title: Arjen Robben

text: In Bayern's opening match of the 2014–15 Bundesliga season, Robben scored one goal and assisted the other as the team defeated VfL Wolfsburg 2–1 on 22 August 2014. On 1 November, he scored the winning goal from a penalty kick against Borussia Dortmund in "Der Klassiker". On 16 December 2014, Robben scored his 100th competitive goal for Bayern Munich in a home match against SC Freiburg.


title: Arjen Robben

text: On 21 February 2015, Robben scored twice in a 6–0 win against SC Paderborn, giving him the record of having scored against every Bundesliga team he had faced. He ended 2014–15 alongside teammate Robert Lewandowski as joint second-top goalscorer in the Bundesliga with 17 goals. This was in spite of the player missing the last two months of the season through injury. He finished the 2014–15 season with 19 goals in 30 appearances.


title: Arjen Robben

text: Robben started his 2015–16 season by scoring the opening goal of the DFL-Supercup against VfL Wolfsburg. The match finished in a 1–1 draw, and Robben scored in the subsequent shootout, however Bayern were defeated.


title: Arjen Robben

text: Robben's first goal of the Bundesliga season came from a penalty kick in a 3–0 defeat of Bayer Leverkusen on 29 August. However, an injury suffered six days later when playing for the Netherlands against Iceland in qualification for UEFA Euro 2016 kept him out of Bayern's next nine fixtures. On 24 October, Robben returned to the Bayern team, playing 65 minutes and scoring the opening goal of a 4–0 win against 1. FC Köln, the club's 1,000th Bundesliga victory. He made his first Champions League appearance of the season on 4 November, scoring after coming on as a substitute in Bayern's 5–1 win over Arsenal. He finished the 2015–16 season with seven goals in 22 appearances.


title: Arjen Robben

text: Robben, recovering from his injury, was chosen for a friendly against SV Lippstadt 08. He scored a goal, shortly before being substituted because of another injury in the 36th minute. Robben's six-week injury caused him to miss several matches. He returned to training and remained on the bench until 21 September Bundesliga match against Hertha BSC, where he came on instead of Thomas Müller and scored in the 3–0 victory. Robben scored in both first and second leg of the round-of-sixteen tie against Arsenal as Bayern won 10–2 on aggregate. He finished the 2016–17 season with 16 goals in 37 appearances.


title: Arjen Robben

text: On 16 January 2017, Robben signed a one-year extension with Bayern, keeping him at the club until the end of the 2017–18 season. He scored a goal with his right foot for the first time since February 2015 in a 4–0 victory over 1. FSV Mainz 05 on 16 September. On 18 October, Robben made his 100th Champions League appearance in a 3–0 victory over Celtic F.C. in the group stage match of the competition. On 4 November, Robben scored a stunning goal in a 3–1 away win over their Borussia Dortmund in Der Klassiker which made him Bayern's most successful non-German scorer in the Bundesliga and the most successful Dutch goal scorer in the Bundesliga. He managed to score twice in a 6–0 away victory over SC Paderborn as Bayern went on through to the semi-final of DFB-Pokal on 6 February 2018. Robben won his seventh Bundesliga title and set a Dutch record by winning his 11th career domestic league title surpassing Dutch legend Johan Cruyff's 10 career domestic league titles. He finished the 2017–18 season with seven goals in 34 appearances.


title: Arjen Robben

text: On 11 May 2018, Robben signed a one-year contract; extending his stay until the end of the 2018–19 season.


title: Arjen Robben

text: On 12 August, he started the season by playing in the 2018 DFL-Supercup for 58 minutes and won the title as Bayern defeated Eintracht Frankfurt with a 5–0 victory. On 27 November, Robben scored two goals in a 5–1 win against S.L. Benfica in the Champions League. On 2 December, Robben said in an interview "I can say that this is my last year [at Bayern] and it is good like that. I think it is the right moment after ten years. The club moves on and I may move on. It is the end of a very good and long period."


title: Arjen Robben

text: On 18 May 2019, he scored his last league goal for Bayern in Bundesliga making it a 5–1 victory against Frankfurt, which meant Bayern winning their 28th Bundesliga title and Robben winning his eighth. On 25 May 2019, Robben won his fifth DFB-Pokal as Bayern defeated RB Leipzig 3–0 in the 2019 DFB-Pokal Final. He finished the 2018–19 season with six goals in 19 appearances. In 201 Bundesliga matches he scored 99 goals.


title: Arjen Robben

text: On 27 June 2020, Robben announced his return to the game, signing with FC Groningen, the club he started his career with, in an effort to help the team recover from COVID-19 setbacks. He scored his first goal for Groningen in a pre-season friendly against Arminia Bielefeld. On 13 September 2020, he made his league debut for Groningen against PSV, but he was substituted after only 28 minutes due to injury. After a nearly seven-month absence, he made his return as a substitute in a loss to SC Heerenveen on 11 April 2021. Robben returned in Groningen's starting line-up on 9 May; he provided two assists in a 4–0 win at Emmen.


title: Arjen Robben

text: In April 2003, he made his Netherlands international debut in a friendly match against Portugal at age 19.


title: Arjen Robben

text: Robben's first participation at an international tournament came at UEFA Euro 2004 when manager Dick Advocaat called up younger players, such as Wesley Sneijder and John Heitinga. During the group stage of the tournament, Advocaat substituted Robben in the 66th minute to defend a 2–1 lead over the Czech Republic. The Czechs, however, then scored two goals and won 3–2, which led to criticism about Advocaat's decision. In the quarter-finals, Robben scored the deciding kick in the penalty shootout between the Netherlands and Sweden, which ensured that at the fifth attempt, the Dutch finally won a shootout.


title: Arjen Robben

text: Robben played in his first 2006 FIFA World Cup qualifier in 2006. In six games for the Netherlands, Robben scored two goals. The Netherlands qualified for the 2006 World Cup and in their opening match against Serbia and Montenegro, Robben scored the winning goal in the 18th minute, and was named the Player of the match. In the match against the Ivory Coast, Robben received this award the second time, thus becoming one of the eight players in the tournament to have won multiple Man of the Match awards.


title: Arjen Robben

text: During Euro 2008, new Dutch manager Marco van Basten changed the side's formation to 4–2–3–1, preferring the midfield trio of Rafael van der Vaart, Wesley Sneijder and Dirk Kuyt, with Robben left to battle for a position with Robin van Persie on the wing. In the group match against France, which the Netherlands won 4–1, Robben was brought on at half-time. He dragged the ball out of reach of a defender near the half-way line and paced his way ahead of three defenders, then crossed straight to Van Persie, who powered a shot into the goal past France goalkeeper Grégory Coupet. At 2–0, Thierry Henry scored a goal for France, making it 2–1, but almost directly after the restart, Robben was played through by Sneijder and he ran at the France goal and was forced into an acute angle, but despite the angle, he shot the ball into the net past Coupet.


title: Arjen Robben

text: Robben was selected by new manager Bert van Marwijk for the final 23-man Dutch squad for the 2010 World Cup. In the team's last friendly match against Hungary, just before its flight to South Africa on 4 June, Robben fell awkwardly and picked up a hamstring injury, causing concern over his fitness for the tournament. On 5 June, Van Marwijk announced that he had "decided not to summon any substitute for Arjen. I want to give him every chance to still participate in the World Cup". On 12 June, Robben arrived in South Africa to join the team. He was an unused substitute for the opening match against Denmark as they coasted to a 2–0 victory, and again in the 1–0 victory against Japan. He came on in the 73rd minute against Cameroon in their 2–1 victory, hitting the post from which Klaas-Jan Huntelaar scored from the rebound.


title: Arjen Robben

text: On 28 June 2010, Robben started his first match against Slovakia in which he went on to score the first goal in the second round as the Netherlands won 2–1. He also won the man of the match award. Robben scored the third goal with a well-placed header in the semi-finals against Uruguay which the Netherlands won 3–2 to reach the final. Robben played for the full match in the final as the Netherlands lost 1–0 to Spain. Robben had Netherlands's best opportunity of the match when played in on goal by Wesley Sneijder in the 62nd minute of the match, but was denied by Iker Casillas. Robben was nominated for the 2010 World Cup Golden Ball, the tournament's best player, which was won by Diego Forlán.


title: Arjen Robben

text: Robben was selected to play for the Netherlands at Euro 2012. He started in the first group stage match, a 1–0 loss to Denmark, in which he hit the post with a curled effort. In the next match, a 2–1 loss to Germany, he hopped over the advertising boards and walked the long way round after being subbed off for Dirk Kuyt because he thought he could have inspired the Netherlands to a win. After a 2–1 loss to Portugal, the Netherlands were knocked out with three straight defeats.


title: Arjen Robben

text: In the Netherlands' first match of the 2014 World Cup, Robben scored two goals in a 5–1 win over Spain. In the Netherlands' second group stage match, a 3–2 win over Australia, Robben scored the team's opening goal as they qualified for the knockout stage. In the round of 16 knock-out match with Mexico, the Netherlands won on a penalty awarded after Robben went down following a challenge from Rafael Márquez. The Associated Press said that Robben's "theatrical" fall "did little to defuse the debate about his reputation for diving". Robben maintained that the penalty was correct but admitted to falling easily earlier in the match, telling a Dutch TV channel that "the one [at the end] was a penalty, but the other one was a dive in the first half. I shouldn't be doing that".


title: Arjen Robben

text: In the quarter-final, Robben scored the Netherlands' second kick in a 4–3 penalty shootout defeat of Costa Rica. On 11 July, Robben was named on the ten-man shortlist for FIFA's Golden Ball award for the tournament's best player.


title: Arjen Robben

text: On 10 October 2017, Robben retired from international football after the team's unsuccessful 2018 World Cup qualifying campaign. He collected 96 caps between 2003 and 2017 for the "Oranje" and scored 37 goals.


title: Arjen Robben

text: Throughout most of his career, Robben was regarded as one of the best wingers in the world, with ESPN stating he is "able to take on and beat defenders at the drop of a hat, Robben's ability to get to the byline and deliver in accurate crosses instils fear into any defence". Usually deployed on the right wing, Robben often cuts inside on his left foot to move to a more central attacking position, and uses his speed and dribbling skills to take on defenders until he finds the space to make an attempt on goal.


title: Arjen Robben

text: He is predominantly a left-footed player. During a match he will often drift to the left wing, with his teammate, in the case of Bayern Munich, Franck Ribéry, switching over to the right. Another factor in his playing style is his relationship with the right-fullback. At Bayern, this partnership with Philipp Lahm has had benefits for the team's efficiency and chemistry. Robben's defensive work improved when he went to Bayern where he often tracks back and covers for his partner when he pushes forward and becomes a temporary right back.


