a common actionable cause based on subjective testimony alone. Other narratives, especially those based on multiple intersections of oppression, are more complex. Davis (2008) asserts that intersectionality is ambiguous and open-ended, and that its "lack of clear-cut definition or even specific parameters has enabled it to be drawn upon in nearly any context of inquiry".
Rekia Jibrin and Sara Salem argue that intersectional theory creates a unified idea of anti-oppression politics that requires a lot out of its adherents, often more than can reasonably be expected, creating difficulties achieving praxis. They also say that intersectional philosophy encourages a focus on the issues inside the group instead of on society at large, and that intersectionality is "a call to complexity and to abandon oversimplification... this has the parallel effect of emphasizing 'internal differences' over hegemonic structures". Writing in the New Statesman, Helen Lewis adds that in emphasizing internal differences over hegemonic structures, and having complex and at times contradictory recommendations, it can create paralysis because it is not very accessible.
The moral psychologist Jonathan Haidt, in a speech at the American conservative think tank Manhattan Institute, criticized the theory by saying:[In intersectionality] the binary dimensions of oppression are said to be interlocking and overlapping. America is said to be one giant matrix of oppression, and its victims cannot fight their battles separately. They must all come together to fight their common enemy, the group that sits at the top of the pyramid of oppression: the straight, white, cis-gendered, able-bodied Christian or Jewish or possibly atheist male. This is why a perceived slight against one victim group calls forth protest from all victim groups.
This is why so many campus groups now align against Israel. Intersectionality is like NATO for social-justice activists. Barbara Tomlinson is employed at the Department of Women's Studies at UC Santa Barbara and has been critical of the applications of intersectional theory. She has identified several ways in which the conventional theory has been destructive to the movement. She asserts that the common practice of using intersectionality to attack other ways of feminist thinking and the tendency of academics to critique intersectionality instead of using intersectionality as a tool to critique other conventional ways of thinking has been a misuse of the ideas it stands for.
Tomlinson argues that in order to use intersectional theory correctly, intersectional feminists must not only consider the arguments but the tradition and mediums through which these arguments are made. Conventional academics are likely to favor writings by authors or publications with prior established credibility instead of looking at the quality of each piece individually, contributing to negative stereotypes associated with both feminism and intersectionality by having weaker arguments in defense of feminism and intersectionality become prominent based on renown. She goes on to argue that this allows critics of intersectionality to attack these weaker arguments, "[reducing] intersectionality's radical critique of power to desires for identity and inclusion, and offer a deradicalized intersectionality as an asset for dominant disciplinary discourses".
Sharon Goldman of the Israel-America Studies Program at Shalem College also criticized intersectionality on the basis of its being too simplistic. Goldman stipulates that many of the people championed by intersectionality truly are victims of oppression, but her reading of the ideology is that it favors the powerless over the powerful regardless of context. Any group that overcomes adversity, achieves independence, or defends itself successfully is seen as corrupt or imperialist by intersectionality adherents. The examples Goldman gives are American Jews who, inspired by the abject victimhood of the Holocaust, engaged in politics to successfully advance their ideas into the American mainstream.
American Jews are not given the benefit of the doubt by intersectionality adherents because they proudly reject victimization. Lisa Downing argues that because intersectionality focuses too much on group identities, which can lead it to ignore the fact that people are individuals, not just members of a class. Ignoring this can cause intersectionality to lead to a simplistic analysis and inaccurate assumptions about how a person's values and attitudes are determined. Psychology Researchers in psychology have incorporated intersection effects since the 1950s. These intersection effects were based on studying the lenses of biases, heuristics, stereotypes, and judgments. Psychologists have extended research in psychological biases to the areas of cognitive and motivational psychology.
What is found, is that every human mind has its own biases in judgment and decision-making that tend to preserve the status quo by avoiding change and attention to ideas that exist outside one's personal realm of perception. Psychological interaction effects span a range of variables, although person-by-situation effects are the most examined category. As a result, psychologists do not construe the interaction effect of demographics such as gender and race as either more noteworthy or less noteworthy than any other interaction effect. In addition, oppression can be regarded as a subjective construct when viewed as an absolute hierarchy. Even if an objective definition of oppression was reached, person-by-situation effects would make it difficult to deem certain persons or categories of persons as uniformly oppressed.
For instance, black men are stereotypically perceived as violent, which may be a disadvantage in police interactions, but also as physically attractive, which may be advantageous in romantic situations. Psychological studies have shown that the effect of multiplying "oppressed" identities is not necessarily additive, but rather interactive in complex ways. For instance, black gay men may be more positively evaluated than black heterosexual men, because the "feminine" aspects of gay stereotypes temper the hypermasculine and aggressive aspect of black stereotypes. Alan Dershowitz, scholar of United States constitutional law and criminal law in answering a question on the criticism of Israel by intersectional movements, he stated that the concept of intersectionality is an oversimplification of reality that makes LGBT activists stand in solidarity with advocates of Sharia, even though Islamic law denies the rights of the former.
He feels that identity politics does not evaluate ideas or individuals on the basis of the quality of their character. Dershowitz argues that in academia, intersectionality is taught with a large influence from antisemitism. He states that Jews are actually more liberal and supportive of equal rights than many other religious sects. Writer and political pro-Israel activist Chloé Valdary considers intersectionality "a rigid system for determining who is virtuous and who is not, based on traits like skin color, gender, and financial status". Valdary also states:Intersectionality's greatest flaw is in reducing human beings to political abstractions, which is never a tendency that turns out well—in part because it so severely flattens our complex human experience, and therefore fails to adequately describe reality.
As it turns out, one can be personally successful and still come from a historically oppressed community—or vice versa. The human experience is complex and multifaceted and deeper than the superficial ways in which intersectionalists describe it. As zealotry Conservative political commentator Andrew Sullivan argues that the practice of intersectionality "manifests itself almost as a religion. It posits a classic orthodoxy through which all of human experience is explained—and through which all speech must be filtered." David A. French, writer for the National Review, states that proponents of intersectionality are "zealots of a new religious faith" intending to fill a "religion-shaped hole in the human heart".
See also Black feminism Caste Humanism Identitarianism Kyriarchy Oppression Olympics Privilege (social inequality) Standpoint theory Transnational feminism Triple oppression Womanism Notes References External links "Demarginalizing the Intersection of Race and Sex: A Black Feminist Critique of Antidiscrimination Doctrine, Feminist Theory and Antiracist Politics", by Kimberlé Crenshaw, 1989 Black Feminist Thought in the Matrix of Domination A Brief History of Black Feminist Thought Intersectionality 101 Category:Feminist theory Category:Social constructionism
De'Anthony Melton (born May 28, 1998) is an American professional basketball player for the Memphis Grizzlies of the National Basketball Association (NBA). He was selected by the Houston Rockets in the second round of the 2018 NBA draft with the 46th pick, but was traded to Phoenix before his rookie season began. He played college basketball for the USC Trojans of the Pac-12 Conference, but did not play in the 2017–18 season due to the events relating to the 2017–18 NCAA Division I men's basketball corruption scandal. High school career Melton attended Crespi Carmelite High School in Encino, California. As a three-star recruit, he committed to the USC Trojans on November 20, 2015.
