❤️ Pavement (band) 🐮

"Pavement was an American indie rock band that formed in Stockton, California in 1989. For most of their career, the group consisted of Stephen Malkmus (vocals and guitar), Scott Kannberg (guitar and vocals), Mark Ibold (bass), Steve West (drums) and Bob Nastanovich (percussion and vocals). Initially conceived as a recording project, the band at first avoided press or live performances, while attracting considerable underground attention with their early releases. Gradually evolving into a more polished band, Pavement recorded five full-length albums and nine EPs over the course of their decade- long career, though they disbanded with some acrimony in 1999 as the members moved on to other projects. In 2010, they undertook a well-received reunion tour, and will perform two 30th anniversary shows in 2021. Though only briefly brushing the mainstream with the single "Cut Your Hair" in 1994, Pavement was a successful indie rock band. Rather than signing with a major label as many of their 1980s forebears had done, they remained signed to independent labels throughout their career and have often been described as one of the most influential bands to emerge from the American underground in the '90s. Some prominent music critics, such as Robert Christgau and Stephen Thomas Erlewine, called them the best band of the 1990s. In their career, they also achieved a significant cult following.Erlewine, Stephen Thomas; Phares, Heather. [ "Pavement biography"], AllMusic. Retrieved on March 26, 2009.From The Rolling Stone Encyclopedia of Rock & Roll (Simon & Schuster, 2001) History = Slanted and Enchanted: 1989–1992 = Pavement formed in Stockton, California in 1989 as a studio project of guitarists and vocalists Stephen Malkmus & Scott Kannberg, known originally only as "S.M." and "Spiral Stairs". Their debut EPs were extremely lo-fi releases titled Slay Tracks: 1933–1969, Demolition Plot J-7, and Perfect Sound Forever. They were recorded at Louder Than You Think, the home studio of Stockton local and former hippie Gary Young who also played drums on the recordings. Upon first hearing the duo's songs, Young was quoted as saying, "this Malkmus idiot is a complete songwriting genius."Sullivan, James. "Pavement Gear Up". Synthesis.net. February 27, 1997. Retrieved on March 26, 2009. During this time the band was often compared to English rock band The Fall, however Kannberg stated in a 1992 interview that he preferred Minneapolis rock band The Replacements. The Fall's Mark E. Smith claimed that Pavement were a "rip-off" of his bandPerry, Andrew. "Alright?". Select. May 1993. and that they didn't "have an original idea in their heads",Herrington, Tony. "Mark E. Smith: Interview". The Wire. September 1996. although other members of The Fall have been more positive about the band.Atwal, Sandy. "Fall in a Hole". Imprint. September 30, 1994. After the release of Slay Tracks, a new drummer, Jason Turner, was drafted to replace Young both live and in the studio. However, after just one tour and a handful of recording sessions, when it became apparent that the percussionist and Malkmus did not get along well, Turner was ousted and Young reinstated.Jovanovic (2004), p.78–82. Malkmus later described Turner as "this depressed guy who might assassinate me one day... He's very competitive."Tape Op: The Book About Creative Music Recording, edited by Larry Crane. Interviews with Stephen Malkmus and Gary Young, pp. 68–78 Around the same time, Bob Nastanovich was incorporated into the live Pavement band as an auxiliary percussionist. Malkmus had been roommates with Nastanovich in New York City, and had told him, "You could be our drummer if we ever played." Around 1992 Pavement became a full-time band, with the addition of bassist Mark Ibold, who had been one of the band's earliest fans, with Malkmus, Kannberg, Young and Nastanovich rounding out the lineup. Their debut album, Slanted and Enchanted, was released commercially in 1992 after copies had been circulated on cassette tape for nearly a year. Though the percussive influence of The Fall was still pervasive, as was that of English post-punk band Swell Maps, many of the songs also exhibited a strong sense of melody. Since its release Slanted and Enchanted has appeared on many critics' best-of lists and is frequently cited as being among the most influential indie rock albums of the 1990s. It is listed on Rolling Stone's 500 Greatest Albums of All Time. Later the same year, the band released the EP Watery, Domestic. =Crooked Rain, Crooked Rain: 1993–1994= During the Slanted & Enchanted tour, Gary Young's eccentric behavior increased, and he would stage incidents such as handing out cabbage and mashed potatoes to fans at the door of the venue, doing handstands, drunkenly falling off his drum stool, and running around the venue while the rest of the band was playing. The band only understood how severe Young's drinking problem was after Pavement's first few tours. Malkmus later told Tape Op, "We knew that he was like a hippie and kinda flaky, but we didn't know he had such a bad drinking problem. We found out on that tour, because he really got sick from being nervous... That's why I let Bob be in the band...'Keep