title: Arjen Robben

text: Robben has been often accused of diving, and he has at times openly admitted to the charges. In December 2011, Robben apologised for a dive against VfL Bochum in a DFB-Pokal match which earned him a yellow card, saying, "I must not do things like this." In July 2014, Robben admitted to diving in the first half of Netherlands' match against Mexico in the last 16 round of the 2014 World Cup, but maintained that he had not dived for the decisive penalty he won in stoppage time which Netherlands converted to win 2–1. Following the incident, Robben's former manager at Chelsea, José Mourinho, said that Robben's speed and creativity cause opponents who cannot stop him to foul him, but "sometimes [Robben] tries to get an advantage, or to get a penalty".


title: Arjen Robben

text: Robben married his girlfriend Bernadien Eillert on 9 June 2007 in Groningen. The two met while in high school in the city, at the Kamerlingh Onnes, and have two sons, Luka (born 2008) and Kai (born 2012) and a daughter, Lynn (born 2010). Robben's father, Hans, works as his agent.


title: Arjen Robben

text: Robben has a sponsorship deal with German sportswear and equipment supplier Adidas. Robben features in EA Sports' "FIFA" video game series, and was the third-highest-rated player in "FIFA 15".


title: Arjen Robben

text: Robben scored his first ever international goal in October 2003 during a UEFA Euro 2004 qualifier against Moldova. He ultimately decided to retire from international duty in 2017 following their unsuccessful qualifying campaign for the 2018 FIFA World Cup.


title: Cornelius Vanderbilt

text: Cornelius Vanderbilt (May 27, 1794 – January 4, 1877), nicknamed "the Commodore", was an American business magnate who built his wealth in railroads and shipping. After working with his father's business, Vanderbilt worked his way into leadership positions in the inland water trade and invested in the rapidly growing railroad industry, effectively transforming the geography of the United States.


title: Cornelius Vanderbilt

text: As one of the richest Americans in history and wealthiest figures overall, Vanderbilt was the patriarch of the wealthy and influential Vanderbilt family. He provided the initial gift to found Vanderbilt University in Nashville, Tennessee. According to historian H. Roger Grant: "Contemporaries, too, often hated or feared Vanderbilt or at least considered him an unmannered brute. While Vanderbilt could be a rascal, combative and cunning, he was much more a builder than a wrecker [...] being honorable, shrewd, and hard-working."


title: Cornelius Vanderbilt

text: Cornelius Vanderbilt's great-great-great-grandfather, Jan Aertson or Aertszoon ("Aert's son"), was a Dutch farmer from the village of De Bilt in Utrecht, Netherlands, who emigrated to New Amsterdam (later New York) as an indentured servant in 1650. The Dutch "van der" ("of the") was eventually added to Aertson's village name to create "van der Bilt" ("of the Bilt"). This was eventually condensed to Vanderbilt.


title: Cornelius Vanderbilt

text: Anthony Janszoon van Salee was another of Cornelius Vanderbilt's great-great-great-great-grandfathers.


title: Cornelius Vanderbilt

text: Cornelius Vanderbilt was born in Staten Island, New York, on May 27, 1794, to Cornelius van Derbilt and Phebe Hand. He began working on his father's ferry in New York Harbor as a boy, quitting school at the age of 11. At the age of 16, Vanderbilt decided to start his own ferry service. According to one version of events, he borrowed () from his mother to purchase a periauger (a shallow draft, two-masted sailing vessel), which he christened the "Swiftsure." However, according to the first account of his life, published in 1853, the periauger belonged to his father and the younger Vanderbilt received half the profit. He began his business by ferrying freight and passengers on a ferry between Staten Island and Manhattan. Such was his energy and eagerness in his trade that other captains nearby took to calling him The Commodore in jest—a nickname that stuck with him all his life.


title: Cornelius Vanderbilt

text: While many Vanderbilt family members had joined the Episcopal Church, Cornelius Vanderbilt remained a member of the Moravian Church to his death. Along with other members of the Vanderbilt family, he helped erect a local Moravian parish church in his city.


title: Cornelius Vanderbilt

text: On December 19, 1813, at age 19 Vanderbilt married his first cousin, Sophia Johnson. They moved into a boarding house on Broad Street in Manhattan.


title: Cornelius Vanderbilt

text: They had 13 children together: Phebe in 1814, Ethelinda in 1817, Eliza in 1819, William in 1821, Emily in 1823, Sophia in 1825, Maria in 1827, Frances in 1828, Cornelius Jeremiah in 1830, George in 1832 (who died in 1836), Mary, in 1834, Catherine in 1836, and another son named George in 1839.


title: Cornelius Vanderbilt

text: In addition to running his ferry, Vanderbilt bought his brother-in-law John De Forest's schooner "Charlotte" and traded in food and merchandise in partnership with his father and others. But on November 24, 1817, a ferry entrepreneur named Thomas Gibbons asked Vanderbilt to captain his steamboat between New Jersey and New York. Although Vanderbilt kept his own businesses running, he became Gibbons's business manager.


title: Cornelius Vanderbilt

text: When Vanderbilt entered his new position, Gibbons was fighting against a steamboat monopoly in New York waters, which had been granted by the New York State Legislature to the politically influential patrician Robert Livingston and Robert Fulton, who had designed the steamboat. Though both Livingston and Fulton had died by the time Vanderbilt started working for Gibbons, the monopoly was held by Livingston's heirs. They had granted a license to Aaron Ogden to run a ferry between New York and New Jersey. Gibbons launched his steamboat venture because of a personal dispute with Ogden, whom he hoped to drive into bankruptcy. To accomplish this, he undercut prices and also brought a landmark legal case—"Gibbons v. Ogden"—to the United States Supreme Court to overturn the monopoly.


title: Cornelius Vanderbilt

text: Working for Gibbons, Vanderbilt learned to operate a large and complicated business. He moved with his family to New Brunswick, New Jersey, a stop on Gibbons' line between New York and Philadelphia. There his wife Sophia operated a very profitable inn, using the proceeds to feed, clothe and educate their children. Vanderbilt also proved a quick study in legal matters, representing Gibbons in meetings with lawyers. He also went to Washington, D.C., to hire Daniel Webster to argue the case before the Supreme Court. Vanderbilt appealed his own case against the monopoly to the Supreme Court, which was next on the docket after "Gibbons v. Ogden". The Court never heard Vanderbilt's case, because on March 2, 1824, it ruled in Gibbons' favor, saying that states had no power to interfere with interstate commerce. The case is still considered a landmark ruling. The protection of competitive interstate commerce is considered the basis for much of the prosperity which the United States has generated.


title: Cornelius Vanderbilt

text: After Thomas Gibbons died in 1826, Vanderbilt worked for Gibbons' son William until 1829. Though he had always run his own businesses on the side, he now worked entirely for himself. Step by step, he started lines between New York and the surrounding region. First he took over Gibbons' ferry to New Jersey, then switched to western Long Island Sound. In 1831, he took over his brother Jacob's line to Peekskill, New York, on the lower Hudson River. That year he faced opposition by a steamboat operated by Daniel Drew, who forced Vanderbilt to buy him out. Impressed, Vanderbilt became a secret partner with Drew for the next thirty years, so that the two men would have an incentive to avoid competing with each other.


title: Cornelius Vanderbilt

text: On November 8, 1833, Vanderbilt was nearly killed in the Hightstown rail accident on the Camden and Amboy Railroad in New Jersey. Also on the train was former president John Quincy Adams.


title: Cornelius Vanderbilt

text: In 1834, Vanderbilt competed on the Hudson River against the Hudson River Steamboat Association, a steamboat monopoly between New York City and Albany. Using the name "The People's Line", he used the populist language associated with Democratic president Andrew Jackson to get popular support for his business. At the end of the year, the monopoly paid him a large amount to stop competing, and he switched his operations to Long Island Sound.


title: Cornelius Vanderbilt

text: During the 1830s, textile mills were built in large numbers in New England as the United States developed its manufacturing base. Some of the first railroads in the United States were built from Boston to Long Island Sound, to connect with steamboats that ran to New York. By the end of the decade, Vanderbilt dominated the steamboat business on the Sound, and began to take over management of the connecting railroads. In the 1840s, he launched a campaign to take over the most attractive of these lines, the New York, Providence and Boston Railroad, popularly known as the Stonington. By cutting fares on competing lines, Vanderbilt drove down the Stonington stock price, and took over the presidency of the company in 1847. It was the first of the many railroads he would head.


title: Cornelius Vanderbilt

text: During these years, Vanderbilt also operated many other businesses. He bought large amounts of real estate in Manhattan and Staten Island, and took over the Staten Island Ferry in 1838. It was in the 1830s when he was first referred to as "commodore", then the highest rank in the United States Navy. A common nickname for important steamboat entrepreneurs, by the end of the 1840s it was applied only to Vanderbilt.


title: Cornelius Vanderbilt

text: When the California gold rush began in 1849, Vanderbilt switched from regional steamboat lines to ocean-going steamships. Many of the migrants to California, and almost all of the gold returning to the East Coast, went by steamship to Panama, where mule trains and canoes provided transportation across the isthmus. (The Panama Railroad was soon built to provide a faster crossing.) Vanderbilt proposed a canal across Nicaragua, which was closer to the United States and was spanned most of the way across by Lake Nicaragua and the San Juan River. In the end, he could not attract enough investment to build the canal, but he did start a steamship line to Nicaragua, and founded the Accessory Transit Company to carry passengers across Nicaragua by steamboat on the lake and river, with a carriage road between the Pacific port of San Juan del Sur and Virgin Bay on Lake Nicaragua.


title: Cornelius Vanderbilt

text: In 1852, a dispute with Joseph L. White, a partner in the Accessory Transit Company, led to a business battle in which Vanderbilt forced the company to buy his ships for an inflated price. In early 1853, he took his family on a grand tour of Europe in his steamship yacht, the "North Star." While he was away, White conspired with Charles Morgan, Vanderbilt's erstwhile ally, to betray him, and deny him money he was owed by the Accessory Transit Company. When Vanderbilt returned from Europe, he retaliated by developing a rival steamship line to California, cutting prices until he forced Morgan and White to pay him off.


title: Cornelius Vanderbilt

text: He then turned to transatlantic steamship lines, running in opposition to the heavily subsidized Collins Line, headed by Edward K. Collins. Vanderbilt eventually drove the Collins Line into extinction. During the 1850s, Vanderbilt also bought control of a major shipyard and the Allaire Iron Works, a leading manufacturer of marine steam engines, in Manhattan.