College career Freshman year Melton played in 36 games, starting in the last 25 games of the season that year. He averaged 8.3 points, 3.5 assists, 4.7 rebounds, and 1.9 steals per game. On January 25, in an 84–76 win over future #2 pick Lonzo Ball and UCLA, Melton had 13 points, 9 rebounds, 5 assists, 4 steals, and a block. He followed this performance with a 16-point, 7 rebound, 6 assist, 6 steal, and 2 block game against future #1 pick Markelle Fultz and Washington on February 1. Melton became the first freshman to record at least 300 points, 150 rebounds, 100 assists, 60 steals, and 35 blocks in their starting season since Dwyane Wade.
As a result of those starts, he was projected to be a key player for USC's upcoming season. Sophomore year On September 26, 2017, federal prosecutors in New York announced charges of fraud and corruption against 10 people involved in college basketball, including USC assistant coach Tony Bland. The charges allege that Bland and other members involved allegedly received benefits from financial advisers and others to influence student-athletes to retain their services. Following the announcement, USC indefinitely suspended Melton in relation to the scandal due to a family member's involvement there. On February 21, 2018, Melton announced he would withdraw from USC and declare for the 2018 NBA draft.
Professional career Phoenix Suns (2018–2019) Melton was selected with the 46th pick by the Houston Rockets in the 2018 NBA draft. He played for the Rockets during the 2018 NBA Summer League in Las Vegas. In five games, Melton recorded 16.4 points, 7.2 rebounds, 4.0 assists, and 3.0 steals per game. His best game during the event was on July 9 where he made 26 points, 10 rebounds, and 5 assists in a 104–90 win over the Los Angeles Clippers. On August 31, Melton was traded alongside Ryan Anderson to the Phoenix Suns in exchange for Brandon Knight and Marquese Chriss.
Melton signed his first NBA contract with the Suns on September 21. He made his professional debut with the Suns on October 22, 2018, in a 123–103 loss to the Golden State Warriors, but did not record significant playing time until October 31 in a 120–90 loss to the San Antonio Spurs. Melton was assigned to the Northern Arizona Suns, Phoenix's development team in the NBA G League, for their first game of the season on November 3, 2018. He played the one game, a 118–108 loss to the Santa Cruz Warriors, and was recalled to Phoenix the next day.
On December 4, Melton recorded a professional high of 21 points in a 122–105 loss to the Sacramento Kings. He also recorded season-high 10 assists in a 102–93 win over the Denver Nuggets on January 12, 2019. Melton shined on the defensive end of the court as well as put his playmaking skills on display in the Suns victory against the Sacramento Kings on January 8. Melton scored 10 points with eight assists, four rebounds, four steals and two blocks in the win. Melton later suffered a right ankle sprain on January 24, leaving him out of action for close to a month before returning briefly for the NBA G League on February 20, and then played in the NBA proper three days later against the Atlanta Hawks.
On March 16, Melton recorded a career-high 8 rebounds in a 138–136 overtime win over the New Orleans Pelicans. Melton was 2nd in the NBA in rookie steals in 2018–19 with 1.36 spg. When Melton is on the floor, he recorded steals more frequently than any other player in the NBA, ranking 1st with 3.3 steals per 100 possessions (min. 700 total minutes played). Among rookies to start in majority of their appearances, Melton's 3.3 steals per 100 possessions are tied for 6th most all-time in Basketball-Reference's database. Memphis Grizzlies (2019–present) On July 7, 2019, the Suns traded Melton, Josh Jackson, and two second round picks to the Memphis Grizzlies in exchange for Jevon Carter and Kyle Korver.
Career statistics NBA Regular season |- | style="text-align:left;"| | style="text-align:left;"| Phoenix | 50 || 31 || 19.7 || .391 || .305 || .750 || 2.7 || 3.2 || 1.4 || .5 || 5.0 |- class="sortbottom" | style="text-align:center;" colspan="2"| Career | 50 || 31 || 19.7 || .391 || .305 || .750 || 2.7 || 3.2 || 1.4 || .5 || 5.0 College |- | style="text-align:left;"| 2016–17 | style="text-align:left;"| USC | 36 || 25 || 27.0 || .437 || .284 || .706 || 4.7 || 3.5 || 1.9 || 1.0 || 8.3 References External links USC Trojans Bio Category:1998 births Category:Living people Category:African-American basketball players Category:American men's basketball players Category:Basketball players from California Category:Houston Rockets draft picks Category:Memphis Grizzlies players Category:Northern Arizona Suns players Category:People from North Hollywood, Los Angeles Category:Phoenix Suns players Category:Point guards Category:Shooting guards Category:Sportspeople from Los Angeles Category:USC Trojans men's basketball players
The small forward (SF), also known as the three, is one of the five positions in a regulation basketball game. Small forwards are typically shorter, quicker, and leaner than power forwards and centers, but typically taller, larger and stronger than either of the guard positions. The small forward is considered to be perhaps the most versatile of the five main basketball positions. In the NBA, small forwards usually range from 6' 4" (1.93 m) to 6' 8" (2.03 m) without shoes while in the WNBA, small forwards are usually between 5' 10" (1.78 m) to 6' 1" (1.85 m). Small forwards are responsible for scoring points, defending and often as secondary or tertiary rebounders behind the power forward and center, although a few have considerable passing responsibilities.
Many small forwards in professional basketball are prolific scorers. The styles with which small forwards amass their points vary widely. Some players at the position are very accurate shooters, others prefer to initiate physical contact with opposing players, and still others are primarily slashers who also possess jump shots. In some cases, small forwards position on the baseline or as off-the-ball specialists. Small forwards who are defensive specialists are very versatile as they can often guard multiple positions using their size, speed, and strength. Small forwards that are inducted in the Naismith Memorial Basketball Hall of Fame include Julius Erving, Cheryl Miller, Larry Bird, Sheryl Swoopes, James Worthy, Elgin Baylor, Scottie Pippen, Dominique Wilkins and Rick Barry.
