title: John F. Kennedy Jr. plane crash

text: On the afternoon of July 21, divers recovered the bodies of Kennedy and the Bessette sisters. Divers found the Bessette sisters near the fuselage, while Kennedy himself was still strapped in his seat. Coast Guard Admiral Richard M. Larrabee said that all three bodies were "near and under" the fuselage, still strapped in. The bodies were taken to the county medical examiner's office by motorcade. Autopsies on the evening of July 21 performed by the county medical examiner found that all three had died upon impact. At the same time, the Kennedy and Bessette families announced their plans for memorial services. After the autopsies were completed, the three bodies were taken from Hyannis to Duxbury, Massachusetts, where they were cremated in the Mayflower Cemetery crematorium.


title: John F. Kennedy Jr. plane crash

text: The National Transportation Safety Board (NTSB) officially declared that Kennedy's plane had crashed into the Atlantic Ocean off the coast of Martha's Vineyard. The probable cause of the crash was "the pilot's failure to maintain control of the airplane during a descent over water at night, which was a result of spatial disorientation". Under the heading "Spatial Disorientation", the report listed, from the FAA Instrument Flying Handbook Advisory Circular 61-27C, six examples:


title: John F. Kennedy Jr. plane crash

text: Kennedy was not qualified to fly his plane by "instruments only". The crash occurred in conditions not legally requiring such qualification. Other pilots flying similar routes on the night of the crash reported no visual horizon due to haze.


title: John F. Kennedy Jr. plane crash

text: Atmospheric conditions along Kennedy's flight path on the night of the crash were occasionally hazy, which can lead to spatial disorientation for pilots. The weather was officially listed as "visual meteorological conditions" (VMC), which allowed Kennedy to fly under Visual Flight Rules (VFR) rather than Instrument Flight Rules (IFR), especially since he lacked an instrument rating. The visibility was very poor in Essex County, New Jersey and airports along Kennedy's flight path reported visibility between five and eight miles with haze and a few clouds. Some pilots flying similar routes as Kennedy on the night of the accident reported no visual horizon over water because of haze. Also, the NTSB reported on one pilot who cancelled a similar flight that evening due to "poor" weather. The conditions near the crash site were: "Clear skies at or below 12,000 feet; visibility 10 miles".


title: John F. Kennedy Jr. plane crash

text: Kennedy first obtained his private pilot license in 1998 and received "high performance airplane" and "complex airplane" endorsements two months before the crash. His estimated total flight experience before the crash was about 310 hours, of which 55 hours were at night. His estimated flight time in the accident airplane was about 36 hours, of which about nine hours were at night. About three hours of that flight time were without a Certified Flight Instructor (CFI) on board, and only 48 minutes of that time was flown at night, which included a night landing.


title: John F. Kennedy Jr. plane crash

text: It is not clear how much of Kennedy's total flight experience was in the plane type that crashed or in his other more basic plane, the Cessna Skylane 182. Fifteen months before the crash, Kennedy had flown about 35 flights either to or from northern New Jersey and the Martha's Vineyard area. Kennedy flew more than 17 of these legs without a CFI on board, including at least five at night. His last known flight in his airplane without a CFI on board happened two months before the crash.


title: John F. Kennedy Jr. plane crash

text: The CFI who prepared Kennedy for his private pilot checkride stated that he had "very good" flying skills for his level of experience. Four months before the crash, Kennedy passed the FAA's written airplane instrument examination and later enrolled in an instrument rating course. He continued to receive flight instruction in New Jersey in his plane, including flights from CDW to MVY. His instructors said Kennedy required help working the rudder pedals to taxi and land the plane because of his ankle injury.


title: John F. Kennedy Jr. plane crash

text: During a training flight at night under instrument conditions, his instructor stated that Kennedy had the ability to fly the airplane without a visible horizon but may have experienced difficulty performing additional tasks under such conditions. He also stated that the pilot was not ready for an instrument evaluation and needed additional training. The instructor at the time of the crash was not aware that Kennedy would be flying in those conditions without an instructor on board. The CFI further stated that he had talked to Kennedy on the night of the accident and offered to fly with him that night. He stated that Kennedy had the capability to conduct a night flight to Martha's Vineyard as long as a visible horizon existed.


title: John F. Kennedy Jr. plane crash

text: The NTSB suggested that Kennedy's rocky marriage may have contributed to a source of stress by the time of the crash. Additionally, Kennedy's magazine, "George", was in serious financial trouble. According to the Aeronautical Information Manual (AIM): "stress from everyday living can impair pilot performance, often in subtle ways. Distractions can so interfere with judgment that unwarranted risks are taken, such as flying into deteriorating weather conditions to keep on schedule."


title: John F. Kennedy Jr. plane crash

text: From 8:49 p.m., about ten minutes after departure, and for a five-minute period lasting until 8:54 p.m., Kennedy's plane flew in the same vicinity as American Airlines Flight 1484, a Fokker 100, which was on approach to Westchester County Airport (HPN). The Traffic collision avoidance system (TCAS) sounded on the Fokker 100, leading to some discussion between the Fokker and the New York approach controller (ATC). However, the interaction ended with American Airlines Flight 1484 reporting to ATC that they believed the other plane was a Piper and that the TCAS warning was resolved. No corrective action was reported to have been taken by ATC nor Flight 1484. It is not known if Kennedy was aware of, or distracted by, the incident.


title: John F. Kennedy Jr. plane crash

text: The flight was originally scheduled for daylight hours but had to be postponed after Kennedy's sister-in-law was delayed at work. Heavy traffic further delayed Kennedy's flight and pushed it back until after dark. Originally planned to depart at 6:00 p.m., Kennedy's plane departed at 8:39 p.m. instead, nearly a half-hour past sunset. At the time of the takeoff, the moon was just above the horizon and provided very little illumination.


title: John F. Kennedy Jr. plane crash

text: Kennedy never received a weather briefing or filed a flight plan with any Flight Service Station. Except for the take-off portion of his flight, Kennedy did not contact any air traffic controllers; during the flight, he never requested help or declared an emergency. Under the conditions of his flight, Kennedy was not required to file a flight plan, and because he did not, no one knew his exact route or expected time of his arrival. According to the Weather Service International, as reported in the investigation by the NTSB, Kennedy made two weather requests before taking off. The information he was provided indicated that visibility ranged from along his route to at Martha's Vineyard.


title: John F. Kennedy Jr. plane crash

text: Shortly after passing Point Judith, Rhode Island, Kennedy's plane headed directly towards Martha's Vineyard. Instead of following the coastline of Rhode Island Sound and Buzzards Bay, which would have provided visible lights on the ground, Kennedy chose the shorter, direct path over a 30-mile (50 km) open stretch of water. According to the FAA Airplane Flying Handbook, crossing large bodies of water at night may be very hazardous, not only from the standpoint of ditching in the water but also because the featureless horizon visually blends with the water, in which case depth perception and orientation become difficult.


title: John F. Kennedy Jr. plane crash

text: Six weeks before the crash, Kennedy had fractured his left ankle in a paragliding accident over the Memorial Day weekend. He had surgery and wore a cast which was removed just the day before the fatal flight. He still walked with a limp on the day of the accident. During his interviews, Kennedy's orthopaedic surgeon stated that, at the time of the crash, he would have been able to apply the type of pressure that would normally be required to drive a car.


title: John F. Kennedy Jr. plane crash

text: While the NTSB examined the wreckage, they soon discovered that both of Kennedy's radios had incorrect frequencies selected. Kennedy had accidentally selected 127.25 for Martha Vineyard's ATIS instead of 126.25; likewise, he selected 135.25 for Essex County ATIS but it should have been 135.5. The NTSB declined to comment on the contribution this factor had in the crash, if any.


title: John F. Kennedy Jr. plane crash

text: On the morning of July 22, relatives brought the cremated remains of Kennedy onto the . His ashes were later scattered into the Atlantic Ocean off the coast of Martha's Vineyard. The ship was used for the public memorial service with the permission of U.S. Secretary of Defense William Cohen. "Briscoe" spent about half an hour off the Vineyard's southwest coast, and was approximately 2.5 miles from the crash site.


title: John F. Kennedy Jr. plane crash

text: President Clinton ordered that the flags at the White House be lowered to half-staff to honor Kennedy. During a public memorial service for Kennedy, his paternal uncle, Senator Ted Kennedy, stated:


title: John F. Kennedy Jr. plane crash

text: Ted Kennedy also used the term that had been said about his brother's presidency, saying that "for a thousand days, he was a husband who adored the wife who became his perfect soul mate."


title: John F. Kennedy Jr. plane crash

text: Kennedy's last will and testament, signed 18 months before his death, stipulated that all of his personal belongings, property, and holdings were to be "evenly distributed" among his two nieces, Rose and Tatiana, and nephew, John, who were among 14 beneficiaries in the will.


title: Endless Love (1981 film)

text: Endless Love is a 1981 American romantic drama film directed by Franco Zeffirelli, starring Brooke Shields and Martin Hewitt. The film marks Tom Cruise's feature film appearance debut.


title: Endless Love (1981 film)

text: Based on the 1979 Scott Spencer novel of the same name, the screenplay was written by Judith Rascoe. The original music score was composed by Jonathan Tunick. Although the novel is set in the summer of 1969, the movie transports the action to the early 1980s.


title: Endless Love (1981 film)

text: Critics compared the film unfavorably to the novel, which showcased the dangers of obsessive love. Despite the poor reviews, its eponymous theme song, performed by Diana Ross and Lionel Richie, became a #1 hit on the "Billboard" Hot 100. The song spent nine weeks at #1 and received Academy Award and Golden Globe Award nominations for "Best Original Song", along with five Grammy Award nominations.


title: Endless Love (1981 film)

text: In suburban Chicago, teenagers Jade Butterfield and David Axelrod fall in love after being introduced by Jade's brother Keith.


title: Endless Love (1981 film)

text: The Butterfields' bohemian lifestyle, for which they're well known in their community, allows Jade and David to develop an all-consuming and passionate relationship, including allowing them to have sex in Jade's room. Where her family is open, David's home life is dull; his parents are wealthy liberal political activists who have little interest in his life.