title: Cornelius Vanderbilt

text: In November 1855, Vanderbilt began to buy control of Accessory Transit once again. That same year, the American military adventurer, William Walker, led an expedition to Nicaragua and briefly took control of the government. Edmund Randolph, a close friend of Walker, coerced the Accessory Transit's San Francisco agent, Cornelius K. Garrison, into opposing Vanderbilt. Randolph convinced Walker to annul the charter of the Accessory Transit Company, and give the transit rights and company steamboats to him; Randolph sold these to Garrison. Garrison brought Charles Morgan in New York into the plan. Vanderbilt took control of the company just before these developments were announced. When he tried to convince the U.S. and English governments to help restore the company to its rights and property, they refused. So he negotiated with Costa Rica, which (along with the other Central American republics) had declared war on Walker. Vanderbilt sent a man to Costa Rica who led a raid that captured the steamboats on the San Juan River, cutting Walker off from his reinforcements from insurgent groups in the United States. Walker was forced to give up, and was driven out of the country by a U.S. Navy officer. But the new Nicaraguan government refused to allow Vanderbilt to restart the transit business, so he started a line by way of Panama, eventually developing a monopoly on the California steamship business.


title: Cornelius Vanderbilt

text: When the Civil War began in 1861, Vanderbilt attempted to donate his largest steamship, the "Vanderbilt," to the Union Navy. Secretary of the Navy Gideon Welles refused it, thinking its operation and maintenance too expensive for what he expected to be a short war. Vanderbilt had little choice but to lease it to the War Department, at prices set by ship brokers. When the Confederate ironclad "Virginia" (popularly known in the North as the "Merrimack") wrought havoc with the Union blockading squadron at Hampton Roads, Virginia, Secretary of War Edwin Stanton and President Abraham Lincoln called on Vanderbilt for help. This time he succeeded in donating the "Vanderbilt" to the Union Navy, equipping it with a ram and staffing it with handpicked officers. It helped bottle up the "Virginia," after which Vanderbilt converted it into a cruiser to hunt for the Confederate commerce raider "Alabama", captained by Raphael Semmes. For donating the "Vanderbilt," he was awarded a Congressional Gold Medal. Vanderbilt also paid to outfit a major expedition to New Orleans. He suffered a grievous loss when George Washington Vanderbilt II, his youngest and favorite son, and heir apparent, a graduate of the United States Military Academy, fell ill and died without ever seeing combat.


title: Cornelius Vanderbilt

text: Though Vanderbilt had relinquished his presidency of the Stonington Railroad during the California gold rush, he took an interest in several railroads during the 1850s, serving on the boards of directors of the Erie Railway, the Central Railroad of New Jersey, the Hartford and New Haven, and the New York and Harlem (popularly known as the Harlem). In 1863, Vanderbilt took control of the Harlem in a famous stockmarket corner, and was elected its president. He later explained that he wanted to show that he could take this railroad, which was generally considered worthless, and make it valuable. It had a key advantage: it was the only steam railroad to enter the center of Manhattan, running down 4th Avenue (later Park Avenue) to a station on 26th Street, where it connected with a horse-drawn streetcar line. From Manhattan it ran up to Chatham Four Corners, New York, where it had a connection to the railroads running east and west.


title: Cornelius Vanderbilt

text: Vanderbilt brought his eldest son, Billy, in as vice-president of the Harlem. Billy had had a nervous breakdown early in life, and his father had sent him to a farm on Staten Island. But he proved himself a good businessman, and eventually became the head of the Staten Island Railway. Though the Commodore had once scorned Billy, he was impressed by his son's success. Eventually he promoted him to operational manager of all his railroad lines. In 1864, the Commodore sold his last ships, in order to concentrate on the railroads.


title: Cornelius Vanderbilt

text: Once in charge of the Harlem, Vanderbilt encountered conflicts with connecting lines. In each case, the strife ended in a battle that Vanderbilt won. He bought control of the Hudson River Railroad in 1864, the New York Central Railroad in 1867, and the Lake Shore and Michigan Southern Railway in 1869. He later bought the Canada Southern as well. In 1870, he consolidated two of his key lines into the New York Central and Hudson River Railroad, one of the first giant corporations in United States history.


title: Cornelius Vanderbilt

text: In 1869, Vanderbilt directed the Harlem to begin construction of the Grand Central Depot on 42nd Street in Manhattan. It was finished in 1871, and served as his lines' terminus in New York. He sank the tracks on 4th Avenue in a cut that later became a tunnel, and 4th Avenue became Park Avenue. The depot was replaced by Grand Central Terminal in 1913.


title: Cornelius Vanderbilt

text: In 1868, Vanderbilt fell into a dispute with Daniel Drew, who had become treasurer of the Erie Railway. To get revenge, he tried to corner Erie stock, which led to the so-called Erie War. This brought him into direct conflict with Jay Gould and financier James Fisk Jr., who had just joined Drew on the Erie board. They defeated the corner by issuing "watered stock" in defiance of state law, which restricted the number of shares a company could issue. But Gould bribed the legislature to legalize the new stock. Vanderbilt used the leverage of a lawsuit to recover his losses, but he and Gould became public enemies.


title: Cornelius Vanderbilt

text: Gould never got the better of Vanderbilt in any other important business matter, but he often embarrassed Vanderbilt, who uncharacteristically lashed out at Gould in public. By contrast, Vanderbilt befriended his other foes after their fights ended, including Drew and Cornelius Garrison.


title: Cornelius Vanderbilt

text: Following his wife Sophia's death in 1868, Vanderbilt went to Canada. On August 21, 1869, in London, Ontario, he married a cousin from Mobile, Alabama, with the name — unusual for a woman — of Frank Armstrong Crawford.


title: Cornelius Vanderbilt

text: Vanderbilt's second wife convinced him to give , the largest charitable gift in American history to that date, to Bishop Holland Nimmons McTyeire, the husband of her cousin, Amelia Townsend, to found Vanderbilt University in Nashville, Tennessee, named in his honor. He also paid $50,000 for a church for his second wife's congregation, the Church of the Strangers. In addition, he donated to churches around New York, including a gift to the Moravian Church on Staten Island of for a cemetery (the Moravian Cemetery). He chose to be buried there.


title: Cornelius Vanderbilt

text: Cornelius Vanderbilt died on January 4, 1877, at his residence, No. 10 Washington Place, after having been confined to his rooms for about eight months. The immediate cause of his death was exhaustion, brought on by long suffering from a complication of chronic disorders. At the time of his death, aged 82, Vanderbilt had an estimated worth of $105 million.


title: Cornelius Vanderbilt

text: In his will, he left 95% of his $105 million estate to his son William (Billy) and four grandsons through him. This left his only other living son, Cornelius Jeremiah Vanderbilt, and 9 daughters (Phebe Jane, Ethelinda, Eliza, Emily Almira, Sophia Johnson, Maria Louisa, Frances Lavinia, Mary Alicia, and Catherine Juliette), to receive comparatively little inheritance; far less than even their young nephews. Corneel, Ethelinda and Mary took the matter of their father's will to court, claiming he was not in his right mind in his old age when he drew up the will; that he had been behaving strangely and was under William's influence as well as that of a corrupt spiritualist in his employ, who'd allegedly been approached and paid off by William to do his bidding, according to eyewitness testimony. Allegedly, William paid the spiritualist (a Mr. Stoddard) to suggest the "spirits"—during a point in the session when said spiritualist would fall into a "trance" in the Commodore's presence—claimed William would be the most trustworthy to inherit the estate and business and that his other children actually hated him and were just waiting for him to die. A "spirit", during this session, came in the form of William's deceased mother, Sophia. Not wanting to risk further public humiliation of the family name in court, William finally settled with his siblings. He gave Corneel an extra $200,000 in cash and a trust fund of $400,000. He gave Mary and Ethelinda the same settlement. Still, all told, this was comparatively very little from the—by far—largest estate in the world at that time.


title: Cornelius Vanderbilt

text: William's eldest son, Cornelius Vanderbilt II, received $5 million in the will, while his three younger sons—William Kissam Vanderbilt, Frederick William Vanderbilt, and George Washington Vanderbilt II—received $2 million apiece. Vanderbilt willed amounts ranging from $250,000 to $500,000 to each of his daughters. His wife received $500,000, their New York City home, and 2,000 shares of common stock in the New York Central Railroad. To his younger surviving son, Cornelius Jeremiah Vanderbilt, whom he regarded as a wastrel, he left the income from a $200,000 trust fund. (Although his daughters and Cornelius received bequests much smaller than those of their brother William, these made them very wealthy by the standards of 1877 and were not subject to inheritance tax.)


title: Cornelius Vanderbilt

text: Vanderbilt's biographer T. J. Stiles says, "He vastly improved and expanded the nation's transportation infrastructure, contributing to a transformation of the very geography of the United States. He embraced new technologies and new forms of business organization, and used them to compete...He helped to create the corporate economy that would define the United States into the 21st century."


title: Cornelius Vanderbilt

text: The Commodore had lived in relative modesty considering his nearly unlimited means, splurging only on race horses. His descendants were the ones who built the Vanderbilt houses that characterize the United States' Gilded Age.


title: Cornelius Vanderbilt

text: According to "The Wealthy 100" by Michael Klepper and Robert Gunther, Vanderbilt would be worth $143 billion in 2007 United States dollars if his total wealth as a share of the nation's gross domestic product (GDP) in 1877 (the year of his death) were taken and applied in that same proportion in 2007. This would make him the second-wealthiest person in United States history, after Standard Oil co-founder John Davison Rockefeller (1839–1937). Another calculation, from 1998, puts him in third place, after Andrew Carnegie. In real terms, however, Vanderbilt's peak wealth of $105 million in 1877 is only worth US$ (in dollars)


title: Cornelius Vanderbilt

text: In 1999, Cornelius Vanderbilt was inducted into the North America Railway Hall of Fame, recognizing his significant contributions to the railroad industry. He was inducted in the "Railway Workers & Builders: North America" category.


title: Cornelius Vanderbilt

text: A statue of Cornelius Vanderbilt is located on the south side of Grand Central Terminal, facing the Park Avenue road viaduct to the south. The bronze statue was sculpted by Ernst Plassmann and was originally sited at the Hudson River Railroad depot at St. John's Park before being moved to Grand Central Terminal in 1929.