References Category:Basketball positions Category:Small forwards
Carl Braun (22 March 1822 – 28 March 1891), sometimes Carl Rudolf Braun alternative spelling: Karl Braun, or Karl von Braun-Fernwald, name after knighthood Carl Ritter von Fernwald Braun was an Austrian obstetrician. He was born 22 March 1822 in Zistersdorf, Austria, son of the medical doctor Carl August Braun. Career Carl Braun studied in Vienna from 1841 and, in 1847, took the position of Sekundararzt (assistant doctor) in the Vienna General Hospital. In 1849 he succeeded Ignaz Semmelweis as assistant to professor Johann Klein at the hospital's first maternity clinic, a position he held until 1853. In 1853, after Braun became a Privatdozent, he was appointed ordinary professor of obstetrics in Trient and vice-director of the Tiroler Landes-Gebär- und Findelanstalt.
In November 1856 he was called to Vienna to succeed Johann Klein as professor of obstetrics. On Braun's recommendation, the hospital's first gynaecology clinic was created in 1858, under his direction. He is credited for establishing gynaecology as an independent field of study In 1867-1871 he was appointed dean of the medical faculty, and in the academic year 1868/69 was made rector of the University of Vienna. He was knighted in 1872 (cf. the title Ritter) and in 1877 became a Hofrat, a title reserved for very eminent professors. His name is associated with a disorder of pregnancy called the "Braun-Fernwald sign".
This sign is described as an asymmetrical enlargement and softening of the uterine fundus at the site of implantation at 4–5 weeks. Views on puerperal fever In full harmony with his contemporaries, Braun identified 30 causes of childbed fever opposing Ignaz Semmelweis's thesis that 'cadaverous poisoning' was the only cause of childbed fever. Despite this scholar opposition, Braun maintained a relatively low mortality rate in the First Division, roughly consistent with the rate Semmelweis himself achieved, as historical mortality rates of puerperal fever in the period April 1849 to end 1953 show. These results suggest that Braun continued, assiduously, to require hand disinfection before attending women and did not let mortality return to the high levels before Semmelweis introduced the chlorine washings.
Works Klinik der Geburtshilfe und Gynäkologie (im Verein mit Chiari und Spaeth, Erlangen 1855) ([The] Maternity and Gynaecology Clinik, together with Chiari and Spaeth, Erlangen 1855) Lehrbuch der Geburtshilfe mit Berücksichtigung der Puerperalprocesse und der Operationstechnik (Wien 1857) (Textbook of obstetrics [also] concerning the puerperal process and surgical technique). Google book search https://books.google.com/books?id=3OOCGAAACAAJ. Lehrbuch der gesammten Gynäkologie (2. Aufl., Ib. 1881) (Textbook of Gynaecology, 2nd ed. 1881). WorldCat entry: http://www.worldcat.org/oclc/8179918 Über 12 Fälle von Kaiserschnitt und Hysterectomie bei engem Becken (mit achtmaligem günstigem Ausgang) (On 12 cases of caesarean section and hysterectomy with narrow pelvis (with eight successful outcomes)) References p92 footnote 15 Braun, Carl Ritter von Fernwald.
Pagel: Biographisches Lexikon hervorragender Ärzte des neunzehnten Jahrhunderts. Berlin, Wien 1901, Sp. 229-231. (in German) Corroborated by source provided in Swedish wiki Nordisk familjebok, 1904–1926 http://runeberg.org/nfbd/0035.html (in Swedish) Österreich-Lexikon http://aeiou.iicm.tugraz.at/aeiou.encyclop.b/b717161.htm, retrieved 28 Aug 2008, Category:1822 births Category:1891 deaths Category:People from Zistersdorf Category:Austrian gynaecologists Category:Austrian obstetricians Category:University of Vienna faculty Category:19th-century Austrian physicians
The Gibson J-160E is one of the first acoustic-electric guitars produced by the Gibson Guitar Corporation. The J-160E was Gibson's second attempt at creating an acoustic-electric guitar (the first being the small-body CF-100E). The basic concept behind the guitar was to fit a single-pickup into a normal-size dreadnought acoustic guitar. The J-160E used plywood for most of the guitar's body, and was ladder-braced, whereas other acoustic Gibsons were X-braced. The rosewood fingerboard had trapezoid inlays, and the guitar had an adjustable bridge. For amplification, a single-coil pickup (an uncovered P-90 pickup) was installed under the top of the body with the pole screws protruding through the top at the end of the fingerboard, with a volume and a tone knob.
John Lennon and George Harrison frequently used one with The Beatles, both on-stage and in the studio. Gibson produces a standard J-160E and a John Lennon J-160E Peace model, based on the J-160E he used during the Bed-In days of 1969. Epiphone makes an EJ-160E John Lennon replica signature model. Notable J-160E users Barry Gibb of the Bee Gees plays a J-160E, and can be seen in several live performances of the band from 1967 to 1968. Sam Lightnin' Hopkins played a J-160E which is on display at the Rock Hall of Fame. Richard Barone plays a J-160E as his primary acoustic guitar on solo and band performances and with The Bongos.
Pete Doherty of the Libertines/babyshambles plays a J-160E during most of his solo appearances. Chad Stuart and Jeremy Clyde of Chad and Jeremy played J-160E guitars from 1964 to 1968. Peter Asher and Gordon Waller of Peter and Gordon played J-160E guitars, and can be seen in live acts in US during the 2000s. Steve Marriott of Small Faces used a J-160E as the main acoustic guitar for the 1967 album Small Faces. Mike Viola of Mike Viola and The Candy Butchers uses a J-160E with a Fishman blend Pickup. Elvis Costello uses a J-160E. Aimee Mann used a vintage J-160E as her primary stage guitar from the early 1990s through 2008, when the guitar was damaged in an auto accident John McClung of Weekend State uses a J-160E.
Dan Healy uses a 2005 Gibson Historic Collection J-160E with a P-100 pickup, solid spruce top with X bracing and solid mahogany back and sides. The guitar can be heard in tracks 2, 3, 4, 6, 7, 9 and 10 on the Ronan Keating 2016 album Time of My Life. George Harrison's Official Facebook Page posted this note on June 4, 2015: "George used his J-160E live, on television, and it's the only instrument used on every Beatles album from Please Please Me to Abbey Road." References J160E Category:The Beatles' musical instruments Category:Products introduced in 1954 Category:1954 in music
Telegraph Island (also known as Jazīrat al Maqlab or جزيرة_تليغراف, and Jazīrat Şaghīr) is located in the Elphinstone Inlet or Khor Ash Sham, the inner inlet of Khasab Bay, less than 400 meters off the shore of the Musandam Peninsula, and less than 500 meters south of much larger but also much lesser known Sham Island, both of which are parts of the Sultanate of Oman. It is 160 meters long, and up to 90 meters wide, yielding an area of 1.1 hectares. The name as "Telegraph" comes from the telegraph-cable repeater station built on the island in 1864. The inlet at the island is a fjord surrounded by high mountains, with notable geology in the rock strata which dip downwards under the immense pressures caused by the Arabian tectonic plate meeting (and subducting beneath) the Eurasian plate.