the beat going if Gary passes out.'" In 1993, Malkmus attempted to record some new songs at Young's studio with unsuccessful results. The singer later said, "We kind of wanted to not record with him anymore, but we were too nice to fire people or even really talk about it... We tried to record there, but it wasn't sounding good and he didn't have his studio ready and he was also in a drinking funk." At the conclusion of a 1993 tour of Australia, Japan, and Europe, the group held a meeting in a hotel room in Copenhagen during which Malkmus, Kannberg and Ibold remained silent while Nastanovich (Young's best friend at the time) argued with the drummer and informed him that his antics were unnecessary. Young ultimately agreed to leave the band. He was replaced by Steve West, who had been a fellow museum security guard at the Whitney Museum of American Art along with Malkmus and David Berman.Crooked Rain, Crooked Rain: LA's Desert Origin's CD. Liner notes by Spiral Stairs West's debut performance was in 1993 at a Drag City festival in Chicago. Also that same year, the band contributed to the AIDS-Benefit Album No Alternative produced by the Red Hot Organization with their song "Unseen Power of the Picket Fence". Pavement's second album Crooked Rain, Crooked Rain was released in 1994. The record was more indebted to the classic rock tradition than their debut. The single "Cut Your Hair" was the band's closest brush with the mainstream, and briefly enjoyed airplay on alternative rock radio and MTV. Pavement performed the song on The Tonight Show with Jay Leno. The video also aired on "Career Day", a season five episode of Beavis and Butt-head, who termed it "buttwipe music" and also wanted the band to "try harder." The lyrics from another single from the album, "Range Life", criticized alternative rock stars The Smashing Pumpkins and the Stone Temple Pilots. Malkmus has insisted over the years that the line is meant to be light-hearted and is sung from the point of view of the aging hippie character in the song. Later live versions of the track had the singer substituting "The Spice Girls", "Counting Crows", or others for "Stone Temple Pilots". In response Smashing Pumpkins leader Billy Corgan threatened to drop his band from their slot headlining the 1994 Lollapalooza Festival if Pavement was allowed to play. Corgan and Malkmus would trade barbs through the press for several years afterwards."Rivalries great fodder for entertainment in music industry". Daily Bruin, April 27, 2003. Like its predecessor, the album received widespread acclaim, including being listed on Rolling Stone magazine's list of the 500 greatest albums of all time =Wowee Zowee and Brighten the Corners: 1995–1997= The next album, Wowee Zowee, recorded in Memphis and released April 11, 1995, covered a wide range of styles including punk, country and balladry across its 18 tracks, which often avoided conventional song structures. On the Slow Century DVD, Malkmus attributed his odd choice of singles to his marijuana smoking, stating that "I was smoking a lot of grass back then but to me they sounded like hits." Although Malkmus has said in recent interviews that the album is the last "classic Pavement record," Kannberg has voiced regrets about Wowee Zowee. "We made some mistakes on that record... we were kind of pressured into putting out a record a little faster than we were ready to. I mean, I'm totally into the record. It's just if we had another six months to think about it, it would've been much different." During the tour for the album, Nastanovich stated on the Slow Century DVD, the band would often not work out a setlist before shows, opting for drug and alcohol fueled jams over hit singles. Some of these shows were held during the 1995 Lollapallooza festival, where the incoherent performances received a hostile reaction from many audiences. Footage from Slow Century shows the band being pelted with mud and rocks. The band then left the stage immediately and dubbed themselves "The Band That Ruined Lollapallooza." Wowee Zowee was followed up by the EP Pacific Trim, which was recorded with only Malkmus and drummers Nastanovich and Steve West. Their studio time was originally reserved for a Silver Jews recording, but frontman David Berman walked out in frustration and the trio decided not to waste prepaid recording time. Brighten the Corners was released in 1997; produced by Mitch Easter it is a shorter, more conventional record than the previous album. Malkmus said on the Slow Century DVD that the album was an attempt to show audiences that Pavement had more mainstream and classic rock influences than it had previously portrayed. The album contained two of the band's best known singles in "Stereo" and "Shady Lane". It was the only Pavement album to include a lyric sheet except Slanted and Enchanted and sold better than its predecessors. Despite increased success, the band continued to fragment, with its members focusing more on other musical projects or on raising families. =Terror Twilight and breakup: 1999–2000= In 1999, the band began work on its final album, Terror Twilight. Bob Nastanovich came up with the title, and has revealed the meaning of it in several interviews: "Twilight Terror is the short span between sunset and dusk; this is considered the most dangerous time in traffic, because half of the people switch on the headlights, and the other half doesn't. It's when most accidents happen." During an interview on the Slow Century documentary DVD, he said that the band was having trouble coming up with names for the record until he put forth Terror Twilight. One of the final contenders was Farewell Horizontal (also the name of a 1989 science fiction novel by K. W. Jeter), and Nastanovich said, "There was no way I was going to be on the Farewell Horizontal tour for the next year." The band originally planned to self-produce Terror Twilight, renting out Jackpot! Studios in Portland, Oregon. The group stalled though, with Malkmus, Ibold, Nastanovich and Jackpot! employee and future Jicks bassist Joanna Bolme usually opting to play Scrabble over getting any sort of work accomplished.Jovanovic (2004), p.177. Kannberg was especially frustrated over the sessions, particularly at Malkmus' refusal to include any of Kannberg's songs on the album. Fan favorite "For Sale: The Preston School of Industry" and one other song penned by the guitarist were briefly worked on during the sessions, but eventually abandoned. At the end of the initial two-week session, very little had been accomplished and it was decided that a producer should be brought in to assist them. Nigel Godrich, best known for his work with Radiohead and Beck, was hired to produce the album."PAVEMENT: THE CRACKS OF THEIR ASPHALT". Chart Attack, June 22, 1999. Interview By Stuart Berman The group first attempted to record in Sonic Youth's lower Manhattan studio, which Godrich took a dislike to as it was within ear-shot of several practice spaces, and also set up more like a home studio.Jovanovic (2004), p.179-180. Godrich eventually convinced the band to move to a more "proper" 24-track studio, where he had previously worked on albums by the Beastie Boys and R.E.M. Though the producer took an immediate shine to Malkmus, Kannberg and Nastanovich were wary of him. Nastanovich believes that though Godrich "took on a pretty substantial challenge and did a good job," that he "focused his attention on Stephen" and produced them "in a way that just sort of had more disregard for the rest of us."Jovanovic (2004), p.181. Nastanovich also later recalled an awkward incident where it became apparent that the producer didn't know the auxiliary percussionist's name. Kannberg, still disillusioned over his songs being rejected by Malkmus, said of Terror Twilight: "It was not fun to do that record from the very beginning. It was the hardest record to make." Malkmus wrote the entirety of the record, and the recording of the music only featured minor contributions from the rest of the band. The group released one last EP, titled Major Leagues. It features three Malkmus songs, two original Spiral Stairs songs and two covers, "The Killing Moon" by Echo & the Bunnymen and "The Classical" by The Fall. Pavement embarked on a six-month world tour in support of the album, during which time relationships within the group frayed, especially between Malkmus and the other members of the band. Steve West later recalled times on the tour bus where Malkmus would put his coat over his head, refuse to speak to anyone, and call himself "The Little Bitch". Critics noted that their setlist had started to include a number of their early songs, much like a band playing their greatest hits on a farewell tour. Malkmus's attitude continued to deteriorate as the tour wore on, finally coming to a head during their slot at the 1999 Coachella Festival. During the concert, Malkmus refused (or possibly was unable, due to illness) to sing, effectively turning their set into a mostly instrumental one. Nastanovich called a band meeting after the show, and Malkmus finally confided to his bandmates: "I just don't want to do this anymore."Jovanovic (2004), p.187. The group's final concert together was at Brixton Academy in London on November 20, 1999."Cracks in the Pavement – Reports elsewhere claimed that we were making it all up". NME. December 1, 1999. Retrieved on March 27, 2009. During the concert, Malkmus had a pair of handcuffs attached to his microphone stand, and at one point told that audience, "These symbolize what it's like being in a band all these years.""Cracks in the Pavement – Malkmus and co look like they'll go separate ways". NME. December 1, 1999. Retrieved on March 27, 2009. After the concert, he confirmed to people at the after-party that Pavement was now done "for the foreseeable future." About two weeks later, a spokesperson for Domino records told NME: "Pavement are retiring for the foreseeable future to: 1. Start families 2. Sail around the world 3. Get into the computer industry 4. Dance 5. Get some attention." In the summer of 2000, Malkmus called Kannberg and told him, "You need to change the website to say we aren't a band anymore. People keep asking me if we're breaking up and you know we're not a band anymore, right?"Jovanovic (2004), p.194. Kannberg told Malkmus that he needed to call the other members of the band to inform them that the band was finally breaking up, but Malkmus