title: Cornelius Vanderbilt

text: Cornelius Vanderbilt was buried in the family vault in the Moravian Cemetery at New Dorp on Staten Island. He was later reburied in a tomb in the same cemetery constructed by his son Billy. Three of his daughters and son, Cornelius Jeremiah Vanderbilt, contested the will on the grounds that their father was of unsound mind and under the influence of his son Billy and spiritualists whom he consulted on a regular basis. The court battle lasted more than a year and was ultimately won outright by Billy, who increased the bequests to his siblings and paid their legal fees.


title: Cornelius Vanderbilt

text: One of Vanderbilt's great-great-granddaughters, Gloria Vanderbilt, was a renowned fashion designer, and her youngest son, Anderson Cooper, is a television news anchor. Through Billy's daughter Emily Thorn Vanderbilt, another descendant is actor Timothy Olyphant.


title: Cornelius Vanderbilt

text: Cornelius Jeremiah Vanderbilt was childless when he committed suicide, in 1882, and George Washington Vanderbilt died during the Civil War, before having any children. All of the Vanderbilt multimillionaires descend through the oldest son Billy and his wife.


title: Cornelius Vanderbilt

text: Cornelius' youngest grandson through William, George Washington Vanderbilt II, built the 250-room Biltmore Estate in the mountains of Asheville, North Carolina, as his main residence with part of his inheritance from his grandfather. It still retains the title of the largest privately owned home in the United States, though it is open to the public. The mansion contains of total floor space and originally sat on of land. It now sits on due to George's final wishes that be sold to the government at —a significantly cut rate and what George had originally paid—in order to form the core of the Pisgah National Forest, as well as George's widow, Edith Stuyvesant Vanderbilt Gerry, being forced to sell off additional land to pay for the estate's upkeep.


title: Rotation matrix

text: In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix


title: Rotation matrix

text: rotates points in the plane counterclockwise through an angle with respect to the positive axis about the origin of a two-dimensional Cartesian coordinate system. To perform the rotation on a plane point with standard coordinates , it should be written as a column vector, and multiplied by the matrix :


title: Rotation matrix

text: If and are the endpoint coordinates of a vector, where is cosine and is sine, then the above equations become the trigonometric summation angle formulae. Indeed, a rotation matrix can be seen as the trigonometric summation angle formulae in matrix form. One way to understand this is say we have a vector at an angle 30° from the axis, and we wish to rotate that angle by a further 45°. We simply need to compute the vector endpoint coordinates at 75°.


title: Rotation matrix

text: The examples in this article apply to "active rotations" of vectors "counterclockwise" in a "right-handed coordinate system" ( counterclockwise from ) by "pre-multiplication" ( on the left). If any one of these is changed (such as rotating axes instead of vectors, a "passive transformation"), then the inverse of the example matrix should be used, which coincides with its transpose.


title: Rotation matrix

text: Since matrix multiplication has no effect on the zero vector (the coordinates of the origin), rotation matrices describe rotations about the origin. Rotation matrices provide an algebraic description of such rotations, and are used extensively for computations in geometry, physics, and computer graphics. In some literature, the term "rotation" is generalized to include improper rotations, characterized by orthogonal matrices with a determinant of −1 (instead of +1). These combine "proper" rotations with "reflections" (which invert orientation). In other cases, where reflections are not being considered, the label "proper" may be dropped. The latter convention is followed in this article.


title: Rotation matrix

text: Rotation matrices are square matrices, with real entries. More specifically, they can be characterized as orthogonal matrices with determinant 1; that is, a square matrix is a rotation matrix if and only if and . The set of all orthogonal matrices of size with determinant +1 is a representation of a group known as the special orthogonal group , one example of which is the rotation group SO(3). The set of all orthogonal matrices of size with determinant +1 or −1 is a representation of the (general) orthogonal group .


title: Rotation matrix

text: The direction of vector rotation is counterclockwise if is positive (e.g. 90°), and clockwise if is negative (e.g. −90°). Thus the clockwise rotation matrix is found as


title: Rotation matrix

text: The two-dimensional case is the only non-trivial (i.e. not one-dimensional) case where the rotation matrices group is commutative, so that it does not matter in which order multiple rotations are performed. An alternative convention uses rotating axes, and the above matrices also represent a rotation of the "axes clockwise" through an angle .


title: Rotation matrix

text: If a standard right-handed Cartesian coordinate system is used, with the to the right and the up, the rotation is counterclockwise. If a left-handed Cartesian coordinate system is used, with directed to the right but directed down, is clockwise. Such non-standard orientations are rarely used in mathematics but are common in 2D computer graphics, which often have the origin in the top left corner and the down the screen or page.


title: Rotation matrix

text: See below for other alternative conventions which may change the sense of the rotation produced by a rotation matrix.


title: Rotation matrix

text: form a ring isomorphic to the field of the complex numbers . Under this isomorphism, the rotation matrices correspond to circle of the unit complex numbers, the complex numbers of modulus .


title: Rotation matrix

text: If one identifies formula_14 with formula_15 through the linear isomorphism formula_16 the action of a matrix of the above form on vectors of formula_14 corresponds to the multiplication by the complex number , and rotations correspond to multiplication by complex numbers of modulus .


title: Rotation matrix

text: A basic rotation (also called elemental rotation) is a rotation about one of the axes of a coordinate system. The following three basic rotation matrices rotate vectors by an angle about the -, -, or -axis, in three dimensions, using the right-hand rule—which codifies their alternating signs. (The same matrices can also represent a clockwise rotation of the axes.)


title: Rotation matrix

text: For column vectors, each of these basic vector rotations appears counterclockwise when the axis about which they occur points toward the observer, the coordinate system is right-handed, and the angle is positive. , for instance, would rotate toward the a vector aligned with the , as can easily be checked by operating with on the vector :


title: Rotation matrix

text: This is similar to the rotation produced by the above-mentioned two-dimensional rotation matrix. See below for alternative conventions which may apparently or actually invert the sense of the rotation produced by these matrices.


title: Rotation matrix

text: Other rotation matrices can be obtained from these three using matrix multiplication. For example, the product


title: Rotation matrix

text: represents a rotation whose yaw, pitch, and roll angles are , and , respectively. More formally, it is an intrinsic rotation whose Tait–Bryan angles are , , , about axes , , , respectively.


title: Rotation matrix

text: These matrices produce the desired effect only if they are used to premultiply column vectors, and (since in general matrix multiplication is not commutative) only if they are applied in the specified order (see Ambiguities for more details). The order of rotation operations is from right to left; the matrix adjacent to the column vector is the first to be applied, and then the one to the left.


title: Rotation matrix

text: Every rotation in three dimensions is defined by its axis (a vector along this axis is unchanged by the rotation), and its angle — the amount of rotation about that axis (Euler rotation theorem).


title: Rotation matrix

text: There are several methods to compute the axis and angle from a rotation matrix (see also axis–angle representation). Here, we only describe the method based on the computation of the eigenvectors and eigenvalues of the rotation matrix. It is also possible to use the trace of the rotation matrix.


title: Rotation matrix

text: since the rotation of around the rotation axis must result in . The equation above may be solved for which is unique up to a scalar factor unless .


title: Rotation matrix

text: Viewed in another way, is an eigenvector of corresponding to the eigenvalue . Every rotation matrix must have this eigenvalue, the other two eigenvalues being complex conjugates of each other. It follows that a general rotation matrix in three dimensions has, up to a multiplicative constant, only one real eigenvector.


title: Rotation matrix

text: The matrix–vector product becomes a cross product of a vector with itself, ensuring that the result is zero:


title: Rotation matrix

text: This does not work if is symmetric. Above, if is zero, then all subsequent steps are invalid. In this case, it is necessary to diagonalize and find the eigenvector corresponding to an eigenvalue of 1.


title: Rotation matrix

text: To find the angle of a rotation, once the axis of the rotation is known, select a vector perpendicular to the axis. Then the angle of the rotation is the angle between and .


title: Rotation matrix

text: A more direct method, however, is to simply calculate the trace: the sum of the diagonal elements of the rotation matrix. Care should be taken to select the right sign for the angle to match the chosen axis:


title: Rotation matrix

text: A derivation of this matrix from first principles can be found in section 9.2 here. The basic idea to derive this matrix is dividing the problem into few known simple steps.


title: Rotation matrix

text: where is the cross product matrix of ; the expression is the outer product, and is the identity matrix. Alternatively, the matrix entries are:


title: Rotation matrix

text: where is the Levi-Civita symbol with . This is a matrix form of Rodrigues' rotation formula, (or the equivalent, differently parametrized Euler–Rodrigues formula) with


title: Rotation matrix

text: If the 3D space is right-handed and , this rotation will be counterclockwise when points towards the observer (Right-hand rule). Explicitly, with formula_39 a right-handed orthonormal basis,


title: Rotation matrix

text: A rotation is termed proper if , and improper (or a roto-reflection) if . For even dimensions , the eigenvalues of a proper rotation occur as pairs of complex conjugates which are roots of unity: for , which is real only for . Therefore, there may be no vectors fixed by the rotation (), and thus no axis of rotation. Any fixed eigenvectors occur in pairs, and the axis of rotation is an even-dimensional subspace.


title: Rotation matrix

text: For odd dimensions , a proper rotation will have an odd number of eigenvalues, with at least one and the axis of rotation will be an odd dimensional subspace. Proof:


title: Rotation matrix

text: Here is the identity matrix, and we use , as well as since is odd. Therefore, , meaning there is a null vector with , that is , a fixed eigenvector. There may also be pairs of fixed eigenvectors in the even-dimensional subspace orthogonal to , so the total dimension of fixed eigenvectors is odd.


title: Rotation matrix

text: For example, in 2-space , a rotation by angle has eigenvalues and , so there is no axis of rotation except when , the case of the null rotation. In 3-space , the axis of a non-null proper rotation is always a unique line, and a rotation around this axis by angle has eigenvalues . In 4-space , the four eigenvalues are of the form . The null rotation has . The case of is called a "simple rotation", with two unit eigenvalues forming an "axis plane", and a two-dimensional rotation orthogonal to the axis plane. Otherwise, there is no axis plane. The case of is called an "isoclinic rotation", having eigenvalues repeated twice, so every vector is rotated through an angle .


title: Rotation matrix

text: The trace of a rotation matrix is equal to the sum of its eigenvalues. For , a rotation by angle has trace . For , a rotation around any axis by angle has trace . For , and the trace is , which becomes for an isoclinic rotation.