title: Endless Love (1981 film)

text: One night, Jade's mother Ann sneaks downstairs, catching Jade and David making love by the fireplace. She starts living vicariously through them but her husband, Hugh, watches them with increasing unease. Jade's nightly trysts begin to negatively impact her grades and her ability to sleep. One night, she tries to steal a prescription sleeping pill from her father but he catches her. As a last straw, Hugh demands that David stop seeing Jade until the end of the school term. Although initially causing a scene, Ann gently coaxes David to agree, telling him not to let Hugh "do something he'll regret".


title: Endless Love (1981 film)

text: Back at school, one of David's friends, Billy, tells him that when he was eight, he tried to burn a pile of newspapers, got scared and put the fire out, and his parents thought he was a hero for saving the house from burning. Inspired by this story, David starts a fire on the Butterfields' front porch and walks away. But by the time he returns, the flame has spread too far. David evacuates the Butterfields from the burning house before he is subsequently apprehended.


title: Endless Love (1981 film)

text: Following trial, David is convicted of second-degree arson, sentenced to five years' probation, sent to a mental hospital for evaluation and forbidden to go anywhere near Jade or her family again. He continues to write her daily, but the letters aren't sent because of the no-contact order. His parents pull strings to have him released early, much to Hugh's chagrin. Meanwhile, David receives his many letters upon his exit, and after realizing why Jade never wrote back, he decides to pursue her although he knows it is a violation of his parole.


title: Endless Love (1981 film)

text: After the loss of their home, the Butterfields moved from Chicago to Manhattan where Ann and Hugh file for divorce. In Manhattan, Ann tries to seduce David but he refuses, leaving her confused. When she isn't looking, he thumbs through her address book, finding out where Jade is and discovers that she now attends the University of Vermont.


title: Endless Love (1981 film)

text: Intent on catching a bus to Vermont, David sees Hugh on the street. He starts chasing him, and Hugh is hit by a car and killed. Hugh's fiancée Ingrid arrives to the scene just in time to see David leaving. He comes close to boarding the bus to Vermont, but is overcome with grief and returns to Ann's apartment and consoles her.


title: Endless Love (1981 film)

text: Later, Jade goes to David's room to say goodbye but he pulls her back as she tries to leave, throwing her on the bed and holding her down until she admits she loves him. Keith goes to David's to find them together again and tells Jade that David is at fault for their father's death. She refuses to believe it at first but when he confirms that David was actually at the scene, she becomes horrified and hides behind Keith. Trying to explain, David shoves Keith out of the way in a desperate bid to get to her. Keith holds him off until the police arrive and arrest David for brawling, disturbing the peace and violating his parole.


title: Endless Love (1981 film)

text: David is sentenced to prison and despairs that he'll never see Jade again. At a lakeside, Jade tells Ann that nobody will ever love her like David does. From behind bars, David sees Jade approach him through his barred cell window.


title: Endless Love (1981 film)

text: "Endless Love" is based on Scott Spencer's 1979 novel of the same name. The film is directed by Franco Zeffirelli and written by Judith Rascoe. The film stars Brooke Shields and Martin Hewitt in the two leading roles. It is also the film debut of Hewitt, Tom Cruise, Jami Gertz, Jeff Marcus and Ian Ziering. Principal photography of this film began on 22nd September 1980 and shot on location in Chicago, New York City and Long Island as well on set at Astoria Studios in Queens, NY. Production of this film was finished on 19th December 1980.


title: Endless Love (1981 film)

text: The MPAA awarded the initial cut of "Endless Love" an X rating. Director Franco Zeffirelli subsequently made several cuts in the love scenes between Brooke Shields and Martin Hewitt to achieve a lower rating. The film was re-submitted to the MPAA five times before they awarded it an R rating.


title: Endless Love (1981 film)

text: The film's theme song, written by Lionel Richie and performed by Richie and Diana Ross and also called "Endless Love", became a number 1 hit on the "Billboard" Hot 100, and was the biggest-selling single in Ross' career. Billboard magazine chose it as "The Best Duet of All Time" in 2011, 30 years after its debut. It spent nine weeks at #1 and received Academy Award and Golden Globe Award nominations for "Best Original Song", along with five Grammy Award nominations. The soundtrack peaked at #9 on the Billboard Top 200 and was certified platinum. It also featured a second duet between Ross and Richie, "Dreaming of You", that received considerable airplay but was never released as a single.


title: Endless Love (1981 film)

text: The film received mostly poor reviews on release. Roger Ebert compared the film unfavorably with the novel, describing Martin Hewitt as miscast and criticizing the narrative, although he did praise Brooke Shields's performance:Is there anything good in the movie? Yes. Brooke Shields is good. She is a great natural beauty, and she demonstrates, in a scene of tenderness and concern for Hewitt and in a scene of rage with her father, that she has a strong, unaffected screen acting manner. But the movie as a whole does not understand the particular strengths of the novel that inspired it, does not convince us it understands adolescent love, does not seem to know its characters very well, and is a narrative and logical mess.


title: Endless Love (1981 film)

text: Janet Maslin in "The New York Times" wrote:There are two sorts of people who'll be going to see "Endless Love" — those who have read the richly imaginative novel on which the movie is based and those who have not. There will be dismay in the first camp, but it may be nothing beside the bewilderment in the second."


title: Endless Love (1981 film)

text: Film historian Leonard Maltin seemed to agree, calling the film a "textbook example of how to do everything wrong in a literary adaptation."


title: Endless Love (1981 film)

text: In 2014, Scott Spencer, the author of the novel on which the film was based, wrote, "I was frankly surprised that something so tepid and conventional could have been fashioned from my slightly unhinged novel about the glorious destructive violence of erotic obsession". Spencer described the film as a "botched" job and wrote that Franco Zeffirelli "egregiously and ridiculously misunderstood" the novel.


title: Endless Love (1981 film)

text: The film premiere for "Endless Love" took place on July 16, 1981, at the Ziegfeld Theatre in New York City. The film was released the next day. Despite the poor critical reception, the film was a box office success. It made $4,163,623 on its opening weekend and went on to gross $31,184,024 in total, becoming the twenty-second highest earning film domestically in 1981. Internationally, the film took in a further $1,308,650 bringing its total worldwide gross to $32,492,674.


title: Endless Love (1981 film)

text: The film was released on Blu-ray on Shout! Factory in mid-August 2019, with enhanced 5.1 surround sound. It is also available in the streaming format via various providers.


title: Type 055 destroyer

text: The Type 055 destroyer (NATO/OSD Renhai-class cruiser) is a class of stealth guided-missile destroyers (guided-missile cruisers per NATO/OSD standard) being constructed for the Chinese People's Liberation Army Navy. It is a multi-mission design; the combination of sensors and weapons suggests a main role of area air defence, with anti-submarine warfare capabilities surpassing previous Chinese surface combatants.


title: Type 055 destroyer

text: The Type 055 is expected to undertake expeditionary missions and form the primary escort for Chinese aircraft carriers.


title: Type 055 destroyer

text: The United States classifies these ships as cruisers. The United States Navy defines a cruiser as a large multi-mission surface combatant with flag facilities; this suggests the U.S. expects the Type 055 to fulfill a similar role as the .


title: Type 055 destroyer

text: The People's Liberation Army Navy was interested in a large destroyer from as early as the late-1960s. A development program, code-named "055", initiated in 1976 was cancelled in 1983 after encountering insurmountable technical obstacles from industrial underdevelopment; for example, the required gas turbine power plants could neither be produced domestically nor imported at acceptable prices.


title: Type 055 destroyer

text: In April 2014, an image emerged of a full-scale mock-up of the Type 055 superstructure—with enclosed integrated mast for radar and other electronics—at the Chinese naval electronic testing range in Wuhan.


title: Type 055 destroyer

text: "Nanchang", the first ship of the class, began construction in 2014 at the Jiangnan Shipyard in Shanghai, and was commissioned on 12 January 2020. Its first public appearance—preceding commissioning—was during the PLAN's 70th-anniversary parade on 23 April 2019. When launched, "Nanchang" was among the largest post-Second World War warships launched in East Asia.


title: Type 055 destroyer

text: The Type 055 adopts a conventional flared hull with distinctive stealthy features including an enclosed forecastle that hides mooring points, anchor chains, and other equipment. The bow and main deckhouse are configured similarly to previous Type 052C/D destroyers. A continuous structure amidship increases internal volume and reduces radar cross-section. The smokestack design reduces both infrared signature and radar cross-section. Chinese sources credit the design as being generally stealthy, with reduced radar, noise, infrared, and electromagnetic radiation signatures.


title: Type 055 destroyer

text: Propulsive power is generated by four 28 MW QC-280 gas turbines in combined gas and gas arrangement. Additional power may be provided by six 5 MW QD-50 gas turbines.


title: Type 055 destroyer

text: Chinese literature suggests that the Type 055 is capable of "facilitat[ing] command management of a battlegroup and supporting elements". The command-and-control and battle management systems are likely comparable to contemporary PLAN systems, which reflect over a decade of intensive interest in information integration from the late 2000s.


title: Type 055 destroyer

text: The class has a dual-band radar system; four S-band Type 346B Dragon Eye active electronically scanned array (AESA) panels are mounted on the superstructure, and four smaller X-band panels are mounted on the mast. The Type 346B panels are estimated to be 40% larger than the Type 052D destroyer's Type 346A panels, for greater transmission power and sensitivity. Chinese sources claim the radar has anti-stealth capability and can be used to guide anti-satellite missiles.


title: Type 055 destroyer

text: A deployment port exists for variable depth and towed array sonar. The large bulbous bow likely contains a bow sonar; the Type 055 may mount a larger bow sonar than previous Chinese surface combatants.


title: Type 055 destroyer

text: Various electronic warfare support measures (ESM), electronic countermeasures (ECM), and electro-optic (EO) sensors and datalinks are mounted. They are likely more advanced than those deployed on previous ships.


title: Type 055 destroyer

text: The primary armament are missiles carried in 112-cell vertical launching system (VLS); 64 cells forward and 48 cells aft. The same VLS model is used on the Type 052D destroyer, which is believed to be an implementation of the GJB 5860-2006 standard; the GJB 5860-2006 is capable of hot and cold launches using concentric canisters. The longest variant, with 9-metre cells, is likely used. The Type 055 is expected to carry HHQ-9 surface-to-air missiles, YJ-18 anti-ship cruise missiles, CJ-10 land-attack cruise missiles, and missile-launched anti-submarine torpedoes upon entering service. Potentially, the larger cells may also carry anti-ship ballistic missiles.