In the 19th century, it was the location of a British repeater station used to boost telegraphic messages along the Persian Gulf submarine cable (see below), which was part of the London to Karachi telegraphic cable. It was not an easy posting for the operators, with the severe summer heat and hostility of local tribes making life extremely uncomfortable. Because of this, the island is, according to some travel agents and journalists, where the expression "go round the bend" comes from, a reference to the heat making British officers desperate to return to civilization, which meant a voyage around the bend in the Strait of Hormuz back to India.
Today, Telegraph Island is an eerie reminder of the British Empire. Abandoned in the mid-1870s, the island has remained deserted and only the crumbling ruins of the repeater station and the operators' quarters can be seen. Tourism has grown in the Persian Gulf region, so the island is regularly visited by dhows carrying tourists to view the ruins and to fish and snorkel in the waters around it. However, the intense heat (particularly in the summer months) endures. Geology of the Musandam Peninsula Telegraph Island is situated in a fjord at the northern end of the Musandam Peninsula, which forms part of the Oman mountain range described by the geologist George Martin Lees as "projecting like a spur into the vitals of Persia" (today's Iran).
Being part of the edge of the Arabian tectonic plate, the rock stata are subjected to massive pressure as the plate subducts beneath the Eurasian plate. The result is that Musandam is being pushed downwards at approximately per year at its northernmost point, with spectacular results. Fresh-water springs that once flowed over the land may have become submerged, possibly giving rise to stories of sailors diving into the sea to collect fresh water in leather bags. The Persian Gulf Telegraphic Cable Background The expansion of the British Empire in the 19th century required a fast and reliable system of communication to enable the British government in London to issue instructions and receive information quickly.
The Indian Mutiny of 1857, followed by the annexation of India in 1858, emphasised this need. By 1856, cables had been laid linking Britain with North Africa and the Ottoman Empire. A scheme to lay a cable through Mesopotamia to the head of the Persian Gulf failed when the Turks refused to grant permission. By 1858, the British government, through grants and subsidies, was actively encouraging schemes to establish a telegraphic link between Britain and India. In 1859, the Red Sea Telegraph Company laid a submarine cable through the Red Sea and Indian Ocean to Bombay. It became apparent that the cable was not sufficiently robust to withstand the conditions, and that too little slack had been built into the cable, leading to breaks in the line.
The cable was a failure, and no messages were passed between London and Bombay. The British government created the Indo-European Telegraph Department in 1862 to connect a telegraphic link between Karachi and lines in the Ottoman Empire. It planned to run the cable along the Makran coast between Fao, Bushire, and Gwadar. A number of local agreements were made with tribal leaders along the proposed route and a cable laid from Karachi as far as Gwadar. However, the Persian government declined to grant permission to extend the line to Ottoman territory. Attention was focused on the Persian Gulf and, following a report by Lieutenant Colonel Patrick Stewart, a decision was taken to lay a cable to Musandam from Gwadar.
Laying the cable In 1864 the Government of India contracted the Gutta Percha Company to manufacture the core. Henley’s Telegraph Works was to construct the armouring, and Sir Charles Bright was appointed the consulting engineer. Because telegraph signals tended to fade over distance, it was necessary to build a series of repeater stations along the cable route to boost them, hence the decision to build a repeater station at Musandam. The cable was landed on a small rocky island in the Elphinstone Inlet (Khor Ash Sham) of the Musandam Peninsula. Hence this island became known as Telegraph Island. A repeater station was built on the island, which was about a mile offshore, because of fears about the volatile tribes on the mainland, primarily the Zahuriyeen tribe, who lived on the nearby Maqtab Isthmus.
The work of laying the Gwadar to Musandam section took a month to complete. The station at Musandam was fitted with telegraphic equipment by Siemens. There were quarters for the telegraph operators, together with those for the servants and two hulks fitted up for staff who wanted a break from the monotony of living on the island. They had a couple of boats for their leisure time, and regular newspaper deliveries. This, together with the work of maintaining the cable, and visits from steamers to change over staff and bring supplies, it was hoped would keep the staff occupied. The Musandam to Bushire section was completed on 25 March 1864, and that between Kurrachee and the head of the Gulf at Fao, on 5 April.
Unfortunately, the Turks experienced problems in completing the link from Baghdad to Fao, primarily due to attacks by hostile tribes in Mesopotamia, which were only stopped after negotiations with local sheikhs and by stationing guards at close intervals along the line. The through connection was eventually achieved in 1865. The telegraph station was abandoned in December 1868 when the cable was diverted to Henjam and Jask. In 1870, after a new cable was laid from Bombay to Aden, there was a significant reduction in the use of the Persian Gulf cable. Operation The Persian Gulf cable was never entirely reliable, with interruptions and errors at the repeater stations.
A message usually took a minimum of five days to reach London from Karachi. Another problem was the destructive influence of the teredo (a wormlike bivalve mollusk) on the gutta percha insulation of the cables, which was more susceptible to them than the india rubber insulation used on other cables in warm water areas. In 1865, to address these problems, it was planned to run a second line onshore through Persia, and although this line was built, it was abandoned in 1868 because of objections from the Turks who regarded it as unnecessary competition with the Fao to Baghdad line.
After the repeater station had been established on Telegraph Island, it became apparent that the location was unsuitable, with extreme heat making life unbearable for the operators and leading to the death of two in two years. The opposition of the tribesmen, although at times subdued, was never overcome entirely with the result that it was constantly necessary to have a gun boat in the vicinity of the island. After it was evacuated, Telegraph Island remained an important strategic point, as evidenced by the decision of the British Government in 1904 to erect flagpoles there. However, an agreement could not be reached about which flag to fly – the Union Jack or the Blue Ensign of the Royal India Marine.
The former would imply sovereignty over the island, which Britain did not have, while the latter might bring responsibility on the British to defend it, which the government did not want. It was eventually decided to fly neither flag, and all the flagpoles were removed except one. The flagpole could not be seen from the open sea but is not clear whether officials in London ever appreciated this fact. Tourism The Musandam Peninsula is a popular tourist destination, with modern hotels in and around the town of Khasab and boat trips to the Elphinstone Inlet. Tourist dhows run day trips to Telegraph Island, tying up beside the island for a few hours while visitors can climb up the steps and visit the crumbling ruins of the repeater station and operators' quarters.