title: Rotation matrix

text: In Euclidean geometry, a rotation is an example of an isometry, a transformation that moves points without changing the distances between them. Rotations are distinguished from other isometries by two additional properties: they leave (at least) one point fixed, and they leave "handedness" unchanged. In contrast, a translation moves every point, a reflection exchanges left- and right-handed ordering, a glide reflection does both, and an improper rotation combines a change in handedness with a normal rotation.


title: Rotation matrix

text: If a fixed point is taken as the origin of a Cartesian coordinate system, then every point can be given coordinates as a displacement from the origin. Thus one may work with the vector space of displacements instead of the points themselves. Now suppose are the coordinates of the vector from the origin to point . Choose an orthonormal basis for our coordinates; then the squared distance to , by Pythagoras, is


title: Rotation matrix

text: A geometric rotation transforms lines to lines, and preserves ratios of distances between points. From these properties it can be shown that a rotation is a linear transformation of the vectors, and thus can be written in matrix form, . The fact that a rotation preserves, not just ratios, but distances themselves, is stated as


title: Rotation matrix

text: Because this equation holds for all vectors, , one concludes that every rotation matrix, , satisfies the orthogonality condition,


title: Rotation matrix

text: Rotations preserve handedness because they cannot change the ordering of the axes, which implies the special matrix condition,


title: Rotation matrix

text: Equally important, it can be shown that any matrix satisfying these two conditions acts as a rotation.


title: Rotation matrix

text: Noting that any identity matrix is a rotation matrix, and that matrix multiplication is associative, we may summarize all these properties by saying that the rotation matrices form a group, which for is non-abelian, called a special orthogonal group, and denoted by , , , or , the group of rotation matrices is isomorphic to the group of rotations in an space. This means that multiplication of rotation matrices corresponds to composition of rotations, applied in left-to-right order of their corresponding matrices.


title: Rotation matrix

text: In most cases the effect of the ambiguity is equivalent to the effect of a rotation matrix inversion (for these orthogonal matrices equivalently matrix transpose).


title: Rotation matrix

text: Two features are noteworthy. First, one of the roots (or eigenvalues) is 1, which tells us that some direction is unaffected by the matrix. For rotations in three dimensions, this is the "axis" of the rotation (a concept that has no meaning in any other dimension). Second, the other two roots are a pair of complex conjugates, whose product is 1 (the constant term of the quadratic), and whose sum is (the negated linear term). This factorization is of interest for rotation matrices because the same thing occurs for all of them. (As special cases, for a null rotation the "complex conjugates" are both 1, and for a 180° rotation they are both −1.) Furthermore, a similar factorization holds for any rotation matrix. If the dimension, , is odd, there will be a "dangling" eigenvalue of 1; and for any dimension the rest of the polynomial factors into quadratic terms like the one here (with the two special cases noted). We are guaranteed that the characteristic polynomial will have degree and thus eigenvalues. And since a rotation matrix commutes with its transpose, it is a normal matrix, so can be diagonalized. We conclude that every rotation matrix, when expressed in a suitable coordinate system, partitions into independent rotations of two-dimensional subspaces, at most of them.


title: Rotation matrix

text: The sum of the entries on the main diagonal of a matrix is called the trace; it does not change if we reorient the coordinate system, and always equals the sum of the eigenvalues. This has the convenient implication for and rotation matrices that the trace reveals the angle of rotation, , in the two-dimensional space (or subspace). For a matrix the trace is , and for a matrix it is . In the three-dimensional case, the subspace consists of all vectors perpendicular to the rotation axis (the invariant direction, with eigenvalue 1). Thus we can extract from any rotation matrix a rotation axis and an angle, and these completely determine the rotation.


title: Rotation matrix

text: with . Therefore, we may set and , for some angle . To solve for it is not enough to look at alone or alone; we must consider both together to place the angle in the correct quadrant, using a two-argument arctangent function.


title: Rotation matrix

text: Although will probably not equal 1, but some value , we can use a slight variation of the previous computation to find a so-called Givens rotation that transforms the column to


title: Rotation matrix

text: zeroing . This acts on the subspace spanned by the - and -axes. We can then repeat the process for the -subspace to zero . Acting on the full matrix, these two rotations produce the schematic form


title: Rotation matrix

text: Shifting attention to the second column, a Givens rotation of the -subspace can now zero the value. This brings the full matrix to the form


title: Rotation matrix

text: entries below the diagonal to zero. We can zero them by extending the same idea of stepping through the columns with a series of rotations in a fixed sequence of planes. We conclude that the set of rotation matrices, each of which has entries, can be parameterized by angles.


title: Rotation matrix

text: In three dimensions this restates in matrix form an observation made by Euler, so mathematicians call the ordered sequence of three angles Euler angles. However, the situation is somewhat more complicated than we have so far indicated. Despite the small dimension, we actually have considerable freedom in the sequence of axis pairs we use; and we also have some freedom in the choice of angles. Thus we find many different conventions employed when three-dimensional rotations are parameterized for physics, or medicine, or chemistry, or other disciplines. When we include the option of world axes or body axes, 24 different sequences are possible. And while some disciplines call any sequence Euler angles, others give different names (Cardano, Tait–Bryan, roll-pitch-yaw) to different sequences.


title: Rotation matrix

text: One reason for the large number of options is that, as noted previously, rotations in three dimensions (and higher) do not commute. If we reverse a given sequence of rotations, we get a different outcome. This also implies that we cannot compose two rotations by adding their corresponding angles. Thus Euler angles are not vectors, despite a similarity in appearance as a triplet of numbers.


title: Rotation matrix

text: This is no illusion; not just one, but many, copies of -dimensional rotations are found within -dimensional rotations, as subgroups. Each embedding leaves one direction fixed, which in the case of matrices is the rotation axis. For example, we have


title: Rotation matrix

text: fixing the -axis, the -axis, and the -axis, respectively. The rotation axis need not be a coordinate axis; if is a unit vector in the desired direction, then


title: Rotation matrix

text: A direction in -dimensional space will be a unit magnitude vector, which we may consider a point on a generalized sphere, . Thus it is natural to describe the rotation group as combining and . A suitable formalism is the fiber bundle,


title: Rotation matrix

text: where for every direction in the base space, , the fiber over it in the total space, , is a copy of the fiber space, , namely the rotations that keep that direction fixed.


title: Rotation matrix

text: Thus we can build an rotation matrix by starting with a matrix, aiming its fixed axis on (the ordinary sphere in three-dimensional space), aiming the resulting rotation on , and so on up through . A point on can be selected using numbers, so we again have numbers to describe any rotation matrix.


title: Rotation matrix

text: In fact, we can view the sequential angle decomposition, discussed previously, as reversing this process. The composition of Givens rotations brings the first column (and row) to , so that the remainder of the matrix is a rotation matrix of dimension one less, embedded so as to leave fixed.


title: Rotation matrix

text: When an rotation matrix , does not include a −1 eigenvalue, thus none of the planar rotations which it comprises are 180° rotations, then is an invertible matrix. Most rotation matrices fit this description, and for them it can be shown that is a skew-symmetric matrix, . Thus ; and since the diagonal is necessarily zero, and since the upper triangle determines the lower one, contains independent numbers.


title: Rotation matrix

text: Conveniently, is invertible whenever is skew-symmetric; thus we can recover the original matrix using the "Cayley transform",


title: Rotation matrix

text: which maps any skew-symmetric matrix to a rotation matrix. In fact, aside from the noted exceptions, we can produce any rotation matrix in this way. Although in practical applications we can hardly afford to ignore 180° rotations, the Cayley transform is still a potentially useful tool, giving a parameterization of most rotation matrices without trigonometric functions.


title: Rotation matrix

text: If we condense the skew entries into a vector, , then we produce a 90° rotation around the -axis for (1, 0, 0), around the -axis for (0, 1, 0), and around the -axis for (0, 0, 1). The 180° rotations are just out of reach; for, in the limit as , does approach a 180° rotation around the axis, and similarly for other directions.


title: Rotation matrix

text: This is useful, for instance, in computer graphics, since shears can be implemented with fewer multiplication instructions than rotating a bitmap directly. On modern computers, this may not matter, but it can be relevant for very old or low-end microprocessors.


title: Rotation matrix

text: Below follow some basic facts about the role of the collection of "all" rotation matrices of a fixed dimension (here mostly 3) in mathematics and particularly in physics where rotational symmetry is a "requirement" of every truly fundamental law (due to the assumption of isotropy of space), and where the same symmetry, when present, is a "simplifying property" of many problems of less fundamental nature. Examples abound in classical mechanics and quantum mechanics. Knowledge of the part of the solutions pertaining to this symmetry applies (with qualifications) to "all" such problems and it can be factored out of a specific problem at hand, thus reducing its complexity. A prime example – in mathematics and physics – would be the theory of spherical harmonics. Their role in the group theory of the rotation groups is that of being a representation space for the entire set of finite-dimensional irreducible representations of the rotation group SO(3). For this topic, see Rotation group SO(3) § Spherical harmonics.


title: Rotation matrix

text: The rotation matrices for each form a group, the special orthogonal group, . This algebraic structure is coupled with a topological structure inherited from formula_85 in such a way that the operations of multiplication and taking the inverse are analytic functions of the matrix entries. Thus is for each a Lie group. It is compact and connected, but not simply connected. It is also a semi-simple group, in fact a simple group with the exception SO(4). The relevance of this is that all theorems and all machinery from the theory of analytic manifolds (analytic manifolds are in particular smooth manifolds) apply and the well-developed representation theory of compact semi-simple groups is ready for use.


title: Rotation matrix

text: and is the space of skew-symmetric matrices of dimension , see classical group, where is the Lie algebra of , the orthogonal group. For reference, the most common basis for is


title: Rotation matrix

text: Connecting the Lie algebra to the Lie group is the exponential map, which is defined using the standard matrix exponential series for For any skew-symmetric matrix , is always a rotation matrix.


title: Rotation matrix

text: An important practical example is the case. In rotation group SO(3), it is shown that one can identify every with an Euler vector , where is a unit magnitude vector.


title: Rotation matrix

text: By the properties of the identification formula_88, is in the null space of . Thus, is left invariant by and is hence a rotation axis.


title: Rotation matrix

text: This is the matrix for a rotation around axis by the angle . For full detail, see exponential map SO(3).


title: Rotation matrix

text: The BCH formula provides an explicit expression for in terms of a series expansion of nested commutators of and . This general expansion unfolds as


title: Rotation matrix

text: for suitable trigonometric function coefficients, detailed in the Baker–Campbell–Hausdorff formula for SO(3).