title: Type 055 destroyer

text: It has been suggested that future variants may be armed with lasers or electromagnetic railguns. Since the current design does not have integrated electric propulsion, installation of integrated electric propulsion will be required for the ship to meet power requirements in the future.


title: Lisa Loring

text: Lisa Loring (born Lisa Ann DeCinces; February 16, 1958) is an American actress. She is best known for having played Wednesday Addams at 6 years old on the 1964–1966 sitcom "The Addams Family".


title: Lisa Loring

text: Her parents both served in the United States Navy, but divorced shortly after her birth. She was born in Kwajalein, Marshall Islands, a United Nations Trust Territory at the time, which had been administered by the United States. She grew up in Hawaii and later moved to Los Angeles with her mother. She began modeling at age three and appeared in an episode of "Dr. Kildare", which aired in 1964. Her mother died of alcoholism in 1974 at age 34.


title: Lisa Loring

text: She joined the cast of ABC sitcom "The Pruitts of Southampton" in 1966. After her childhood career, she also was a cast member of the CBS soap opera "As the World Turns" from 1980 to 1983, where she created the role of Cricket Montgomery.


title: Lisa Loring

text: Loring married her childhood sweetheart Farrell Foumberg in 1973 when she was 15, and had a daughter. The marriage ended a year later. Loring married her second husband, actor Doug Stevenson, in 1981. He was a contract performer on another CBS/Procter & Gamble soap opera "Search for Tomorrow." Loring had a second daughter with Stevenson, and their marriage ended in 1983.


title: Lisa Loring

text: In 1987, Loring married adult film actor Jerry Butler. They met on the set of the 1987 adult film "Traci's Big Trick", on which Loring worked as a make-up artist and uncredited writer. In the ensuing years of their marriage, she voiced her dissatisfaction with Butler's continued involvement in pornography, and eventually Butler began secretly participating in porn shoots without her knowledge. In an interview with "Dateline NBC" in the 1990s, Butler described himself as "addicted to the lifestyle", ashamed of his clandestine behavior and its effect on his marriage. The couple later appeared on the "Sally Jessy Raphael Show", again discussing the damage Butler's porn career was causing to their marriage. Butler and Loring divorced in 1992, which was also the year he began to exit the hardcore industry, and evidently by choice, virtually disappearing from the public eye for many years. Loring married for a fourth time in 2003 to Graham Rich. The couple separated in 2008, and as of June 2014, were in the process of a divorce.


title: Philip Baker Hall

text: Hall is known for his collaborations with Paul Thomas Anderson, including "Hard Eight" (1996), "Boogie Nights" (1997) and "Magnolia" (1999). He also starred in leading roles in films, such as "Secret Honor" (1984) and "Duck" (2005). Hall had supporting roles in many films, including "Say Anything..." (1989), "The Truman Show" (1998), "The Talented Mr. Ripley" (1999), "The Insider" (1999), "Lost Souls" (2000), "The Contender" (2000), "Bruce Almighty" (2003), "Dogville" (2003), "Zodiac" (2007), "50/50" (2011) and "Argo" (2012). He received an Independent Spirit Award nomination for Best Male Lead for his role in "Hard Eight" and two Screen Actors Guild Award nominations for Outstanding Performance by a Ensemble Cast in a Motion Picture for "Boogie Nights" and "Magnolia".


title: Philip Baker Hall

text: Hall is also known for his prolific work on television. His early television work included "M*A*S*H", "Murder, She Wrote", and "Cheers". One of his most memorable television roles was as Lt. Joe Bookman in "Seinfeld". He had recurring roles in "The Practice", "The West Wing", "Curb Your Enthusiasm", "Modern Family", and "BoJack Horseman".


title: Philip Baker Hall

text: Hall was born in Toledo, Ohio. His mother was Alice Birdene (née McDonald), and his father, William Alexander Hall, was a factory worker from Montgomery, Alabama. He attended the University of Toledo. He served in Germany as a United States Army translator and as a high school teacher.


title: Philip Baker Hall

text: After his film debut "Cowards", he joined the Los Angeles Theatre Center. His first television role came for an episode of "Good Times". Hall guest starred in episodes of "M*A*S*H" and "Man from Atlantis". He had over 200 guest roles since 1977. He played Richard Nixon in the one-character film "Secret Honor" and reprised his role he had created during the play's original Off-Broadway run. Roger Ebert said about Hall and the film: "Nixon is portrayed by Philip Baker Hall, an actor previously unknown to me, with such savage intensity, such passion, such venom, such scandal, that we cannot turn away. Hall looks a little like the real Nixon; he could be a cousin, and he sounds a little like him. That's close enough. This is not an impersonation, it's a performance." Vincent Canby of "The New York Times" also said about Hall: "Mr. Hall's immense performance, which is as astonishing and risky – for the chances the actor takes and survives – as that of the Oscar-winning F. Murray Abraham in "Amadeus.""


title: Philip Baker Hall

text: In the 1980s, Hall co-starred in various films in supporting roles, including "Nothing in Common" (1986), "Midnight Run" (1988), "Say Anything..." and "Ghostbusters II" (both 1989). He played "Lt. Joe Bookman", a detective pursuing a long-overdue library book in the "Seinfeld" episodes, "The Library" and "The Finale". His first "Seinfeld" appearance led him to be widely lauded as one of the best guest stars on the series, and led to many other jobs.


title: Philip Baker Hall

text: Hall starred in Paul Thomas Anderson's short film "Cigarettes & Coffee", which was adapted into Anderson's directorial debut film "Hard Eight" (1996). For the film, Hall played a senior gambler who mentors a homeless man (John C. Reilly). Ebert of the "Chicago Sun-Times" said about Hall, "Here is another great performance. He is a man who has been around, who knows casinos and gambling, who finds himself attached to three people he could easily have avoided, who thinks before he acts." Hall was nominated for the Independent Spirit Award for Best Male Lead. He later starred in Anderson's other films, "Boogie Nights" (1997) and "Magnolia" (1999). He was nominated for two Screen Actors Guild Awards for Outstanding Performance by a Cast in a Motion Picture. Hall starred with Philip Seymour Hoffman in four films.


title: Philip Baker Hall

text: Hall had turns in a variety of films in the 1990s, including "The Rock", "The Truman Show", "The Talented Mr. Ripley", and "The Insider". He co-starred in other films in the 2000s, including "Dogville", "Zodiac", and "Argo". He played Captain Diel in the "Rush Hour" trilogy (though his scenes were cut from "Rush Hour 2" and he was uncredited for the scene in "Rush Hour 3"). Hall had prominent roles in "Bruce Almighty", "In Good Company", "The Amityville Horror", "The Matador", "You Kill Me", "All Good Things", "50/50", and "The Sum of All Fears".


title: Philip Baker Hall

text: Hall starred in the sitcom "The Loop". He guest starred in the animated series "The Life & Times of Tim". He played a physician in "Curb Your Enthusiasm", and appeared in "Modern Family". He appeared in an episode of "The Newsroom" and in a Holiday Inn commercial. For the short film "Dear Chickens", he won best actor at "Los Angeles Short Festival" and at "Filmets Badalona Film Festival" in Barcelona.


title: Philip Baker Hall

text: Hall had two daughters, Patricia and Darcy, with his first wife, Mary-Ella Holst. He later married Holly Wolfle, with whom he had two daughters, Adella and Anna. He also had four grandchildren and a brother.


title: Naked and Afraid

text: Naked and Afraid is an American reality series that airs on the Discovery Channel. Each episode chronicles the lives of two survivalists who meet for the first time naked and are given the task of surviving a stay in the wilderness for 21 days. Each survivalist is allowed to bring one helpful item, such as a machete or a fire starter. After they meet in the assigned locale, the partners must build a shelter and find water and food.


title: Naked and Afraid

text: The events of each couple's quest play out in a single episode. Partners strip down and meet each other. They are provided with rough cross-body satchels containing a personal diary/camera for use when the camera crew is not there at night, and a map. They all wear identical necklaces with a center bead which is a microphone with a wire that connects to a wireless audio transmitter hidden in the satchel. Some personal jewelry is allowed. Each survivalist is allowed to bring one helpful item, such as a hatchet or a fire starter. There is a camera crew who are not allowed to intervene except for medical emergencies when it is "absolutely necessary". Castmate Alison Teal reports that she was granted an "emergency" tampon. At the beginning of the fourth season, the survivalists were given a third "surprise" item. A team is allowed to "tap out" at any point during the 21-day challenge. If they do not tap out, they must arrive at the designated extraction point on the final day to be picked up by a helicopter, plane, automobile or boat or other vehicle suitable to the environment. Viewers are updated throughout with statistics including how many days have passed, the time, and the temperature. "Naked and Afraid" computes and then updates the cast members' PSR (Primitive Survival Rating), which is based on predictions and observations of survival fitness in skill, experience, and mental strengths. Before and after weight measurements are also revealed at the end of an episode.


title: Naked and Afraid

text: The production's home base is in the United States (the Louisiana Bayou, Florida (Everglades and Seminole County) and Alabama). Panama was used as a setting twice in the first and fifth seasons, Nicaragua was also used twice as well in the third season. Like the first season, the second season visited six countries as well, using Malaysia as a setting once again (Season 1 shot on the Malaysian Borneo, while Season 2's survivalists were stranded in Peninsular Malaysia). Season 10 took contestants to the frozen tundra of Alaska and an African Kill Zone.


title: Naked and Afraid

text: On July 31, 2013, Discovery Channel posted a casting call and a dare to "survive the 21-day challenge" via their Twitter account.


title: Naked and Afraid

text: Following the success of the live talk show "Shark After Dark", Discovery aired "Naked After Dark" in season two, which aired after each new episode. Host Josh Wolf talked with the cast from the episode, showed behind-the-scenes and never before seen episode footage and has in-studio stunts. It aired for 1 season with 6 episodes.


title: Naked and Afraid

text: Starting in June 2014, Discovery started airing repeat episodes of the show with "Pop-Up" text on the screen. These graphics shed light on the countries and the survivalists with interesting and often humorous factoids. These shows premiere on Wednesday nights.


title: Naked and Afraid

text: By the time of the 2015 season premiere, Brian Moylan of "The Guardian" called it the "best reality show on television" as it "takes the outrageousness of other reality genres, but applies it to celebrate human strength and fortitude rather than exploiting the frailty and narcissism of those that just want to be noticed."