However, changes in the tide of can make it difficult to land passengers at certain times of day. Visitors may also swim and snorkel in the surrounding waters. The inlet is calm and sheltered, and it is possible to see dolphins swimming alongside the dhows that make their way to and from Khasab. Cultural references In an episode of the fantasy television show Warehouse 13, titled "Around the Bend", aired on the Syfy Channel on 10 August 2010, a fictional military telegraph from the island is recognised by a character as causing "violent insanity". Agent Lattimer goes "around the bend" after tapping its lever.
The island is featured in Telegraph Island: Jason Smiley Stewart – My Life Story, Volume 2, a novel loosely based on fact. References External links History of the Atlantic Cable & Undersea Communications Cable&Wireless Worldwide Category:Islands of Oman Category:Musandam Governorate
"Black Coffee" is a funk rock song written by Tina Turner. It originally recorded by R&B duo Ike & Tina Turner for their 1972 album Feel Good on United Artists Records. English rock band Humble Pie released a popular rendition of the song in 1973. Humble Pie version Humble Pie covered "Black Coffee" for their 1973 album Eat It on A&M Records. Their version features the Blackberries singing backing vocals. Steve Marriott adjusted some of the lyrics. In the original version, Tina Turner sings, "My skin is brown but my mind is black." Marriott sings, "My skin is white but my soul is black."
When questioned about the lyrics by journalist James Johnson of NME, Marriott said: "I just sang it 'cos I loved the song and it was an interpretation of somebody else's lyrics. People should have known that I've been into black music for years anyway." Humble Pie promoted the song on the British TV program The Old Grey Whistle Test in March 1973. The single didn't make an impression on the charts, but it became one of Humble Pie's best known songs, and is considered one of Marriott's best vocal performances. In 1989, Marriott and Clem Clempson recorded the song as a jingle for Nescafé coffee's new product – Blend 37.
They won a Gold Medal Award for the top Commercial 1989. Critical reception Cash Box (March 3, 1973): "From their forthcoming album, 'Eat It' comes this enticing Ike & Tina Turner composition tailor made for Steve Marriott's raspy vocals. This is a slice of the Pie you won't want to miss. Chart performance References Category:1972 songs Category:1973 singles Category:Ike & Tina Turner songs Category:Humble Pie (band) songs Category:Songs written by Tina Turner Category:Song recordings produced by Ike Turner Category:A&M Records singles Category:Funk rock songs
Oxprenolol (brand names Trasacor, Trasicor, Coretal, Laracor, Slow-Pren, Captol, Corbeton, Slow-Trasicor, Tevacor, Trasitensin, Trasidex) is a non-selective beta blocker with some intrinsic sympathomimetic activity. It is used for the treatment of angina pectoris, abnormal heart rhythms and high blood pressure. Oxprenolol is a lipophilic beta blocker which passes the blood–brain barrier more easily than water-soluble beta blockers. As such, it is associated with a higher incidence of CNS-related side effects than beta blockers with more hydrophilic molecules such as atenolol, sotalol and nadolol. Oxprenolol is a potent beta blocker and should not be administered to asthmatics under any circumstances due to their low beta levels as a result of depletion due to other asthma medication, and because it can cause irreversible, often fatal, airway failure and inflammation.
Pharmacology Pharmacodynamics Oxprenolol is a beta blocker. In addition, it has been found to act as an antagonist of the serotonin 5-HT1A and 5-HT1B receptors with respective Ki values of 94.2 nM and 642 nM in rat brain tissue. Chemistry Stereochemistry Oxprenolol is a chiral compound, the beta blocker is used as a racemate, e. g. a 1:1 mixture of (R)-(+)-oxprenolol and (S)-(–)-oxprenolol. Analytical methods (HPLC) for the separation and quantification of (R)-(+)-oxprenolol and (S)-(–)-oxprenolol in urine and in pharmaceutical formulations have been described in the literature. References Category:5-HT1A antagonists Category:5-HT1B antagonists Category:Abandoned drugs Category:Allyl compounds Category:Beta blockers Category:N-isopropyl-phenoxypropanolamines Category:Sympathomimetic amines Category:Catechol ethers
In Euclidean plane geometry, Apollonius's problem is to construct circles that are tangent to three given circles in a plane (Figure 1). Apollonius of Perga (c. 262 190 BC) posed and solved this famous problem in his work (, "Tangencies"); this work has been lost, but a 4th-century AD report of his results by Pappus of Alexandria has survived. Three given circles generically have eight different circles that are tangent to them (Figure 2), a pair of solutions for each way to divide the three given circles in two subsets (there are 4 ways to divide a set of cardinality 3 in 2 parts).
In the 16th century, Adriaan van Roomen solved the problem using intersecting hyperbolas, but this solution does not use only straightedge and compass constructions. François Viète found such a solution by exploiting limiting cases: any of the three given circles can be shrunk to zero radius (a point) or expanded to infinite radius (a line). Viète's approach, which uses simpler limiting cases to solve more complicated ones, is considered a plausible reconstruction of Apollonius' method. The method of van Roomen was simplified by Isaac Newton, who showed that Apollonius' problem is equivalent to finding a position from the differences of its distances to three known points.
This has applications in navigation and positioning systems such as LORAN. Later mathematicians introduced algebraic methods, which transform a geometric problem into algebraic equations. These methods were simplified by exploiting symmetries inherent in the problem of Apollonius: for instance solution circles generically occur in pairs, with one solution enclosing the given circles that the other excludes (Figure 2). Joseph Diaz Gergonne used this symmetry to provide an elegant straightedge and compass solution, while other mathematicians used geometrical transformations such as reflection in a circle to simplify the configuration of the given circles. These developments provide a geometrical setting for algebraic methods (using Lie sphere geometry) and a classification of solutions according to 33 essentially different configurations of the given circles.
Apollonius' problem has stimulated much further work. Generalizations to three dimensions—constructing a sphere tangent to four given spheres—and beyond have been studied. The configuration of three mutually tangent circles has received particular attention. René Descartes gave a formula relating the radii of the solution circles and the given circles, now known as Descartes' theorem. Solving Apollonius' problem iteratively in this case leads to the Apollonian gasket, which is one of the earliest fractals to be described in print, and is important in number theory via Ford circles and the Hardy–Littlewood circle method. Statement of the problem The general statement of Apollonius' problem is to construct one or more circles that are tangent to three given objects in a plane, where an object may be a line, a point or a circle of any size.
These objects may be arranged in any way and may cross one another; however, they are usually taken to be distinct, meaning that they do not coincide. Solutions to Apollonius' problem are sometimes called Apollonius circles, although the term is also used for other types of circles associated with Apollonius. The property of tangency is defined as follows. First, a point, line or circle is assumed to be tangent to itself; hence, if a given circle is already tangent to the other two given objects, it is counted as a solution to Apollonius' problem. Two distinct geometrical objects are said to intersect if they have a point in common.