title: Rotation matrix

text: As a group identity, the above holds for "all faithful representations", including the doublet (spinor representation), which is simpler. The same explicit formula thus follows straightforwardly through Pauli matrices; see the derivation for SU(2). For the general case, one might use Ref.


title: Rotation matrix

text: The Lie group of rotation matrices, , is not simply connected, so Lie theory tells us it is a homomorphic image of a universal covering group. Often the covering group, which in this case is called the spin group denoted by , is simpler and more natural to work with.


title: Rotation matrix

text: In the case of planar rotations, SO(2) is topologically a circle, . Its universal covering group, Spin(2), is isomorphic to the real line, , under addition. Whenever angles of arbitrary magnitude are used one is taking advantage of the convenience of the universal cover. Every rotation matrix is produced by a countable infinity of angles, separated by integer multiples of 2. Correspondingly, the fundamental group of is isomorphic to the integers, .


title: Rotation matrix

text: In the case of spatial rotations, SO(3) is topologically equivalent to three-dimensional real projective space, . Its universal covering group, Spin(3), is isomorphic to the , . Every rotation matrix is produced by two opposite points on the sphere. Correspondingly, the fundamental group of SO(3) is isomorphic to the two-element group, .


title: Rotation matrix

text: We can also describe Spin(3) as isomorphic to quaternions of unit norm under multiplication, or to certain real matrices, or to complex special unitary matrices, namely SU(2). The covering maps for the first and the last case are given by


title: Rotation matrix

text: Many features of these cases are the same for higher dimensions. The coverings are all two-to-one, with , , having fundamental group . The natural setting for these groups is within a Clifford algebra. One type of action of the rotations is produced by a kind of "sandwich", denoted by . More importantly in applications to physics, the corresponding spin representation of the Lie algebra sits inside the Clifford algebra. It can be exponentiated in the usual way to give rise to a representation, also known as projective representation of the rotation group. This is the case with SO(3) and SU(2), where the representation can be viewed as an "inverse" of the covering map. By properties of covering maps, the inverse can be chosen ono-to-one as a local section, but not globally.


title: Rotation matrix

text: The matrices in the Lie algebra are not themselves rotations; the skew-symmetric matrices are derivatives, proportional differences of rotations. An actual "differential rotation", or "infinitesimal rotation matrix" has the form


title: Rotation matrix

text: The computation rules are as usual except that infinitesimals of second order are routinely dropped. With these rules, these matrices do not satisfy all the same properties as ordinary finite rotation matrices under the usual treatment of infinitesimals. It turns out that "the order in which infinitesimal rotations are applied is irrelevant". To see this exemplified, consult infinitesimal rotations SO(3).


title: Rotation matrix

text: We have seen the existence of several decompositions that apply in any dimension, namely independent planes, sequential angles, and nested dimensions. In all these cases we can either decompose a matrix or construct one. We have also given special attention to rotation matrices, and these warrant further attention, in both directions .


title: Rotation matrix

text: Given the unit quaternion , the equivalent pre-multiplied (to be used with column vectors) rotation matrix is


title: Rotation matrix

text: Now every quaternion component appears multiplied by two in a term of degree two, and if all such terms are zero what is left is an identity matrix. This leads to an efficient, robust conversion from any quaternion – whether unit or non-unit – to a rotation matrix. Given:


title: Rotation matrix

text: Freed from the demand for a unit quaternion, we find that nonzero quaternions act as homogeneous coordinates for rotation matrices. The Cayley transform, discussed earlier, is obtained by scaling the quaternion so that its component is 1. For a 180° rotation around any axis, will be zero, which explains the Cayley limitation.


title: Rotation matrix

text: The sum of the entries along the main diagonal (the trace), plus one, equals , which is . Thus we can write the trace itself as ; and from the previous version of the matrix we see that the diagonal entries themselves have the same form: , , and . So we can easily compare the magnitudes of all four quaternion components using the matrix diagonal. We can, in fact, obtain all four magnitudes using sums and square roots, and choose consistent signs using the skew-symmetric part of the off-diagonal entries:


title: Rotation matrix

text: This is numerically stable so long as the trace, , is not negative; otherwise, we risk dividing by (nearly) zero. In that case, suppose is the largest diagonal entry, so will have the largest magnitude (the other cases are derived by cyclic permutation); then the following is safe.


title: Rotation matrix

text: If the matrix contains significant error, such as accumulated numerical error, we may construct a symmetric matrix,


title: Rotation matrix

text: and find the eigenvector, , of its largest magnitude eigenvalue. (If is truly a rotation matrix, that value will be 1.) The quaternion so obtained will correspond to the rotation matrix closest to the given matrix (Note: formulation of the cited article is post-multiplied, works with row vectors).


title: Rotation matrix

text: If the matrix is nonsingular, its columns are linearly independent vectors; thus the Gram–Schmidt process can adjust them to be an orthonormal basis. Stated in terms of numerical linear algebra, we convert to an orthogonal matrix, , using QR decomposition. However, we often prefer a closest to , which this method does not accomplish. For that, the tool we want is the polar decomposition (; ).


title: Rotation matrix

text: To measure closeness, we may use any matrix norm invariant under orthogonal transformations. A convenient choice is the Frobenius norm, , squared, which is the sum of the squares of the element differences. Writing this in terms of the trace, , our goal is,


title: Rotation matrix

text: Though written in matrix terms, the objective function is just a quadratic polynomial. We can minimize it in the usual way, by finding where its derivative is zero. For a matrix, the orthogonality constraint implies six scalar equalities that the entries of must satisfy. To incorporate the constraint(s), we may employ a standard technique, Lagrange multipliers, assembled as a symmetric matrix, . Thus our method is:


title: Rotation matrix

text: where is orthogonal and is symmetric. To ensure a minimum, the matrix (and hence ) must be positive definite. Linear algebra calls the polar decomposition of , with the positive square root of .


title: Rotation matrix

text: When is non-singular, the and factors of the polar decomposition are uniquely determined. However, the determinant of is positive because is positive definite, so inherits the sign of the determinant of . That is, is only guaranteed to be orthogonal, not a rotation matrix. This is unavoidable; an with negative determinant has no uniquely defined closest rotation matrix.


title: Rotation matrix

text: To efficiently construct a rotation matrix from an angle and a unit axis , we can take advantage of symmetry and skew-symmetry within the entries. If , , and are the components of the unit vector representing the axis, and


title: Rotation matrix

text: Determining an axis and angle, like determining a quaternion, is only possible up to the sign; that is, and correspond to the same rotation matrix, just like and . Additionally, axis–angle extraction presents additional difficulties. The angle can be restricted to be from 0° to 180°, but angles are formally ambiguous by multiples of 360°. When the angle is zero, the axis is undefined. When the angle is 180°, the matrix becomes symmetric, which has implications in extracting the axis. Near multiples of 180°, care is needed to avoid numerical problems: in extracting the angle, a two-argument arctangent with equal to avoids the insensitivity of arccos; and in computing the axis magnitude in order to force unit magnitude, a brute-force approach can lose accuracy through underflow .


title: Rotation matrix

text: The -, -, and -components of the axis would then be divided by . A fully robust approach will use a different algorithm when , the trace of the matrix , is negative, as with quaternion extraction. When is zero because the angle is zero, an axis must be provided from some source other than the matrix.


title: Rotation matrix

text: Complexity of conversion escalates with Euler angles (used here in the broad sense). The first difficulty is to establish which of the twenty-four variations of Cartesian axis order we will use. Suppose the three angles are , , ; physics and chemistry may interpret these as


title: Rotation matrix

text: One systematic approach begins with choosing the rightmost axis. Among all permutations of , only two place that axis first; one is an even permutation and the other odd. Choosing parity thus establishes the middle axis. That leaves two choices for the left-most axis, either duplicating the first or not. These three choices gives us variations; we double that to 24 by choosing static or rotating axes.


title: Rotation matrix

text: This is enough to construct a matrix from angles, but triples differing in many ways can give the same rotation matrix. For example, suppose we use the convention above; then we have the following equivalent pairs:


title: Rotation matrix

text: The problem of singular alignment, the mathematical analog of physical gimbal lock, occurs when the middle rotation aligns the axes of the first and last rotations. It afflicts every axis order at either even or odd multiples of 90°. These singularities are not characteristic of the rotation matrix as such, and only occur with the usage of Euler angles.


title: Rotation matrix

text: The singularities are avoided when considering and manipulating the rotation matrix as orthonormal row vectors (in 3D applications often named the right-vector, up-vector and out-vector) instead of as angles. The singularities are also avoided when working with quaternions.


title: Rotation matrix

text: In some instances it is interesting to describe a rotation by specifying how a vector is mapped into another through the shortest path (smallest angle). In formula_37 this completely describes the associated rotation matrix. In general, given , the matrix


title: Rotation matrix

text: We sometimes need to generate a uniformly distributed random rotation matrix. It seems intuitively clear in two dimensions that this means the rotation angle is uniformly distributed between 0 and 2. That intuition is correct, but does not carry over to higher dimensions. For example, if we decompose rotation matrices in axis–angle form, the angle should "not" be uniformly distributed; the probability that (the magnitude of) the angle is at most should be , for .


title: Rotation matrix

text: Since is a connected and locally compact Lie group, we have a simple standard criterion for uniformity, namely that the distribution be unchanged when composed with any arbitrary rotation (a Lie group "translation"). This definition corresponds to what is called "Haar measure". show how to use the Cayley transform to generate and test matrices according to this criterion.


title: Rotation matrix

text: We can also generate a uniform distribution in any dimension using the "subgroup algorithm" of . This recursively exploits the nested dimensions group structure of , as follows. Generate a uniform angle and construct a rotation matrix. To step from to , generate a vector uniformly distributed on the -sphere , embed the matrix in the next larger size with last column , and rotate the larger matrix so the last column becomes .


title: Rotation matrix

text: As usual, we have special alternatives for the case. Each of these methods begins with three independent random scalars uniformly distributed on the unit interval. takes advantage of the odd dimension to change a Householder reflection to a rotation by negation, and uses that to aim the axis of a uniform planar rotation.


title: Rotation matrix

text: Another method uses unit quaternions. Multiplication of rotation matrices is homomorphic to multiplication of quaternions, and multiplication by a unit quaternion rotates the unit sphere. Since the homomorphism is a local isometry, we immediately conclude that to produce a uniform distribution on SO(3) we may use a uniform distribution on . In practice: create a four-element vector where each element is a sampling of a normal distribution. Normalize its length and you have a uniformly sampled random unit quaternion which represents a uniformly sampled random rotation. Note that the aforementioned only applies to rotations in dimension 3. For a generalised idea of quaternions, one should look into Rotors.