title: Naked and Afraid

text: Reviewing the premiere in 2013, Brian Lowry of "Variety" found "Naked and Afraid" to be "typical of the genre, but it's still kind of a risible kick, if only for how seriously the show takes itself." Melissa Camacho of Common Sense Media, and David Hinckley of the New York "Daily News", gave the show 3 out of 5 stars.


title: Naked and Afraid

text: Mike Hale of "The New York Times" expressed skepticism that the dangers implied by the situations faced by the couples in each episode were authentic, even by the standards of reality television, observing that the couple in the series premiere appeared to be cleaner and more shaven than he would have expected. Hale also was not impressed with the interaction between the two participants, whose personalities he found uninteresting, stating that he found the snake bite suffered by the field producer shown in the beginning of the episode to be more riveting.


title: Naked and Afraid

text: Discovery Channel issued a press release which announced some Nielsen ratings milestones for the series. "Naked And Afraid" was the "#1 Ad-Supported program in cable among men on Sunday, July 28." They also noted that the program now shares the Discovery Channel record for the highest-rated survival telecast in the network's history since June 2009. The "Surthrive" episode in season 4, featuring Bo Stuart and Debbie Harris is the highest rated episode in the history of the show.


title: Naked and Afraid

text: There has been controversy related to how real the show is. Former contestants have said that the locations are not as isolated as portrayed. A contestant was treated with IV fluids so she could continue, but this was not shown. Les Stroud, of the survival show "Survivorman", has been very vocal with his criticism.


title: Philip IV of Spain

text: Philip IV (, ; 8 April 160517 September 1665), also called the Planet King (Spanish: "Rey Planeta"), was King of Spain from 1621 to his death and (as Philip III) King of Portugal from 1621 to 1640. Philip is remembered for his patronage of the arts, including such artists as Diego Velázquez, and his rule over Spain during the Thirty Years' War.


title: Philip IV of Spain

text: By the time of his death, the Spanish Empire had reached approximately 12.2 million square kilometers (4.7 million square miles) in area but in other aspects was in decline, a process to which Philip contributed with his inability to achieve successful domestic and military reform.


title: Philip IV of Spain

text: Philip IV was born in the Royal Palace of Valladolid, and was the eldest son of Philip III and his wife, Margaret of Austria. In 1615, at the age of 10, Philip was married to 13-year-old Elisabeth of France. Although the relationship does not appear to have been close, some have suggested that Olivares, his key minister, later deliberately tried to keep the two apart to maintain his influence, encouraging Philip to take mistresses instead, by whom he is known to have had at least 30 progeny. Philip had ten children by Elisabeth, with only one being a son, Balthasar Charles, who died at the age of sixteen in 1646. The death of his son deeply shocked the king, who appears to have been a good father by the standards of the day. Elisabeth was able to conspire with other Spanish nobles to remove Olivares from the court in 1643, and for a brief period she held considerable influence over Philip; by the time of her death, however, she was out of favour, following manoeuvering by Olivares' successor and nephew, Luis de Haro.


title: Philip IV of Spain

text: At age of 44 in 1649, Philip remarried, following the deaths of both Elisabeth and his only son. His choice of his second wife, 14-year-old Maria Anna, also known as Mariana, Philip's niece and the daughter of Emperor Ferdinand III, was guided by politics and Philip's desire to strengthen the relationship with Habsburg Austria. They were married on 7 October 1649. Maria Anna bore him five children, but only two survived to adulthood, a daughter Margarita Teresa, born in 1651, and the future Charles II of Spain in 1661 – but the latter was sickly and considered in frequent danger of dying, making the line of inheritance potentially uncertain.


title: Philip IV of Spain

text: Perceptions of Philip's personality have altered considerably over time. Victorian authors were inclined to portray him as a weak individual, delegating excessively to his ministers, and ruling over a debauched Baroque court. Victorian historians even attributed the early death of Balthasar to debauchery, encouraged by the gentlemen entrusted by the king with his education. The doctors who treated the Prince at that time in fact diagnosed smallpox, although modern scholars attribute his death to appendicitis. Historians' estimation of Philip gradually improved in the 20th century, with comparisons between Philip and his father being increasingly positive – some noting that he possessed much more energy, both mental and physical, than his diffident father.


title: Philip IV of Spain

text: Philip was idealized by his contemporaries as the model of Baroque kingship. Outwardly he maintained a bearing of rigid solemnity; foreign visitors described him as being so impassive in public he resembled a statue, and he was said to have been seen to laugh only three times in the course of his entire public life. Philip certainly had a strong sense of his 'royal dignity', but was also extensively coached by Olivares in how to resemble the Baroque model of a sovereign, which would form a key political tool for Philip throughout his reign. Philip was a fine horseman, a keen hunter and a devotee of bull-fighting, all central parts of royal public life at court during the period.


title: Philip IV of Spain

text: Privately, Philip appears to have had a lighter persona. When he was younger, he was said to have a keen sense of humor and a 'great sense of fun'. He privately attended 'academies' in Madrid throughout his reign – these were lighthearted literary salons, aiming to analyze contemporary literature and poetry with a humorous touch. A keen theatre-goer, he was sometimes criticized by contemporaries for his love of these 'frivolous' entertainments. Others have captured his private personality as 'naturally kind, gentle and affable'. Those close to him claimed he was academically competent, with a good grasp of Latin and geography, and could speak French, Portuguese and Italian well. Like many of his contemporaries, including Olivares, he had a keen interest in astrology. His handwritten translation of Francesco Guicciardini's texts on political history still exist.


title: Philip IV of Spain

text: Although interpretations of Philip's role in government have improved in recent years, Diego Velázquez's contemporary description of Philip's key weakness – that 'he mistrusts himself, and defers to others too much' — remains relevant. Although Philip's Catholic beliefs no longer attract criticism from English language writers, Philip is still felt to have been 'unduly pious' in his personal life. Notably, from the 1640s onwards he sought the advice and counsel of a noted cloistered abbess, Sor María de Ágreda, exchanging many letters with her. This did not stop Philip for becoming known for his numerous affairs, particularly with actresses, as encouraged by the Count-Duke Olivares. The most famous of these affairs was with the actress María Inés Calderón (La Calderona), with whom he had a son in 1629, Juan José, who was brought up as a royal prince. By the end of the reign, and with the health of Carlos José in doubt, there was a real possibility of Juan José's making a claim on the throne, which added to the instability of the regency years.


title: Philip IV of Spain

text: During the reign of Philip's father, Philip III, the royal court had been dominated by the Sandoval noble family, most strikingly by the Duke of Lerma, Philip III's principal favorite and chief minister for almost all of his reign. Philip IV came to power as the influence of the Sandovals was being undermined by a new noble coalition, led by Don Baltasar de Zúñiga. De Zúñiga regarded it as essential that the Sandovals be unable to gain an influence over the future king; de Zúñiga first began to develop his own influence over Prince Philip, and then introduced his nephew, Olivares, to the prince, ten years old at the time. At first, Philip did not particularly take to Olivares. Over the course of at least a year, however, the relationship became very close, with Philip's tendency towards underconfidence and diffidence counteracted by Olivares' drive and determination. Olivares rapidly became Philip's most trusted advisor, and when Philip ascended the throne in 1621, at the age of sixteen, he showed his confidence in Olivares by ordering that all papers requiring the royal signature should first be sent to the count-duke. Philip retained Olivares as his confidant and chief minister for the next twenty years.


title: Philip IV of Spain

text: Early in his reign, Philip would be woken by Olivares in the morning to discuss the day's affairs and would meet with him twice more during the day, although later this routine declined until the king would hold only one short meeting on policy with Olivares each day. Philip intervened far more in policies during 1641–42, however, and it has been suggested that Philip paid more attention to policymaking than has traditionally been depicted; some recent histories go so far as to describe him as 'conscientious' in policymaking, although he is still criticised for his failure to make timely decisions. Philip himself argued that it was hardly appropriate for the king himself to go house to house amongst his ministers to see if his instructions were being carried out. The close relationship between Philip and Olivares was demonstrated by their portraits' being placed side by side at the Buen Retiro palace — an act unheard-of in Europe at the time. Philip's relationship with Olivares, however, was not a simplistic one. The pair had many rows and arguments over the course of their relationship, both as a result of their different personalities and differences of opinion over policies.


title: Philip IV of Spain

text: Initially, Philip chose to confirm the reappointment of his father's household to assuage grandee opinion. Under the influence of de Zúñiga and Olivares, however, Philip was then quick to place de Lerma's estates – expanded considerably during his long period as favourite – under administration, and to remove from office Cristóbal de Sandoval, Duke of Uceda, de Lerma's son, who had initially helped de Zúñiga remove his own father from office to advance his own position. Philip's initial announcements reflected an intent to reform the monarchy to the sober, moral position it had been under his grandfather, including selecting ministers whose grandfathers had served under Philip II.


title: Philip IV of Spain

text: Philip has in the past been considered to be 'unimaginative' in his politics, but recent histories have stressed the more radical elements of his first two decades in power. There was a febrile atmosphere in Spain in the early 17th century, with numerous "arbitrista" offering various advice on how to solve Spain's various ills; this advice could, and would, be given in person by those of the lower classes to the king on suitable occasions, provided it was presented with the aim of strengthening the crown. Those debates extended to the nature of the monarchy. It has been suggested that the writers of the period who best capture Philip's view of royal authority were Justus Lipsius and Giovanni Botero, who promoted religiously inspired, stoic self-sacrifice and a view of Habsburg family-led hegemony respectively. Whilst at one level conservative – harking back in foreign policy to the period of Phillip II, invoking traditional values at home – Philip's policies were also radical, rejecting the policy towards the rebellious Dutch that had held since 1609, entering into the Thirty Years' War, and introducing a system of "junta", or small committee, government across Spain in competition to the traditional system of royal councils.


title: Philip IV of Spain

text: Following Olivares' fall from power amidst the crisis of 1640–1643, the victim of failed policies and jealousy from the nobles excluded from power, Philip initially announced that he would rule alone, becoming in effect his own first minister. The "junta" system of government began to be dismantled in favor of the older council system. In due course, however, this personal rule reverted to rule through a royal favorite, initially Luis de Haro, a nephew of Olivares and a childhood playmate of Philip's, and the counter-reform of the committee system halted. De Haro has not been highly regarded by historians; the comment of one, that de Haro was the 'embodiment of mediocrity', is not atypical. After de Haro's death in 1661, Olivares' son-in-law, the Duke of Medina de las Torres, became royal favourite in his place.