By definition, a point is tangent to a circle or a line if it intersects them, that is, if it lies on them; thus, two distinct points cannot be tangent. If the angle between lines or circles at an intersection point is zero, they are said to be tangent; the intersection point is called a tangent point or a point of tangency. (The word "tangent" derives from the Latin present participle, tangens, meaning "touching".) In practice, two distinct circles are tangent if they intersect at only one point; if they intersect at zero or two points, they are not tangent.
The same holds true for a line and a circle. Two distinct lines cannot be tangent in the plane, although two parallel lines can be considered as tangent at a point at infinity in inversive geometry (see below). The solution circle may be either internally or externally tangent to each of the given circles. An external tangency is one where the two circles bend away from each other at their point of contact; they lie on opposite sides of the tangent line at that point, and they exclude one another. The distance between their centers equals the sum of their radii.
By contrast, an internal tangency is one in which the two circles curve in the same way at their point of contact; the two circles lie on the same side of the tangent line, and one circle encloses the other. In this case, the distance between their centers equals the difference of their radii. As an illustration, in Figure 1, the pink solution circle is internally tangent to the medium-sized given black circle on the right, whereas it is externally tangent to the smallest and largest given circles on the left. Apollonius' problem can also be formulated as the problem of locating one or more points such that the differences of its distances to three given points equal three known values.
Consider a solution circle of radius rs and three given circles of radii r1, r2 and r3. If the solution circle is externally tangent to all three given circles, the distances between the center of the solution circle and the centers of the given circles equal , and , respectively. Therefore, differences in these distances are constants, such as ; they depend only on the known radii of the given circles and not on the radius rs of the solution circle, which cancels out. This second formulation of Apollonius' problem can be generalized to internally tangent solution circles (for which the center-center distance equals the difference of radii), by changing the corresponding differences of distances to sums of distances, so that the solution-circle radius rs again cancels out.
The re-formulation in terms of center-center distances is useful in the solutions below of Adriaan van Roomen and Isaac Newton, and also in hyperbolic positioning or trilateration, which is the task of locating a position from differences in distances to three known points. For example, navigation systems such as LORAN identify a receiver's position from the differences in arrival times of signals from three fixed positions, which correspond to the differences in distances to those transmitters. History A rich repertoire of geometrical and algebraic methods have been developed to solve Apollonius' problem, which has been called "the most famous of all" geometry problems.
The original approach of Apollonius of Perga has been lost, but reconstructions have been offered by François Viète and others, based on the clues in the description by Pappus of Alexandria. The first new solution method was published in 1596 by Adriaan van Roomen, who identified the centers of the solution circles as the intersection points of two hyperbolas. Van Roomen's method was refined in 1687 by Isaac Newton in his Principia, and by John Casey in 1881. Although successful in solving Apollonius' problem, van Roomen's method has a drawback. A prized property in classical Euclidean geometry is the ability to solve problems using only a compass and a straightedge.
Many constructions are impossible using only these tools, such as dividing an angle in three equal parts. However, many such "impossible" problems can be solved by intersecting curves such as hyperbolas, ellipses and parabolas (conic sections). For example, doubling the cube (the problem of constructing a cube of twice the volume of a given cube) cannot be done using only a straightedge and compass, but Menaechmus showed that the problem can be solved by using the intersections of two parabolas. Therefore, van Roomen's solution—which uses the intersection of two hyperbolas—did not determine if the problem satisfied the straightedge-and-compass property. Van Roomen's friend François Viète, who had urged van Roomen to work on Apollonius' problem in the first place, developed a method that used only compass and straightedge.
Prior to Viète's solution, Regiomontanus doubted whether Apollonius' problem could be solved by straightedge and compass. Viète first solved some simple special cases of Apollonius' problem, such as finding a circle that passes through three given points which has only one solution if the points are distinct; he then built up to solving more complicated special cases, in some cases by shrinking or swelling the given circles. According to the 4th-century report of Pappus, Apollonius' own book on this problem—entitled (, "Tangencies"; Latin: De tactionibus, De contactibus)—followed a similar progressive approach. Hence, Viète's solution is considered to be a plausible reconstruction of Apollonius' solution, although other reconstructions have been published independently by three different authors.
Several other geometrical solutions to Apollonius' problem were developed in the 19th century. The most notable solutions are those of Jean-Victor Poncelet (1811) and of Joseph Diaz Gergonne (1814). Whereas Poncelet's proof relies on homothetic centers of circles and the power of a point theorem, Gergonne's method exploits the conjugate relation between lines and their poles in a circle. Methods using circle inversion were pioneered by Julius Petersen in 1879; one example is the annular solution method of HSM Coxeter. Another approach uses Lie sphere geometry, which was developed by Sophus Lie. Algebraic solutions to Apollonius' problem were pioneered in the 17th century by René Descartes and Princess Elisabeth of Bohemia, although their solutions were rather complex.
Practical algebraic methods were developed in the late 18th and 19th centuries by several mathematicians, including Leonhard Euler, Nicolas Fuss, Carl Friedrich Gauss, Lazare Carnot, and Augustin Louis Cauchy. Solution methods Intersecting hyperbolas The solution of Adriaan van Roomen (1596) is based on the intersection of two hyperbolas. Let the given circles be denoted as C1, C2 and C3. Van Roomen solved the general problem by solving a simpler problem, that of finding the circles that are tangent to two given circles, such as C1 and C2. He noted that the center of a circle tangent to both given circles must lie on a hyperbola whose foci are the centers of the given circles.
To understand this, let the radii of the solution circle and the two given circles be denoted as rs, r1 and r2, respectively (Figure 3). The distance d1 between the centers of the solution circle and C1 is either or , depending on whether these circles are chosen to be externally or internally tangent, respectively. Similarly, the distance d2 between the centers of the solution circle and C2 is either or , again depending on their chosen tangency. Thus, the difference between these distances is always a constant that is independent of rs. This property, of having a fixed difference between the distances to the foci, characterizes hyperbolas, so the possible centers of the solution circle lie on a hyperbola.
A second hyperbola can be drawn for the pair of given circles C2 and C3, where the internal or external tangency of the solution and C2 should be chosen consistently with that of the first hyperbola. An intersection of these two hyperbolas (if any) gives the center of a solution circle that has the chosen internal and external tangencies to the three given circles. The full set of solutions to Apollonius' problem can be found by considering all possible combinations of internal and external tangency of the solution circle to the three given circles. Isaac Newton (1687) refined van Roomen's solution, so that the solution-circle centers were located at the intersections of a line with a circle.