title: Rotation matrix

text: For the axis–angle form, the axis is uniformly distributed over the unit sphere of directions, , while the angle has the nonuniform distribution over noted previously .


title: Temple of Artemis

text: The Temple of Artemis or Artemision (; ), also known as the Temple of Diana, was a Greek temple dedicated to an ancient, local form of the goddess Artemis (identified with Diana, a Roman goddess). It was located in Ephesus (near the modern town of Selçuk in present-day Turkey). By 401 AD it had been ruined or destroyed. Only foundations and fragments of the last temple remain at the site.


title: Temple of Artemis

text: The earliest version of the temple (a Bronze Age "temenos") antedated the Ionic immigration by many years. Callimachus, in his "Hymn to Artemis", attributed it to the Amazons. In the 7th century BC, it was destroyed by a flood. Its reconstruction, in more grandiose form, began around 550 BC, under Chersiphron, the Cretan architect, and his son Metagenes. The project was funded by Croesus of Lydia, and took 10 years to complete. This version of the temple was destroyed in 356 BC by Herostratus in an act of arson.


title: Temple of Artemis

text: The next, greatest, and last form of the temple, funded by the Ephesians themselves, is described in Antipater of Sidon's list of the world's Seven Wonders:


title: Temple of Artemis

text: I have set eyes on the wall of lofty Babylon on which is a road for chariots, and the statue of Zeus by the Alpheus, and the hanging gardens, and the colossus of the Sun, and the huge labour of the high pyramids, and the vast tomb of Mausolus; but when I saw the house of Artemis that mounted to the clouds, those other marvels lost their brilliancy, and I said, "Lo, apart from Olympus, the Sun never looked on aught so grand".


title: Temple of Artemis

text: The Temple of Artemis ("Artemision") was located near the ancient city of Ephesus, about south from the modern port city of İzmir, in Turkey. Today the site lies on the edge of the modern town of Selçuk.


title: Temple of Artemis

text: The sacred site ("temenos") at Ephesus was far older than the "Artemision" itself. Pausanias was certain that it antedated the Ionic immigration by many years, being older even than the oracular shrine of Apollo at Didyma. He said that the pre-Ionic inhabitants of the city were Leleges and Lydians. Callimachus, in his "Hymn to Artemis" attributed the earliest "temenos" at Ephesus to the Amazons, legendary warrior-women whose religious practise he imagined already centered upon an image ("bretas") of Artemis, their matron goddess. Pausanias believed that the temple pre-dated the Amazons.


title: Temple of Artemis

text: Pausanias's estimation of the site's antiquity seems well-founded. Before World War I, site excavations by David George Hogarth seemed to identify three successive temple buildings. Re-excavations in 1987–88 and re-appraisal of Hogarth's account confirmed that the site was occupied as early as the Bronze Age, with a sequence of pottery finds that extend forward to Middle Geometric times, when a peripteral temple with a floor of hard-packed clay was constructed in the second half of the 8th century BC. The peripteral temple at Ephesus offers the earliest example of a peripteral type on the coast of Asia Minor, and perhaps the earliest Greek temple surrounded by colonnades anywhere.


title: Temple of Artemis

text: In the 7th century BC, a flood destroyed the temple, depositing over half a meter of sand and flotsam over the original clay floor. Among the flood debris were the remains of a carved ivory plaque of a griffin and the Tree of Life, apparently North Syrian, and some drilled tear-shaped amber drops of elliptical cross-section. These probably once dressed a wooden effigy ("xoanon") of the Lady of Ephesus, which must have been destroyed or recovered from the flood. Bammer notes that though the site was prone to flooding, and raised by silt deposits about two metres between the 8th and 6th centuries, and a further 2.4 m between the sixth and the fourth, its continued use "indicates that maintaining the identity of the actual location played an important role in the sacred organization".


title: Temple of Artemis

text: The new temple was sponsored at least in part by Croesus, who founded Lydia's empire and was overlord of Ephesus. It was designed and constructed from around 550 BC by the Cretan architect Chersiphron and his son Metagenes. It was 115 m (377 ft) long and 46 m (151 ft) wide, supposedly the first Greek temple built of marble. Its peripteral columns stood some 13 m (40 ft) high, in double rows that formed a wide ceremonial passage around the "cella" that housed the goddess's cult image. Thirty-six of these columns were, according to Pliny, decorated by carvings in relief. A new ebony or blackened grapewood cult statue was sculpted by Endoios, and a "naiskos" to house it was erected east of the open-air altar.


title: Temple of Artemis

text: A rich foundation deposit from this era, also called the "Artemision deposit", yielded more than a thousand items, including what may be the earliest coins made from the silver-gold alloy electrum. The deposit contains some of the earliest inscribed coins, those of Phanes, dated to 625–600 BC from Ephesus, with the legend ΦΑΕΝΟΣ ΕΜΙ ΣΗΜΑ (or similar) (“I am the badge of Phanes”), or just bearing the name ΦΑΝΕΟΣ (“of Phanes”).


title: Temple of Artemis

text: Fragments of bas-relief on the lowest drums of the temple columns, preserved in the British Museum, show that the enriched columns of the later temple, of which a few survive ("illustration below") were versions of this earlier feature. Pliny the Elder, seemingly unaware of the ancient continuity of the sacred site, claims that the new temple's architects chose to build it on marshy ground as a precaution against earthquakes, with lower foundation layers of fleeces and pounded charcoal.


title: Temple of Artemis

text: The temple became an important attraction, visited by merchants, kings, and sightseers, many of whom paid homage to Artemis in the form of jewelry and various goods. It also offered sanctuary to those fleeing persecution or punishment, a tradition linked in myth to the Amazons who twice fled there seeking the goddess's protection from punishment, firstly by Dionysus and later, by Heracles. Diogenes Laertius claims that the misanthropic philosopher Heraclitus, thoroughly disapproving of civil life at Ephesus, played knucklebones in the temple with the boys, and later deposited his writings there.


title: Temple of Artemis

text: In 356 BC, the temple burned down. Various sources describe this as a vainglorious act of arson by a man, Herostratus, who set fire to the wooden roof-beams, seeking fame at any cost; thus the term "herostratic fame". For this outrage, the Ephesians sentenced the perpetrator to death and forbade anyone from mentioning his name; but Theopompus later noted it. Aristotle's "Meteorology" describes the temple's conflagration, but not its cause. In Greek and Roman historical tradition, the temple's destruction coincided with the birth of Alexander the Great (around 20/21 July 356 BC). Plutarch remarks that Artemis was too preoccupied with Alexander's delivery to save her burning temple; he does not specify a cause for the fire.


title: Temple of Artemis

text: Herostratus' part in the temple's destruction has been questioned in modern scholarship. Stefan Karweise notes that any arsonist would have needed access to the wooden roof framing;


title: Temple of Artemis

text: The fire might even have been deliberately and covertly set by the temple's administrators, aware of that the temple's foundation was sinking, but prevented from re-siting it elsewhere by religious constraints; Bammer has noted the conservation of the original sacred location throughout successive rebuildings, despite continued problems with flooding and foundations. Karwiese questions the motive of Herostratus since he only divulged his purpose under torture, which does not fit a man seeking fame. considers Herostratus a "useful idiot in the service of the priesthood."


title: Temple of Artemis

text: Alexander offered to pay for the temple's rebuilding; the Ephesians tactfully refused, saying "it would be improper for one god to build a temple to another", and eventually rebuilt it after his death, at their own expense. Work started in 323 BC and continued for many years. The third temple was larger than the second; 137 m (450 ft) long by 69 m (225 ft) wide and 18 m (60 ft) high, with more than 127 columns. Athenagoras of Athens names Endoeus, a pupil of Daedalus, as sculptor of Artemis' main cult image.


title: Temple of Artemis

text: Pausanias (c. 2nd century AD) reports another image and altar in the temple, dedicated to Artemis "Protothronia" (Artemis "of the first seat") and a gallery of images above this altar, including an ancient figure of Nyx (the primordial goddess of Night) by the sculptor Rhoecus (6th century BC). Pliny describes images of Amazons, the legendary founders of Ephesus and Ephesian Artemis' original protégés, carved by Scopas. Literary sources describe the temple's adornment by paintings, columns gilded with gold and silver, and religious works of renowned Greek sculptors Polyclitus, Pheidias, Cresilas, and Phradmon.


title: Temple of Artemis

text: This reconstruction survived for 600 years, and appears multiple times in early Christian accounts of Ephesus. According to the New Testament, the appearance of the first Christian missionary in Ephesus caused locals to fear for the temple's dishonor. The 2nd-century "Acts of John" includes an apocryphal tale of the temple's destruction: the apostle John prayed publicly in the Temple of Artemis, exorcising its demons and "of a sudden the altar of Artemis split in many pieces... and half the temple fell down," instantly converting the Ephesians, who wept, prayed or took flight.


title: Temple of Artemis

text: Against this, a Roman edict of 162 AD acknowledges the importance of "Artemesion", the annual Ephesian festival to Artemis, and officially extends it from a few holy days over March–April to a whole month, "one of the largest and most magnificent religious festivals in Ephesus' liturgical calendar".


title: Temple of Artemis

text: In 268 AD, according to Jordanes, a raid by the Goths, under their leaders "Respa, Veduc and Thurar", "laid waste many populous cities and set fire to the renowned temple of Diana at Ephesus." ; the extent and severity of the damage are unknown; the temple may have lain derelict until its official closure during the persecution of pagans in the late Roman Empire, Ammonius of Alexandria comments on its closure, perhaps as early as 407, or no later than the mid 5th century. After closure and after the city had become Christian, the name of Artemis appears to have been erased from inscriptions throughout Ephesus.


title: Temple of Artemis

text: Cyril of Alexandria credited Archbishop of Constantinople John Chrysostom with destroying the temple, referring to him as "the destroyer of the demons and overthrower of the temple of Diana". A later Archbishop of Constantinople, Proclus, noted the achievements of John, saying "In Ephesus, he despoiled the art of Midas," but there is little evidence to support this claim.


title: Temple of Artemis

text: At least some of the stone from the abandoned temple was used in construction of other buildings. A late medieval legend claims that some of the columns in the Hagia Sophia were taken from the Temple of Artemis at Ephesus, but there is no truth to this story.


title: Temple of Artemis

text: The main primary sources for the Temple of Artemis at Ephesus are Pliny the Elder's "Natural History", Pomponius Mela i:17, and Plutarch's "Life of Alexander" (referencing the burning of the "Artemiseum").