title: Philip IV of Spain

text: Philip was to reign through the majority of the Thirty Years' War in Europe, a turbulent period of military history. In Philip III's final years, Baltasar de Zúñiga had convinced him to intervene militarily in Bohemia and the Electorate of the Palatinate on the side of Emperor Ferdinand II. Once Philip himself came to power, he was convinced by de Zúñiga, appointed his principal foreign minister, and Olivares that he should commit Spain to a more aggressive foreign policy in alliance with the Holy Roman Empire. This would lead Philip to renew hostilities with the Dutch in 1621 in an attempt to bring the provinces to the negotiating table with the aim of achieving a peace treaty favourable to Spanish global interests. Philip's government would pursue a 'Netherlands first' strategy throughout the war until 1643. Despite this shift in policy, Philip does not seem to have been particularly bellicose; early on he noted that having inherited such a large empire, war somewhere across his domains was an inevitable condition, and he appeared genuinely upset when he came to power and contemplated how much the people of Castile had paid 'in blood' to support the wars of his royal predecessors.


title: Philip IV of Spain

text: The 1620s were good years for Spanish foreign policy: the war with the Dutch went well, albeit at great expense, culminating in the retaking of the key city of Breda in 1624. By the end of the decade, however, Philip's government was faced with the question of whether to prioritise the war in Flanders or Spain's relationship with France during the War of the Mantuan Succession (1628–1631). Philip's advisors recommended prioritising the war in Flanders, taking action to safeguard the Spanish Road to the Netherlands but at the cost of antagonising Louis XIII. Strategically this was to prove a disaster. Despite fresh Spanish successes in the mid-1630s – in particular, the triumph of Philip's government in raising a fresh Spanish army, marching it into Germany to defeat the Swedish-led Protestant forces at the Battle of Nördlingen in 1634 – the increased tensions with France made war between the two Catholic states increasingly inevitable. Olivares advised Philip that the coming war with France would be all or nothing; Spain would win or fall by the result.


title: Philip IV of Spain

text: The Spanish-French war that ensued from 1635 onwards was not a foregone conclusion. Early Spanish successes threatened Paris, and even after the Spanish defeat at Rocroi, Spain remained a strong opponent. But from 1640 onwards, a period which saw large-scale revolts across Spanish territories in protest against the rising costs of the conflict, Spain was finding it difficult to sustain the war. Philip reacted to the increased French threat by finally abandoning his 'Netherlands first' strategy; resources for the Army of Flanders were savagely cut, and the fight against the French-supported rebels Catalonia took the first priority. Shortly after Rocroi, Philip – now having had to dismiss his favourite, Olivares – issued instructions to his ambassadors to seek a peace treaty. The Peace of Westphalia, delivered by Olivares' replacement Luis de Haro, resolved the long running Eighty Years' War in the Netherlands and the wars in Germany, but the conflict with France dragged on. Philip responded to the perceived weakness of France during the Fronde rebellions of 1648 by continuing the fight; he took personal responsibility for the decision to start a fresh, and ultimately successful, offensive against the French in Catalonia in 1651. True victory over France never emerged, however, and by 1658, after the loss of Dunkirk to an Anglo-French force, Philip was personally desperate for peace. The Treaty of the Pyrenees in 1659, and the marriage of Philip's daughter Maria Theresa to the young King Louis XIV finally brought the war with France to a conclusion. The war against Portugal continued however, as Philip fruitlessly attempted to regain control over his lost kingdom.


title: Philip IV of Spain

text: By the late 1620s, the Spanish army was no longer as dominant on the battlefield as it once had been. The feared "tercio" regiments, composed of well-disciplined pikemen, were increasingly appearing inflexible and outmoded in the face of the new Swedish and Dutch formations with a higher proportion of musketeers. Philip and Olivares attempted to address the perceived weaknesses of the army, which they concluded were primarily due to the "falta de cabezas", or a lack of leadership. In keeping with their wider agenda of renewing the concepts of duty, service and aristocratic tradition, the king agreed to efforts to introduce more grandees into the higher ranks of the military, working hard to overcome the reluctance of many to take up field appointments in the Netherlands and elsewhere.


title: Philip IV of Spain

text: The results were not entirely as hoped. The grandees dragooned into service in this way were disinclined to spend years learning the normal professional military skill set; they wished 'to start out as generals and soldiers on the same day', to quote one disgruntled career soldier. By the 1630s, the king was waiving the usual rules to enable promotion to higher ranks on a shorter timescale, and having to pay significantly inflated salaries to get grandees to take up even these appointments. The performance of these officers at battles such as Rocroi left much to be desired.


title: Philip IV of Spain

text: Philip was also notable for his interest in the Spanish navy. Shortly after taking power he began to increase the size of his fleets, rapidly doubling the size of the naval budget from the start of his reign, then tripling it. Philip is credited with a 'sensible, pragmatic approach' to provisioning and controlling it. He was prepared to involve himself in considerable details of naval policy; he was commenting on the detail of provisions for the "armada" in 1630, for example. The "Junta de Armadas" was the only "junta" committee to survive the fall of Olivares intact. Even after the disastrous Battle of the Downs, Philip remained closely interested in his navy, including ensuring ministerial attention. In 1646, de Haro was personally involved in supplying and equipping the Atlantic fleet from Cadiz. Throughout the period there was no 'weakening of the importance attached to naval forces' by the king, who argued that joint land and naval operations were essential. Some of his conclusions on naval policy were quite advanced: after the peace of 1648, Philip argued that the Dutch fleets off the Spanish peninsula were actually good for trade, despite concerns from his senior officials, since they provided protection against the English and French navies.


title: Philip IV of Spain

text: Philip had inherited a huge empire from his father, spanning the known world, but many of his most difficult challenges as king would stem from domestic problems in Spain itself. Spain in the early 17th century was a collection of possessions – the kingdoms of Castile, Aragon, Valencia and Portugal, the autonomous provinces of Catalonia and Andalusia, complete with the wider provinces of Naples, the Netherlands, Milan etc. – all loosely joined together through the institution of the Castile monarchy and the person of Philip IV. Each part had different taxation, privileges and military arrangements; in practice, the level of taxation in many of the more peripheral provinces was less than that in Castile, but the privileged position of the Castilian nobility at all senior levels of royal appointment was a contentious issue for the less favoured provinces. This loose system had successfully resisted reform and higher taxation before, ironically resulting in Spain's having had historically, up until the 1640s at least, fewer than the usual number of fiscal revolts for an early modern European state.


title: Philip IV of Spain

text: In the first years of his reign, heavily influenced by his royal favourite Olivares, Philip focused on efforts to reform the most chaotic aspects of this system. Frustrated by the notorious slowness of the system of royal councils, Philip supported Olivares' establishment of "juntas" – small committees designed to circumvent the more formal system and to enact policies quickly. Although successful, these "juntas" excluded many of the traditional grandees and caused resentment. Olivares put forward the idea of a "Unión de Armas", or 'Union of Arms'. This would have involved establishing a force of 140,000 paid soldiers, supported by equitable taxes from across the Empire, and has been termed 'the most far-sighted proposal of any statesman of the age'; in practice, however, it met fierce opposition from the various regional assemblies and the plan was withdrawn. During the 1620s, again influenced by a desire to reform Spanish life for the better, Philip also passed considerable legislation with puritanical overtones. In 1623, he closed all the legal brothels in Spain, extended the dormant sumptuary laws on luxury goods and supported Papal efforts to regulate priests' sexual behaviour more tightly.


title: Philip IV of Spain

text: Philip had clear intentions to try to control the Spanish currency, which had become increasingly unstable during the reign of his father and grandfather, but in practice, inflation soared. Partly this was because in 1627 Olivares had attempted to deal with the problem of Philip's Genoese bankers – who had proved uncooperative in recent years – by declaring a state bankruptcy. With the Genoese debt now removed, Olivares hoped to turn to indigenous bankers for renewed funds. In practice, the plan was a disaster. The Spanish treasure fleet of 1628 was captured by the Dutch, and Spain's ability to borrow and transfer money across Europe declined sharply.


title: Philip IV of Spain

text: By the 1630s, Philip's domestic policies were being increasingly impacted by the financial pressures of the Thirty Years' War, and in particular the growing war with France. The costs of the war were huge, and whilst they had largely fallen upon Castile, the ability of the crown to raise more funds and men from this source was increasingly limited. Philip and his government were desperately trying to reduce the responsibilities of central government in response to the overstretch of the war, and various reform ideas that might have been pursued during the 1620s were rejected on this basis. Financial restraints and higher taxes were put in place, but Philip was increasingly selling off regalian and feudal rights, along with much of the royal estate to fund the conflict. It has been argued that the fiscal stringencies of the 1630s, combined with the strength and role of Olivares and the "juntas", effectively cut Philip off from the three traditional pillars of support for the monarchy: the grandees, the Church and the Council of Castile.


title: Philip IV of Spain

text: Crisis came in 1640. An attempt by Olivares to intervene in Catalonia to deal with the French invasion threat resulted in revolt. An alliance of Catalan rebels and French royal forces proved challenging to suppress, and in trying to mobilise Portuguese noble support for the war, Olivares triggered a second uprising. Lisbon's nobles expelled Philip, and gave the throne to the Braganzas, marking the end of sixty years of the Iberian Union and the beginning of the Portuguese Restoration War. The next year, the Duke of Medina Sidonia attempted another rebellion against Philip from Andalusia, possibly attempting to reproduce the Braganzas' success in Portugal. Although Philip and Olivares were able to repress the ducal revolt, Philip had found himself increasingly isolated. On his return from Zaragoza, where he had been commanding the army, he found only one of the Castilian nobility arrived at court on Easter Day 1641. The threat of Philip's being deposed by the grandees of Castile seemed increasingly real.