Newton formulates Apollonius' problem as a problem in trilateration: to locate a point Z from three given points A, B and C, such that the differences in distances from Z to the three given points have known values. These four points correspond to the center of the solution circle (Z) and the centers of the three given circles (A, B and C). Instead of solving for the two hyperbolas, Newton constructs their directrix lines instead. For any hyperbola, the ratio of distances from a point Z to a focus A and to the directrix is a fixed constant called the eccentricity.
The two directrices intersect at a point T, and from their two known distance ratios, Newton constructs a line passing through T on which Z must lie. However, the ratio of distances TZ/TA is also known; hence, Z also lies on a known circle, since Apollonius had shown that a circle can be defined as the set of points that have a given ratio of distances to two fixed points. (As an aside, this definition is the basis of bipolar coordinates.) Thus, the solutions to Apollonius' problem are the intersections of a line with a circle. Viète's reconstruction As described below, Apollonius' problem has ten special cases, depending on the nature of the three given objects, which may be a circle (C), line (L) or point (P).
By custom, these ten cases are distinguished by three letter codes such as CCP. Viète solved all ten of these cases using only compass and straightedge constructions, and used the solutions of simpler cases to solve the more complex cases. Viète began by solving the PPP case (three points) following the method of Euclid in his Elements. From this, he derived a lemma corresponding to the power of a point theorem, which he used to solve the LPP case (a line and two points). Following Euclid a second time, Viète solved the LLL case (three lines) using the angle bisectors.
He then derived a lemma for constructing the line perpendicular to an angle bisector that passes through a point, which he used to solve the LLP problem (two lines and a point). This accounts for the first four cases of Apollonius' problem, those that do not involve circles. To solve the remaining problems, Viète exploited the fact that the given circles and the solution circle may be re-sized in tandem while preserving their tangencies (Figure 4). If the solution-circle radius is changed by an amount Δr, the radius of its internally tangent given circles must be likewise changed by Δr, whereas the radius of its externally tangent given circles must be changed by −Δr.
Thus, as the solution circle swells, the internally tangent given circles must swell in tandem, whereas the externally tangent given circles must shrink, to maintain their tangencies. Viète used this approach to shrink one of the given circles to a point, thus reducing the problem to a simpler, already solved case. He first solved the CLL case (a circle and two lines) by shrinking the circle into a point, rendering it a LLP case. He then solved the CLP case (a circle, a line and a point) using three lemmas. Again shrinking one circle to a point, Viète transformed the CCL case into a CLP case.
He then solved the CPP case (a circle and two points) and the CCP case (two circles and a point), the latter case by two lemmas. Finally, Viète solved the general CCC case (three circles) by shrinking one circle to a point, rendering it a CCP case. Algebraic solutions Apollonius' problem can be framed as a system of three equations for the center and radius of the solution circle. Since the three given circles and any solution circle must lie in the same plane, their positions can be specified in terms of the (x, y) coordinates of their centers. For example, the center positions of the three given circles may be written as (x1, y1), (x2, y2) and (x3, y3), whereas that of a solution circle can be written as (xs, ys).
Similarly, the radii of the given circles and a solution circle can be written as r1, r2, r3 and rs, respectively. The requirement that a solution circle must exactly touch each of the three given circles can be expressed as three coupled quadratic equations for xs, ys and rs: The three numbers s1, s2 and s3 on the right-hand side, called signs, may equal ±1, and specify whether the desired solution circle should touch the corresponding given circle internally (s = 1) or externally (s = −1). For example, in Figures 1 and 4, the pink solution is internally tangent to the medium-sized given circle on the right and externally tangent to the smallest and largest given circles on the left; if the given circles are ordered by radius, the signs for this solution are .
Since the three signs may be chosen independently, there are eight possible sets of equations , each set corresponding to one of the eight types of solution circles. The general system of three equations may be solved by the method of resultants. When multiplied out, all three equations have on the left-hand side, and rs2 on the right-hand side. Subtracting one equation from another eliminates these quadratic terms; the remaining linear terms may be re-arranged to yield formulae for the coordinates xs and ys where M, N, P and Q are known functions of the given circles and the choice of signs.
Substitution of these formulae into one of the initial three equations gives a quadratic equation for rs, which can be solved by the quadratic formula. Substitution of the numerical value of rs into the linear formulae yields the corresponding values of xs and ys. The signs s1, s2 and s3 on the right-hand sides of the equations may be chosen in eight possible ways, and each choice of signs gives up to two solutions, since the equation for rs is quadratic. This might suggest (incorrectly) that there are up to sixteen solutions of Apollonius' problem. However, due to a symmetry of the equations, if (rs, xs, ys) is a solution, with signs si, then so is (−rs, xs, ys), with opposite signs −si, which represents the same solution circle.
Therefore, Apollonius' problem has at most eight independent solutions (Figure 2). One way to avoid this double-counting is to consider only solution circles with non-negative radius. The two roots of any quadratic equation may be of three possible types: two different real numbers, two identical real numbers (i.e., a degenerate double root), or a pair of complex conjugate roots. The first case corresponds to the usual situation; each pair of roots corresponds to a pair of solutions that are related by circle inversion, as described below (Figure 6). In the second case, both roots are identical, corresponding to a solution circle that transforms into itself under inversion.
In this case, one of the given circles is itself a solution to the Apollonius problem, and the number of distinct solutions is reduced by one. The third case of complex conjugate radii does not correspond to a geometrically possible solution for Apollonius' problem, since a solution circle cannot have an imaginary radius; therefore, the number of solutions is reduced by two. Apollonius' problem cannot have seven solutions, although it may have any other number of solutions from zero to eight. Lie sphere geometry The same algebraic equations can be derived in the context of Lie sphere geometry. That geometry represents circles, lines and points in a unified way, as a five-dimensional vector X = (v, cx, cy, w, sr), where c = (cx, cy) is the center of the circle, and r is its (non-negative) radius.
If r is not zero, the sign s may be positive or negative; for visualization, s represents the orientation of the circle, with counterclockwise circles having a positive s and clockwise circles having a negative s. The parameter w is zero for a straight line, and one otherwise. In this five-dimensional world, there is a bilinear product similar to the dot product: The Lie quadric is defined as those vectors whose product with themselves (their square norm) is zero, (X|X) = 0.
Let X1 and X2 be two vectors belonging to this quadric; the norm of their difference equals The product distributes over addition and subtraction (more precisely, it is bilinear): Since (X1|X1) = (X2|X2) = 0 (both belong to the Lie quadric) and since w1 = w2 = 1 for circles, the product of any two such vectors on the quadric equals where the vertical bars sandwiching represent the length of that difference vector, i.e., the Euclidean norm. This formula shows that if two quadric vectors X1 and X2 are orthogonal (perpendicular) to one another—that is, if (X1|X2)=0—then their corresponding circles are tangent.