title: Temple of Artemis

text: After six years of searching, the site of the temple was rediscovered in 1869 by an expedition led by John Turtle Wood and sponsored by the British Museum. These excavations continued until 1874. A few further fragments of sculpture were found during the 1904–1906 excavations directed by David George Hogarth. The recovered sculptured fragments of the 4th-century rebuilding and a few from the earlier temple, which had been used in the rubble fill for the rebuilding, were assembled and displayed in the "Ephesus Room" of the British Museum. In addition, the museum has part of possibly the oldest pot-hoard of coins in the world (600 BC) that had been buried in the foundations of the Archaic temple.


title: Temple of Artemis

text: Today the site of the temple, which lies just outside Selçuk, is marked by a single column constructed of dissociated fragments discovered on the site.


title: Temple of Artemis

text: The archaic "temeton" beneath the later temples clearly housed some form of "Great Goddess" but nothing is known of her cult. The literary accounts that describe it as "Amazonian" refer to the later founder-myths of Greek emigrés who developed the cult and temple of "Artemis Ephesia". The wealth and splendor of temple and city were taken as evidence of Artemis Ephesia's power, and were the basis for her local and international prestige: despite the successive traumas of Temple destruction, each rebuilding – a gift and honor to the goddess – brought further prosperity. Large numbers of people came to Ephesus in March and in the beginning of May to attend the main Artemis Procession.


title: Temple of Artemis

text: Artemis' shrines, temples and festivals ("Artemisia") could be found throughout the Greek world, but Ephesian Artemis was unique. The Ephesians considered her theirs, and resented any foreign claims to her protection. Once Persia ousted and replaced their Lydian overlord Croesus, the Ephesians played down his contribution to the temple's restoration. On the whole, the Persians dealt fairly with Ephesus, but removed some religious artifacts from Artemis' Temple to Sardis and brought Persian priests into her Ephesian cult; this was not forgiven. When Alexander conquered the Persians, his offer to finance the temple's second rebuilding was politely but firmly refused. Ephesian Artemis lent her city's diplomacy a powerful religious edge.


title: Temple of Artemis

text: Under Hellenic rule, and later, under Roman rule, the Ephesian "Artemisia" festival was increasingly promoted as a key element in the pan-Hellenic festival circuit. It was part of a definitively Greek political and cultural identity, essential to the economic life of the region, and an excellent opportunity for young, unmarried Greeks of both sexes to seek out marriage partners. Games, contests and theatrical performances were held in the goddess's name, and Pliny describes her procession as a magnificent crowd-puller; it was shown in one of Apelles' best paintings, which depicted the goddess's image carried through the streets and surrounded by maidens. In the Roman Imperial era, the emperor Commodus lent his name to the festival games, and might have sponsored them.


title: Temple of Artemis

text: From the Greek point of view, the Ephesian Artemis is a distinctive form of their goddess Artemis. In Greek cult and myth, Artemis is the twin sister of Apollo, a virgin goddess of the hunt, the wilderness and the moon, who, despite being a goddess of childbirth was nevertheless known for her chastity. At Ephesus, a goddess whom the Greeks associated with Artemis was venerated in an archaic, pre-Hellenic cult image that was carved of wood (a "xoanon") and kept decorated with jewelry. The features are most similar to Near-Eastern and Egyptian deities, and least similar to Greek ones. The body and legs are enclosed within a tapering pillar-like term, from which the goddess' feet protrude. On the coins minted at Ephesus, the goddess wears a mural crown (like a city's walls), an attribute of Cybele as a protector of cities (see "polos").


title: Temple of Artemis

text: The traditional interpretation of the oval objects covering the upper part of the Ephesian Artemis is that they represent multiple breasts, symbolizing her fertility. This interpretation began in late antiquity and resulted in designations of the Ephesian goddess as "Diana Efesia Multimammia" and other related descriptions. This interpretation was rooted in Minucius Felix and Jerome's Christian attacks on pagan popular religion, and modern scholarship has cast doubt on the traditional interpretation that the statue depicts a many-breasted goddess. Evidence suggests that the oval objects were not intended to depict part of the goddess' anatomy at all. In some versions of the statue, the goddess' skin has been painted black (likely to emulate the aged wood of the original), while her clothes and regalia, including the so-called "breasts", were left unpainted or cast in different colors. Robert Fleischer suggested that instead of breasts, the oval objects were decorations that would have been hung ceremonially on the original wood statue (possibly eggs or the scrotal sacs of sacrificed bulls), and which were incorporated as carved features on later copies. The "breasts" of the Lady of Ephesus, it now appears, were likely based on amber gourd-shaped drops, elliptical in cross-section and drilled for hanging, that were rediscovered in the archaeological excavations of 1987–1988. These objects remained in place where the ancient wooden statue of the goddess had been caught by an 8th-century flood. This form of jewelry, then, had already been developed by the Geometric Period.


title: Temple of Artemis

text: On the coins she rests either arm on a staff formed of entwined serpents or of a stack of "ouroboroi", the eternal serpent with its tail in its mouth. In some accounts, the Lady of Ephesus was attended by eunuch priests called "Megabyzoi"; this could have been a proper name or a title. The practise of ritual self-emasculation as qualification to serve a deity is usually identified with Cybele's eunuch mendicant priests, the Galli. The Megabyzoi of Ephesian Artemis were assisted by young, virgin girls (korai).


title: Temple of Artemis

text: A votive inscription mentioned by Florence Mary Bennett, which dates probably from about the 3rd century BC, associates Ephesian Artemis with Crete: "To the Healer of diseases, to Apollo, Giver of Light to mortals, Eutyches has set up in votive offering [a statue of] the Cretan Lady of Ephesus, the Light-Bearer."


title: Temple of Artemis

text: The Greek habits of syncretism assimilated all foreign gods under some form of the Olympian pantheon familiar to them—in "interpretatio graeca"—and it is clear that at Ephesus, the identification with Artemis that the Ionian settlers made of the "Lady of Ephesus" was slender. Nevertheless, later Greeks and Romans identified her with both Artemis and Diana, and there was a tradition in ancient Rome that identified her with the goddess Isis as well.


title: Temple of Artemis

text: The Christian approach was at variance with the syncretistic approach of pagans to gods who were not theirs. A Christian inscription at Ephesus suggests why so little remains at the site:


title: Temple of Artemis

text: Destroying the delusive image of the demon Artemis, Demeas has erected this symbol of Truth, the God that drives away idols, and the Cross of priests, deathless and victorious sign of Christ.


title: Temple of Artemis

text: The assertion that the Ephesians thought that their cult image had fallen from the sky, though it was a familiar origin-myth at other sites, is only known at Ephesus from Acts 19:35:


title: Temple of Artemis

text: What man is there that knoweth not how that the city of the Ephesians is a worshipper of the great goddess Diana, and of the [image] which fell down from Jupiter?


title: Temple of Artemis

text: Lynn LiDonnici observes that modern scholars are likely to be more concerned with origins of the Lady of Ephesus and her iconology than her adherents were at any point in time, and are prone to creating a synthetic account of the Lady of Ephesus by drawing together documentation that ranges over more than a millennium in its origins, creating a falsified, unitary picture, as of an unchanging icon.


title: Richard Linklater

text: Richard Stuart Linklater (; born July 30, 1960) is an American film director, producer, and screenwriter. He is known for films that revolve mainly around suburban culture and the effects of the passage of time. His films include the comedies "Slacker" (1990) and "Dazed and Confused" (1993); the "Before" trilogy of romance films, "Before Sunrise" (1995), "Before Sunset" (2004), and "Before Midnight" (2013); the music-themed comedy "School of Rock" (2003); the adult animated films "Waking Life" (2001), "A Scanner Darkly" (2006), and "" (2022); the coming-of-age drama "Boyhood" (2014); and the comedy film "Everybody Wants Some!!" (2016).


title: Richard Linklater

text: Linklater is known to have a distinct style and method of filmmaking. Many of his films are noted for their loosely structured narrative. The "Before" trilogy and "Boyhood" both feature the same actors filmed over an extended period of years. He has received several Academy Award nominations and won the Silver Bear for Best Director at the Berlin International Film Festival for his work on "Before Sunrise". He also won a Golden Globe Award for directing "Boyhood". In 2015, Linklater was included on the annual "Time" 100 list of the most influential people in the world.


title: Richard Linklater

text: Linklater was born in Houston, Texas, the son of Diane Margaret (née Krieger), who taught at Sam Houston State University, and Charles W. Linklater, III. He attended Huntsville High School in Huntsville, Texas, during grades 9–11, where he played football for Joe Clements as a backup quarterback for the #1 ranked team in the state. For his senior year, he moved to Bellaire High School in Bellaire, Texas because he was better at baseball than football and Bellaire had a better baseball coach. As a teen, Linklater won a Scholastic Art and Writing Award.


title: Richard Linklater

text: Linklater studied at Sam Houston State University (where he also played baseball), until dropping out to work on an offshore oil rig in the Gulf of Mexico. He frequently read novels on the rig, and upon returning to land, developed a love of film through repeated visits to a repertory cinema in Houston. At this point, Linklater realized he wanted to be a filmmaker. He used his savings to buy a Super-8 camera, a projector, and editing equipment, and moved to Austin, Texas.


title: Richard Linklater

text: Linklater founded the Austin Film Society in 1985 with his college professor Chale Nafus, University of Texas professor Charles Ramirez-Berg, SXSW founder Louis Black, and his frequent collaborator Lee Daniel. One of the mentors for the Film Society was former New York City critic for the "SoHo Weekly News" George Morris, who had relocated to Austin and taught film there.


title: Richard Linklater

text: For several years, Linklater made many short films that were exercises and experiments in film techniques. He finally completed his first feature, "It's Impossible to Learn to Plow by Reading Books" (which is available in The Criterion Collection edition of his second feature, "Slacker"), a Super-8 feature that took a year to shoot and another year to edit. The film is significant in the sense that it establishes most of Linklater's preoccupations. The film has his trademark style of minimal camera movements and lack of narrative, while it examines the theme of traveling with no particular direction in mind. These idiosyncrasies would be explored in greater detail in future projects.


title: Richard Linklater

text: Linklater created Detour Filmproduction (an homage to the 1945 low budget film noir by Edgar G. Ulmer), and subsequently made "Slacker" for only $23,000. It went on to gross more than $1.25 million. The film shows an aimless day in the life of the city of Austin, Texas showcasing its more eccentric characters.