title: Philip IV of Spain

text: Much shaken by events, Philip's solution was to remove his royal favourite Olivares from office in 1643 in an attempt to compromise with the Spanish elite. He announced he would rule alone, rejecting both the concept of a royal favourite as first minister and the system of "junta" government, which he began to dismantle in favour of the older system of royal councils. Clemency was shown to the Duke of Medina Sidonia. The situation began to stabilise, and before long Philip felt secure enough to revert to his preferred method of government. Luis de Haro, Olivares' nephew, took over as favourite and minister and the counter-reform of the "juntas" halted. The spark of reform from Philip's earlier years never returned, however. The Catalan rebellion dragged on for several years. In 1652, the Spanish army retook Barcelona and Philip issued an amnesty for the rebels, promising to respect traditional customs and rights in the future.


title: Philip IV of Spain

text: Philip has been remembered both for the 'astonishing enthusiasm' with which he collected art and for his love of theatre. On the stage, he favoured Lope de Vega, Pedro Calderón de la Barca, and other distinguished dramatists. Philip has been credited with a share in the composition of several comedies. Court theatre used perspective scenery, a new invention from Italy not used in commercial theatre at the time. Some writers have likened the illusion of Baroque royal theatre to the illusion of kingly power the performances were designed to reinforce. Some recent scholarship has suggested that Philip's financial sponsorship of playwrights, however, may have been less extensive than once thought.


title: Philip IV of Spain

text: Artistically, Philip became famous for his patronage of his court painter Diego Velázquez, who originated from Seville; mutual contacts caused him to become known to Olivares, who came from the same region. Velázquez was summoned to Madrid by the king in 1624. Despite some jealousy from the existing court painters, Velázquez rapidly became a success with Philip, being retained for the rest of his career until his death, painting a celebration of the Treaty of the Pyrenees for Philip. The king and Velázquez shared common interests in horses, dogs and art, and in private they formed an easy, relaxed relationship over the years. Philip supported a number of other prominent painters, including Eugenio Caxés, Vicente Carducho, Gonzales and Nardi. Philip obtained paintings from across Europe, especially Italy, accumulating over 4,000 by the time of his death; some have termed this unparalleled assemblage a 'mega-collection'.


title: Philip IV of Spain

text: Philip was nicknamed "el Rey Planeta", the 'Planet King', by his contemporaries, and much of the art and display at his court has been interpreted in the context of his need to project power and authority, over both Spaniards and foreigners alike. Older interpretations, which perceived Philip's court as being completely decadent, have been largely superseded, but the art and symbolism of the period certainly did not reflect the wider threat and decline of Spanish power. Indeed, the limited Spanish military successes of the period were celebrated by royal artists to a disproportionate extent. Numerous artists from the Spanish Netherlands produced work extolling the Army of Flanders, including Vrancx, Snaeyers, Molenaer and de Hondt. The re-capture of Breda alone resulted in major works by Velázquez and the French etcher Jacques Callot, in addition to various plays and books.


title: Philip IV of Spain

text: The 'Planet King' also invested in a new palace to display both his art and the ritual of court. Through Olivares, Philip commenced the building of the "Buen Retiro palace" in Madrid, parts of which still remain near the Prado. Work began modestly in 1631, with the magnificent, if costly, 'Hall of Thrones', completed by 1635. The palace included its own 'theatre, ballroom, galleries, bull ring, gardens, and artificial lakes', and became the centre for artists and dramatists from across Europe. The palace was built during one of the more difficult periods of Philip's reign. Given both its cost, in a time of stringent wartime savings, and the protest that ensued from a disgruntled public, it is considered to have been an important part of the attempt to communicate royal grandeur and authority.


title: Philip IV of Spain

text: The Catholic religion and its rituals played an important part in Philip's life, especially towards the end of his reign. Depressed by events across his domains, he became increasingly concerned with religious affairs. In particular, Philip paid special devotions to a painting of the "Nuestra Señora del Milagro", the Virgin of Miracles; the painting was said to miraculously raise and lower its eyes in response to prayer. Whilst married to Elisabeth, Philip had placed their children under the protection of this image; married to Mariana, they undertook special religious ceremonies together under the gaze of the painting. Philip also had a large standard made with the image of the painting on one side and the royal coat of arms on the other, brought out in processions each year on 12 July. As well as marking a strong personal religious belief, this increasingly visible link between the crown, the Church and national symbols such as the Virgin of Miracles, represented a key pillar of support for Philip as king.


title: Philip IV of Spain

text: Monarchs during the period also had a key role in the canonization process and could utilise this for domestic or international political effect. Philip, for example, keen to reach out to his Portuguese subjects, put his considerable influence behind the case for Isabella of Portugal, a 14th-century role model of a 'perfect queen', to great effect, ultimately paying for a lavish celebration in Lisbon after her canonisation in 1625. Internationally, it was important for Spanish prestige for her to receive at least a proportionate, and ideally greater, share of new saints than other Catholic kingdoms, and Philip sponsored a flurry of texts and books supporting Spain's candidates, particularly in competition with Catholic France.


title: Philip IV of Spain

text: During the emergency of 1640–1643, Philip appears to have had a crisis of faith. Philip genuinely believed the success or failure of his policies represented God's favour and judgement on his actions. The combination of the revolts, the French advances and the loss of his trusted favourite Olivares appears to have deeply shaken him. Queen Isabella and the new president of the Council of Castile, Don Juan Chumacero – both involved in the removal of Olivares – encouraged the king to invite mystics and visionaries from across Europe to his court at Zaragoza. The mystics' principal advice centred on the importance of the king's rejecting Olivares' replacement, de Haro and the remaining pro-Olivares nobles at court. The various mystics were not acceptable to broader Spanish noble opinion and, with de Haro's encouragement, they were ultimately dismissed.


title: Philip IV of Spain

text: Instead, Philip turned to a better established female mystic, Sister María de Ágreda, a prioress known for her religious writings. He asked her to correspond with him and to advise him in spiritual matters. The two became regular correspondents throughout the remainder of their lives. This is documented in over 600 confidential letters between them over a period of twenty-two years. Philip clearly believed that Maria could intercede with God on his behalf and provide advice on what God wished him to do, to improve Spain's failing fortunes. Most believe that Philip was involved in protecting Maria from the Inquisition's investigation of 1650. Philip's son, as Charles II, protected her writings from later censorship.


title: Philip IV of Spain

text: In the 1630 Treaty of Madrid, Philip was styled "Philip, by the grace of God king of the Spains, Both the Sicilies, Jerusalem, the Indies, etc., archduke of Austria, duke of Burgundy, Milan, etc., count of Habsburg, Tyrol, etc." in full and "the Most Serene Philip IV, Catholic King of the Spains," for short.


title: Philip IV of Spain

text: In the 1648 Treaty of Münster, he was styled "Don Philip the Fourth, by the grace of God king of Castile, Leon, Aragon, the Two Sicilies, Jerusalem, Navarre, Granada, Toledo, Valencia, Galicia, Majorca, Minorca, Seville, Sardinia, Cordoba, Corsica, Murcia, Jaen, Algeciras, Gibraltar, the Canary Islands, the Eastern and Western Indies, the islands and "terra firma" of the Ocean, archduke of Austria, duke of Burgundy, Brabant, Milan, count of Habsburg, Flanders, Tyrol, Barcelona, lord of Biscay and Molina, etc." in full and "King of the Spains, Don Philip the Fourth and King of Portugal and the Algarves Don Philip the Third ", for short.


title: Philip IV of Spain

text: Philip IV's reign, after a few years of inconclusive successes, was characterized by political and military adversity. He has been held responsible for the decline of Spain, which was mainly due to organic causes largely beyond the control of any one ruler. Philip IV died in 1665, expressing the pious hope that his surviving son, Charles II, who was only 4 years old at the time, would be more fortunate than himself. On his death, a catafalque was built in Rome to commemorate his life. In his will, Philip left political power as regent on behalf of the young Charles II to his wife Mariana, with instructions that she heed the advice of a small "junta" committee established for this purpose. This committee excluded John Joseph, Philip's illegitimate son, resulting in a chaotic powerplay between Mariana and John Joseph until his death in 1679.


title: Lady Dimitrescu

text: Alcina Dimitrescu (, "Oruchīna Dimitoresuku"), better known as Lady Dimitrescu, is a character in the 2021 survival horror game "Resident Evil Village", developed and published by Capcom. One of the major antagonistic figures of "Village", she is presented as a gigantic noblewoman with vampire-like traits who resides with her three daughters in Castle Dimitrescu, her stronghold within the vicinity of the titular Eastern European village encountered by protagonist Ethan Winters. Dimitrescu governs the village alongside three mutant lords under the oversight of a supreme leader known as Mother Miranda. Lady Dimitrescu is portrayed by American theatre actress Maggie Robertson.


title: Lady Dimitrescu

text: Following initial previews of "Village", Lady Dimitrescu unexpectedly rose in popularity and has become subject to a significant amount of fan-driven labor, including fan art, cosplay, memes, and erotica. Journalists and commentators took note of the trend, which began well before the release of "Village" in May 2021, attributing the rapid surge of fan interest in the character to various aspects of her appearance and sex appeal. Robertson's performance as Lady Dimitrescu was positively-received, and won her several awards.


title: Lady Dimitrescu

text: Created by Capcom for the 2021 video game "Resident Evil Village", Lady Dimitrescu was born from the desire to create a charismatic femme fatale character. She was designed as a "bewitching vampire" antagonist as part of the developer's efforts to revisit the "Resident Evil" franchise's survival horror roots. Her character design was inspired by the 16th century Hungarian noblewoman Elizabeth Báthory, the Japanese urban legend and internet ghost story of "Hasshaku-sama" (or "Hachishaku-sama"), and Anjelica Huston's portrayal of the character Morticia Addams from "The Addams Family". Aspects of Dimitrescu's characterization also references Ramon Salazar from "Resident Evil 4", a diminutive villain said to have resonated with many players as a memorable antagonist. Dimitrescu has been falsely pronounced as "Dimitreesk" in English, with the final "u" letter being silent. This approach has been criticized by Romanian speakers who have said that the correct pronunciation is "dimi-tres-ku", for her name is Romanian.


title: Lady Dimitrescu

text: According to Tomonori Takano, the art director for "Resident Evil Village", the developmental team intended to move away from using simplistic elements like zombies to scare players and instead focused on creating unique situations and memorable characters that would create fear in new ways, continuing a trend which began in "", the predecessor to "Village". The boss character of the Castle Dimitrescu area, Lady Dimitrescu is presented as a nigh invulnerable enemy with long retractable sharp claws that extend from her finger tips. Dimitrescu is programmed to stalk the player-controlled Ethan Winters throughout her castle; she is meant to be constantly evaded by the player, much like Mr. X from the 2019 remake of "Resident Evil 2" or members of the Baker family from "Resident Evil 7".