For if the two signs s1 and s2 are the same (i.e. the circles have the same "orientation"), the circles are internally tangent; the distance between their centers equals the difference in the radii Conversely, if the two signs s1 and s2 are different (i.e. the circles have opposite "orientations"), the circles are externally tangent; the distance between their centers equals the sum of the radii Therefore, Apollonius' problem can be re-stated in Lie geometry as a problem of finding perpendicular vectors on the Lie quadric; specifically, the goal is to identify solution vectors Xsol that belong to the Lie quadric and are also orthogonal (perpendicular) to the vectors X1, X2 and X3 corresponding to the given circles.
The advantage of this re-statement is that one can exploit theorems from linear algebra on the maximum number of linearly independent, simultaneously perpendicular vectors. This gives another way to calculate the maximum number of solutions and extend the theorem to higher-dimensional spaces. Inversive methods A natural setting for problem of Apollonius is inversive geometry. The basic strategy of inversive methods is to transform a given Apollonius problem into another Apollonius problem that is simpler to solve; the solutions to the original problem are found from the solutions of the transformed problem by undoing the transformation. Candidate transformations must change one Apollonius problem into another; therefore, they must transform the given points, circles and lines to other points, circles and lines, and no other shapes.
Circle inversion has this property and allows the center and radius of the inversion circle to be chosen judiciously. Other candidates include the Euclidean plane isometries; however, they do not simplify the problem, since they merely shift, rotate, and mirror the original problem. Inversion in a circle with center O and radius R consists of the following operation (Figure 5): every point P is mapped into a new point P' such that O, P, and P' are collinear, and the product of the distances of P and P' to the center O equal the radius R squared Thus, if P lies outside the circle, then P' lies within, and vice versa.
When P is the same as O, the inversion is said to send P to infinity. (In complex analysis, "infinity" is defined in terms of the Riemann sphere.) Inversion has the useful property that lines and circles are always transformed into lines and circles, and points are always transformed into points. Circles are generally transformed into other circles under inversion; however, if a circle passes through the center of the inversion circle, it is transformed into a straight line, and vice versa. Importantly, if a circle crosses the circle of inversion at right angles (intersects perpendicularly), it is left unchanged by the inversion; it is transformed into itself.
Circle inversions correspond to a subset of Möbius transformations on the Riemann sphere. The planar Apollonius problem can be transferred to the sphere by an inverse stereographic projection; hence, solutions of the planar Apollonius problem also pertain to its counterpart on the sphere. Other inversive solutions to the planar problem are possible besides the common ones described below. Pairs of solutions by inversion Solutions to Apollonius' problem generally occur in pairs; for each solution circle, there is a conjugate solution circle (Figure 6). One solution circle excludes the given circles that are enclosed by its conjugate solution, and vice versa.
For example, in Figure 6, one solution circle (pink, upper left) encloses two given circles (black), but excludes a third; conversely, its conjugate solution (also pink, lower right) encloses that third given circle, but excludes the other two. The two conjugate solution circles are related by inversion, by the following argument. In general, any three distinct circles have a unique circle—the radical circle—that intersects all of them perpendicularly; the center of that circle is the radical center of the three circles. For illustration, the orange circle in Figure 6 crosses the black given circles at right angles. Inversion in the radical circle leaves the given circles unchanged, but transforms the two conjugate pink solution circles into one another.
Under the same inversion, the corresponding points of tangency of the two solution circles are transformed into one another; for illustration, in Figure 6, the two blue points lying on each green line are transformed into one another. Hence, the lines connecting these conjugate tangent points are invariant under the inversion; therefore, they must pass through the center of inversion, which is the radical center (green lines intersecting at the orange dot in Figure 6). Inversion to an annulus If two of the three given circles do not intersect, a center of inversion can be chosen so that those two given circles become concentric.
Under this inversion, the solution circles must fall within the annulus between the two concentric circles. Therefore, they belong to two one-parameter families. In the first family (Figure 7), the solutions do not enclose the inner concentric circle, but rather revolve like ball bearings in the annulus. In the second family (Figure 8), the solution circles enclose the inner concentric circle. There are generally four solutions for each family, yielding eight possible solutions, consistent with the algebraic solution. When two of the given circles are concentric, Apollonius' problem can be solved easily using a method of Gauss. The radii of the three given circles are known, as is the distance dnon from the common concentric center to the non-concentric circle (Figure 7).
The solution circle can be determined from its radius rs, the angle θ, and the distances ds and dT from its center to the common concentric center and the center of the non-concentric circle, respectively. The radius and distance ds are known (Figure 7), and the distance dT = rs ± rnon, depending on whether the solution circle is internally or externally tangent to the non-concentric circle. Therefore, by the law of cosines, Here, a new constant C has been defined for brevity, with the subscript indicating whether the solution is externally or internally tangent. A simple trigonometric rearrangement yields the four solutions This formula represents four solutions, corresponding to the two choices of the sign of θ, and the two choices for C. The remaining four solutions can be obtained by the same method, using the substitutions for rs and ds indicated in Figure 8.
Thus, all eight solutions of the general Apollonius problem can be found by this method. Any initial two disjoint given circles can be rendered concentric as follows. The radical axis of the two given circles is constructed; choosing two arbitrary points P and Q on this radical axis, two circles can be constructed that are centered on P and Q and that intersect the two given circles orthogonally. These two constructed circles intersect each other in two points. Inversion in one such intersection point F renders the constructed circles into straight lines emanating from F and the two given circles into concentric circles, with the third given circle becoming another circle (in general).
This follows because the system of circles is equivalent to a set of Apollonian circles, forming a bipolar coordinate system. Resizing and inversion The usefulness of inversion can be increased significantly by resizing. As noted in Viète's reconstruction, the three given circles and the solution circle can be resized in tandem while preserving their tangencies. Thus, the initial Apollonius problem is transformed into another problem that may be easier to solve. For example, the four circles can be resized so that one given circle is shrunk to a point; alternatively, two given circles can often be resized so that they are tangent to one another.
Thirdly, given circles that intersect can be resized so that they become non-intersecting, after which the method for inverting to an annulus can be applied. In all such cases, the solution of the original Apollonius problem is obtained from the solution of the transformed problem by undoing the resizing and inversion. Shrinking one given circle to a point In the first approach, the given circles are shrunk or swelled (appropriately to their tangency) until one given circle is shrunk to a point P. In that case, Apollonius' problem degenerates to the CCP limiting case, which is the problem of finding a solution circle tangent to the two remaining given circles that passes through the point P. Inversion in a circle centered on P transforms the two given circles into new circles, and the solution circle into a line.