title: Lady Dimitrescu

text: Dimitrescu commands her "daughters" Bela, Cassandra, and Daniela as her subordinates, products of her experiments with the Cadou parasite whose bodies are composed of imitation flies that aggregate into the forms of the corpses they devoured and can only thrive within the castle walls. Castle Dimitrescu, inspired by Peleș Castle in Romania, was originally conceived as being inhabited by "dozens upon dozens" of Dimitrescu's daughters, but the family was ultimately shortlisted to Alcina Dimitrescu and her three daughters following trial and error and testing of the game's pacing. Dimitrescu and her daughters, who compete with each other for their mother's attention and approval, feed off human blood to sustain themselves, with the remains of their male victims discarded and left in a crucified state outside of Castle Dimitrescu. Dimitrescu is said to be highly protective of her daughters in a maternal fashion, which the developers approached by presenting her story as paralleling that of Ethan's own family whom he is also trying to protect.


title: Lady Dimitrescu

text: Dimitrescu stands at , taking into consideration her hat and high heels. As Takano wanted to avoid typical gothic imagery associated with past "Resident Evil" games as well as the horror genre as a whole, the Dimitrescu women's visual design reference fashion trends from the 1930s era instead, while the sisters' clothing are finely embroidered with floral patterns of the Dimitrescu family crest. Her appearance first came about in an early prototype that used the model of Ethan's wife Mia Winters from "Resident Evil 7" but with a hat and a dress, which gave her a ghost-like appearance, but not one that comes across as overly scary. The team scaled-up the model in game and discovered that the additional accessories provided that necessary element. Based on this prototype, one of the first pieces of concept art Takano drew of Lady Dimitrescu involved her imposing visage leaning down to step through a doorway and demonstrating her over nine-foot height, which he recognized needed to be a scene in the final game, as well as being present in the game's first trailer.


title: Lady Dimitrescu

text: Theatrical actress Maggie Robertson provided both voice acting and motion capture for Dimitrescu. She had auditioned for the role on a whim having recently moved from eastern United States to Los Angeles, and was surprised by the role, her first in video games.


title: Lady Dimitrescu

text: In the DLC expansion "The Mercenaries: Additional Orders" for "Resident Evil: Village" released in October 2022, Lady Dimitrescu is one of the playable characters, alongside Chris Redfield and Karl Heisenberg.


title: Lady Dimitrescu

text: Alcina was born into the noble Dimitrescu family sometime before World War I. Although her family traced their origins to Cesare, one of the four founders of an isolated mountain village in Europe, Alcina lived elsewhere, perhaps through a cadet branch. In the aftermath of World War II and the abolition of the nobility, Dimitrescu returned to her family's former lands, which had fallen under the control of a neopagan cult worshipping the Black God. Dimitrescu suffered a hereditary blood disease prior to meeting Miranda who infected her with the Cadou parasite to see if she was compatible for her plans to revive her daughter. Although bestowed with profound regenerative capabilities and eternal life, Alcina's preexisting blood disease required her to ingest large quantities of human blood and flesh on a regular basis to maintain her condition, increasing her size while siring her "daughters" Daniela, Bela and Cassandra. Upon inhabiting the estate, she took over her family's vineyard and wine-distribution business as a means of supporting herself. The Dimitrescu family was closely associated with the noble Beneviento, Moreau, and Heisenberg families, and maintained an alliance with them in controlling the region. This control allowed Dimitrescu's family to rule her castle with barbarous cruelty, regularly taking in new staff to replace those who were taken to the dungeon to be killed with their blood used to create an enriched red wine. The umbrella shaped symbol that represents Dimitrescu, Heisenberg, Moreau, and Beneviento was borrowed by Oswell E. Spencer as the logo of the Umbrella Corporation.


title: Lady Dimitrescu

text: Ethan Winters was captured by Dimitrescu's daughters and strung up on a wall to be drained of blood, but manages to free himself and kill the daughters one at a time. This infuriates her as she ignores Miranda's orders to personally kill Ethan, forced into mutating into her true dragon-like form while confronting Ethan within her estate's family crypt and ultimately dies at his hand.


title: Lady Dimitrescu

text: To promote "Resident Evil Village", Capcom distributed life-size standees of Lady Dimitrescu in video game stores. In Hong Kong, promotional material featuring the character was displayed on public transportation in early 2021. Notable merchandise which feature the character include life-sized towels which were offered as prizes for Japanese participants in a competition held on Twitter. Professional cosplayer Yaya Han uploaded a video sponsored by Capcom on her YouTube channel, which documents the process behind her cosplay activities as the character in April 2021. On April 30, 2021, a puppet show miniseries featuring Dimitrescu alongside her peers was released on the official "biohazard" YouTube channel. Her likeness has been co-opted by third party brands such as Domino's Pizza Malaysia to promote their products. A three foot tall statue of Lady Dimitrescu made by PureArts in collaboration with Capcom is scheduled to be released in second quarter 2023.


title: Lady Dimitrescu

text: Lady Dimitrescu received a very positive reception from critics and players following her initial reveal in a promotional trailer for "Resident Evil Village" which debuted in 2020, and for her appearance in the PlayStation 5-exclusive demo "Maiden" released on January 21, 2021. By February 2021, Takano publicly acknowledged what he later described as an unexpectedly enthusiastic response to the character. Even though "Village" was scheduled for launch on May 7, 2021, Dimitrescu's surge in popularity in early 2021 led to a significant proliferation of fan exploration of the character's concepts through fan art; received a fan-made action-figure commercial, cosplay activities by notable individuals such as Han or Olympic bronze medalist Yekaterina Lisina; and internet memes, which sometimes involved participation by other video game developers on social media. Dimitrescu has been the subject of fan mods for video games; examples include a mod for "Fallout 4" which recreates her long-sleeved ivory gown to be worn by the player character, or a mod for "Village" which replaces Dimitrescu's face with that of Thomas the Tank Engine, with "Eurogamer" describing it as the "headline act" of the game's "burgeoning mod scene". In March 2021, Epic Games included Lady Dimitrescu along with series protagonists Jill Valentine and Ethan Winters in a list of characters and brands as part of a 2021 survey it distributed to "Fortnite" players in order to gauge interest in future crossover promotions.


title: Lady Dimitrescu

text: Several commentators discussed at length about the phenomenon behind the character's popularity, with many identifying her unusually tall height as a key element which drew fan interest. Leon Hurley from "GamesRadar" observed that Dimitrescu is not the main villain of "Village" or its most important character, but nevertheless has become more prominent, popular, discussed, and imitated than was originally intended or expected. In light of the character's positive reception, Jess Kinghorn also from "GamesRadar" suggested that the video game industry should be less fixated with youthfulness by featuring more older female characters who are nuanced in characterization.


title: Lady Dimitrescu

text: Certain commentators have suggested that some of the fan interest in the character is erotic or sexual in nature, specifically feminine dominance. Dimistrescu's popularity has indeed inspired a substantial amount of fan-made pornography or expressions of desire to participate in sexual or erotic fetishism. The character is said to have generated some interest in trampling, the act of being stepped on by a partner playing a sexually dominant role, in particular. Professional erotica writer Gemma Glitter attributed interest in the character by some corners to macrophilia, in this case a fascination of giant-sized women. Steven T. Wright from "Input Mag" reported that Lady Dimitrescu's sex appeal represents the fetishization of their body height for several real-life tall women, as she is considered to be "simultaneously a beacon of their own power and allure, and a painful reminder of the complex social stigmas society attaches to femme body shape". In December 2021, Ed Nightingale of "Eurogamer" claimed that she is the "Icon of the year".


title: Lady Dimitrescu

text: Maggie Robertson has earned multiple awards for her performance as Lady Dimitrescu. She won Best Performer at the 2021 Golden Joystick Awards, Best Performance at The Game Awards 2021, the Great White Way Award for Best Acting in a Game at the 11th Annual New York Game Awards, and the Outstanding Performance in a Drama, Supporting award at the 21st annual NAVGTR Awards. At the 25th Annual D.I.C.E. Awards, Robertson accepted the award for Outstanding Achievement in Character on behalf of the "Village" development team in recognition for their collective work with Lady Dimitrescu.


title: Lady Dimitrescu

text: Rob Dolen of "Game Rant" noted that Dimitrescu's role in the game and fandom is similar to that of Vaas Montenegro in "Far Cry 3", as both are early villains that far overshadow the later characters.


title: Geometric series

text: In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series


title: Geometric series

text: is geometric, because each successive term can be obtained by multiplying the previous term by formula_2. In general, a geometric series is written as formula_3, where formula_4 is the coefficient of each term and formula_5 is the common ratio between adjacent terms. The geometric series had an important role in the early development of calculus, is used throughout mathematics, and can serve as an introduction to frequently used mathematical tools such as the Taylor series, the complex Fourier series, and the matrix exponential.


title: Geometric series

text: The name geometric series indicates each term is the geometric mean of its two neighboring terms, similar to how the name arithmetic series indicates each term is the arithmetic mean of its two neighboring terms. The sequence of geometric series terms (without any of the additions) is called a "geometric sequence" or "geometric progression".


title: Geometric series

text: The geometric series "a" + "ar" + "ar" + "ar" + ... is written in expanded form. Every coefficient in the geometric series is the same. In contrast, the power series written as "a" + "a""r" + "a""r" + "a""r" + ... in expanded form has coefficients "a" that can vary from term to term. In other words, the geometric series is a special case of the power series. The first term of a geometric series in expanded form is the coefficient "a" of that geometric series.


title: Geometric series

text: In addition to the expanded form of the geometric series, there is a generator form of the geometric series written as


title: Geometric series

text: The derivation of the closed form from the expanded form is shown in this article's Sum section. However even without that derivation, the result can be confirmed with long division: "a" divided by (1 - "r") results in "a" + "ar" + "ar" + "ar" + ... , which is the expanded form of the geometric series.


title: Geometric series

text: It is often a convenience in notation to set the series equal to the sum "s" and work with the geometric series


title: Geometric series

text: Changing even one of the coefficients to something other than coefficient "a" would change the resulting sum of functions to some function other than "a" / (1 - "r") within the range |"r"| < 1. As an aside, a particularly useful change to the coefficients is defined by the Taylor series, which describes how to change the coefficients so that the sum of functions converges to any user selected, sufficiently smooth function within a range.


title: Geometric series

text: The geometric series "a" + "ar" + "ar" + "ar" + ... is an infinite series defined by just two parameters: coefficient "a" and common ratio "r". Common ratio "r" is the ratio of any term with the previous term in the series. Or equivalently, common ratio "r" is the term multiplier used to calculate the next term in the series. The following table shows several geometric series:


title: Geometric series

text: The rate of convergence also depends on the value of the common ratio "r". Specifically, the rate of convergence gets slower as "r" approaches 1 or −1. For example, the geometric series with "a" = 1 is 1 + "r" + "r" + "r" + ... and converges to 1 / (1 - "r") when |"r"| < 1. However, the number of terms needed to converge approaches infinity as "r" approaches 1 because "a" / (1 - "r") approaches infinity and each term of the series is less than or equal to one. In contrast, as "r" approaches −1 the sum of the first several terms of the geometric series starts to converge to 1/2 but slightly flips up or down depending on whether the most recently added term has a power of "r" that is even or odd. That flipping behavior near "r" = −1 is illustrated in the adjacent image showing the first 11 terms of the geometric series with "a" = 1 and |"r"| < 1.


title: Geometric series

text: The common ratio "r" and the coefficient "a" also define the geometric progression, which is a list of the terms of the geometric series but without the additions. Therefore the geometric series "a" + "ar" + "ar" + "ar" + ... has the geometric progression (also called the geometric sequence) "a", "ar", "ar", "ar", ... The geometric progression - as simple as it is - models a surprising number of natural phenomena,


title: Geometric series

text: As an aside, the common ratio "r" can be a complex number such as |"r"|e where |"r"| is the vector's magnitude (or length), "θ" is the vector's angle (or orientation) in the complex plane and i = -1. With a common ratio |"r"|e, the expanded form of the geometric series is "a" + "a"|"r"|e + "a"|"r"|e + "a"|"r"|e + ... Modeling the angle "θ" as linearly increasing over time at the rate of some angular frequency "ω" (in other words, making the substitution "θ" = "ω""t"), the expanded form of the geometric series becomes "a" + "a"|"r"|e + "a"|"r"|e + "a"|"r"|e + ... , where the first term is a vector of length "a" not rotating at all, and all the other terms are vectors of different lengths rotating at harmonics of the fundamental angular frequency "ω". The constraint |"r"|<1 is enough to coordinate this infinite number of vectors of different lengths all rotating at different speeds into tracing a circle, as shown in the adjacent video. Similar to how the Taylor series describes how to change the coefficients so the series converges to a user selected sufficiently smooth function within a range, the Fourier series describes how to change the coefficients (which can also be complex numbers in order to specify the initial angles of vectors) so the series converges to a user selected periodic function.


title: Geometric series

text: where is the common ratio. One can derive that closed-form formula for the partial sum, "s", by subtracting out the many self-similar terms as follows:


title: Geometric series

text: As approaches infinity, the absolute value of must be less than one for the series to converge. The sum then becomes


title: Geometric series

text: The formula also holds for complex , with the corresponding restriction that the modulus of is strictly less than one.


title: Geometric series

text: As an aside, the question of whether an infinite series converges is fundamentally a question about the distance between two values: given enough terms, does the value of the partial sum get arbitrarily close to the finite value it is approaching? In the above derivation of the closed form of the geometric series, the interpretation of the distance between two values is the distance between their locations on the number line. That is the most common interpretation of the distance between two values. However, the p-adic metric, which has become a critical notion in modern number theory, offers a definition of distance such that the geometric series 1 + 2 + 4 + 8 + ... with "a" = 1 and "r" = 2 actually does converge to "a" / (1 - "r") = 1 / (1 - 2) = -1 even though "r" is outside the typical convergence range |"r"| < 1.


title: Geometric series

text: Alternatively, a geometric interpretation of the convergence is shown in the adjacent diagram. The area of the white triangle is the series remainder = "s" - "s" = "ar" / (1 - "r"). Each additional term in the partial series reduces the area of that white triangle remainder by the area of the trapezoid representing the added term. The trapezoid areas (i.e., the values of the terms) get progressively thinner and shorter and closer to the origin. In the limit, as the number of trapezoids approaches infinity, the white triangle remainder vanishes as it is filled by trapezoids and therefore "s" converges to "s", provided |"r"|<1. In contrast, if |"r"|>1, the trapezoid areas representing the terms of the series instead get progressively wider and taller and farther from the origin, not converging to the origin and not converging as a series.


title: Geometric series

text: After knowing that a series converges, there are some applications in which it is also important to know how quickly the series converges. For the geometric series, one convenient measure of the convergence rate is how much the previous series remainder decreases due to the last term of the partial series. Given that the last term is "ar" and the previous series remainder is "s" - "s" = "ar" / (1 - "r")), this measure of the convergence rate of the geometric series is "ar" / ("ar" / (1 - "r")) = 1 - "r", if 0 ≤ "r" < 1.


title: Geometric series

text: If "r" < 0, adjacent terms in the geometric series alternate between being positive and negative. A geometric interpretation of a converging alternating geometric series is shown in the adjacent diagram in which the areas of the negative terms are shown below the x axis. Pairing and summing each positive area with its negative smaller area neighbor results in non-overlapped trapezoids separated by gaps. To remove the gaps, broaden each trapezoid to cover the rightmost 1 - "r" of the original triangle area instead of just the rightmost 1 - |"r"|. However, to maintain the same trapezoid areas during this broadening transformation, scaling is needed: scale*(1 - "r") = (1 - |"r"|), or scale = (1 - |"r"|) / (1 - "r") = (1 + "r") / (1 - "r") = (1 + "r") / ((1 + "r")(1 - "r")) = 1 / (1 - "r") where -1 < "r" ≤ 0. Note that because "r" < 0 this scale decreases the amplitude of the separated trapezoids in order to fill in the separation gaps. In contrast, for the case "r" > 0 the same scale 1 / (1 - "r") increases the amplitude of the non-overlapped trapezoids in order to account for the loss of the overlapped areas.


title: Geometric series

text: With the gaps removed, pairs of terms in a converging alternating geometric series become a converging (non-alternating) geometric series with common ratio "r" to account for the pairing of terms, coefficient "a" = 1 / (1 - "r") to account for the gap filling, and the degree (i.e., highest powered term) of the partial series called m instead of n to emphasize that terms have been paired. Similar to the "r" > 0 case, the "r" < 0 convergence rate = "ar" / ("s" - s) = 1 - "r", which is the same as the convergence rate of a non-alternating geometric series if its terms were similarly paired. Therefore, the convergence rate does not depend upon n or m and, perhaps more surprising, does not depend upon the sign of the common ratio. One perspective that helps explain the variable rate of convergence that is symmetric about "r" = 0 is that each added term of the partial series makes a finite contribution to the infinite sum at "r" = 1 and each added term of the partial series makes a finite contribution to the infinite slope at "r" = -1.


title: Geometric series

text: 2,500 years ago, Greek mathematicians had a problem when walking from one place to another: they thought that an infinitely long list of numbers greater than zero summed to infinity. Therefore, it was a paradox when Zeno of Elea pointed out that in order to walk from one place to another, you first have to walk half the distance, and then you have to walk half the remaining distance, and then you have to walk half of that remaining distance, and you continue halving the remaining distances an infinite number of times because no matter how small the remaining distance is you still have to walk the first half of it. Thus, Zeno of Elea transformed a short distance into an infinitely long list of halved remaining distances, all of which are greater than zero. And that was the problem: how can a distance be short when measured directly and also infinite when summed over its infinite list of halved remainders? The paradox revealed something was wrong with the assumption that an infinitely long list of numbers greater than zero summed to infinity.


title: Geometric series

text: "Euclid's Elements of Geometry" Book IX, Proposition 35, proof (of the proposition in adjacent diagram's caption):


title: Geometric series

text: The terseness of Euclid's propositions and proofs may have been a necessity. As is, the "Elements of Geometry" is over 500 pages of propositions and proofs. Making copies of this popular textbook was labor intensive given that the printing press was not invented until 1440. And the book's popularity lasted a long time: as stated in the cited introduction to an English translation, "Elements of Geometry" "has the distinction of being the world's oldest continuously used mathematical textbook." So being very terse was being very practical. The proof of Proposition 35 in Book IX could have been even more compact if Euclid could have somehow avoided explicitly equating lengths of specific line segments from different terms in the series. For example, the contemporary notation for geometric series (i.e., "a" + "ar" + "ar" + "ar" + ... + "ar") does not label specific portions of terms that are equal to each other.


title: Geometric series

text: Most of the theorems appearing in the Elements were not discovered by Euclid himself, but were the work of earlier Greek mathematicians such as Pythagoras (and his school), Hippocrates of Chios, Theaetetus of Athens, and Eudoxus of Cnidos. However, Euclid is generally credited with arranging these theorems in a logical manner, so as to demonstrate (admittedly, not always with the rigour demanded by modern mathematics) that they necessarily follow from five simple axioms. Euclid is also credited with devising a number of particularly ingenious proofs of previously discovered theorems (e.g., Theorem 48 in Book 1).


title: Geometric series

text: To help translate the proposition and proof into a form that uses current notation, a couple modifications are in the diagram. First, the four horizontal line lengths representing the values of the first four terms of a geometric series are now labeled a, ar, ar, ar in the diagram's left margin. Second, new labels A' and D' are now on the first and third lines so that all the diagram's line segment names consistently specify the segment's starting point and ending point.


title: Geometric series

text: Archimedes used the sum of a geometric series to compute the area enclosed by a parabola and a straight line. His method was to dissect the area into an infinite number of triangles.


title: Geometric series

text: Archimedes' Theorem states that the total area under the parabola is 4/3 of the area of the blue triangle.


title: Geometric series

text: Archimedes determined that each green triangle has 1/8 the area of the blue triangle, each yellow triangle has 1/8 the area of a green triangle, and so forth.


title: Geometric series

text: The first term represents the area of the blue triangle, the second term the areas of the two green triangles, the third term the areas of the four yellow triangles, and so on. Simplifying the fractions gives


title: Geometric series

text: This computation uses the method of exhaustion, an early version of integration. Using calculus, the same area could be found by a definite integral.