refused and Kannberg was left with the task of informing them. Steve West later admitted that he never received any official call about the breakup from anyone in the band, and discovered that Pavement had dissolved via the internet. Nastanovich later commented that "There was too much exhaustion for heavy emotion." =Reunion: 2010= Stephen Malkmus in concert with Pavement in 2010. On September 15, 2009, Brooklyn Vegan reported that Pavement were scheduled to perform multiple benefit show dates in New York City's Central Park from September 21, 2010.BV. (September 15, 2009). "the Pavement reunion is on – multiple nights @ Central Park Summerstage in 2010 + a tour? + Coachella? + ATP?". Brooklyn Vegan. Official statements by the band, label, venue and promoter were released on September 17, 2009 confirming the reunion. The announcement included one concert in Central Park and the promise of a tour, but said that the reunion may only be a one-off event. It said, "Please be advised this tour is not a prelude to additional jaunts and/or a permanent reunion." Tickets for the first Central Park concert sold out in two minutes,"Pavement Add Reunion Shows". Pitchfork Media. September 18, 2009. leading to the announcement of three more shows at the same venue. The band since confirmed a world-wide tour which started in Auckland, New Zealand on March 1 at the Auckland Town Hall, before heading on to Australia then heading to the UK, including the All Tomorrow's Parties festival in Minehead, Somerset (May 14 – 16) and several European shows. They also performed at the Coachella Valley Music and Arts Festival in April 2010, the Sasquatch! and Primavera Sound Festivals in May, the Toronto Island Concert in June with Broken Social Scene, Band Of Horses, and others, Open'er Festival, Roskilde Festival and Les Ardentes and Pitchfork Music Festival in July. The band released a "best-of" compilation album in March 2010, entitled Quarantine the Past: The Best of Pavement. On June 24, 2010, Pavement performed at the Bob Hope Theatre in Stockton, their first-ever hometown show. Original drummer Gary Young joined the band for an encore, playing three songs from Slanted and Enchanted. Young also joined the band during six songs the next night at The Greek Theater in Berkeley. In September 2010, Pavement appeared on The Colbert Report and Late Night with Jimmy Fallon. After concluding their proper US tour at the Hollywood Bowl on September 30, the band played the following night in Las Vegas at the Palms Casino as a part of the Matador At 21 festival.NME article: "Pavement drop hint about their future during Las Vegas gig". During their set, Scott Kannberg kicked his monitor and smashed his guitar into his amp in frustration of not being able to hear himself while performing. NME noted the band had an "icy atmosphere onstage". The band honored a further two South American show commitments in November 2010. =Second reunion: 2021= Scott Kannberg joined Stephen Malkmus and The Jicks onstage at the El Rey Theater in Los Angeles on March 28, 2014, for an encore performance of the Pavement song "Stereo." On October 1 and 2, 2016 Stephen Malkmus and the Jicks performed at Spiral Stairs' 50th birthday celebration at The Chapel in San Francisco. On the first night Kannberg joined Malkmus and The Jicks onstage for four songs. The following night Kannberg played three more songs with the group, joined by Gary Young on the final two. On June 1, 2019, Pavement announced that they would be reuniting to perform two 30th anniversary shows at the 2020 Primavera Sound festivals in Barcelona and Porto, and were scheduled to be the band's only performances that year. Due to logistical issues resulting from the COVID-19 pandemic, the festival was cancelled, and their performances were rescheduled and announced as part of Primavera Sound's June 2021 lineup. Musical style Pavement is considered to be one of the bands to have invented the modern "indie" sound and was a large presence in "slacker culture," a subculture most prevalent during the 1990s. The band was also noted for Malkmus' humorous and often cryptic lyrical themes which was a key factor in gaining their cult following. Malkmus rarely wrote ballads or love songs and only a small portion of their discography contained personal reflections or similar lyrical themes, most satirical or laced with sarcasm. Pavement was also noted for having no designated rhythm and lead guitar player. Malkmus and Kannberg switched roles frequently although Malkmus played lead for the majority of their career. The band was also noted for their use of a foil, or a 'hype-man' usually present in hip hop groups. Bob Nastanovich filled the role although he has detested the term himself. Nastanovich also served as a second drummer during their live performances and served as lead vocalist on select songs that required screaming to prevent strain on Malkmus' voice. Other media In 2002, Slow Century, a documentary by Lance Bangs coupled with all of the band's music videos, was released as a 2 DVD set. Included was extensive footage, both professionally shot and taped by fans on camcorders, from the band's earliest shows in 1989 forward. The three final songs from the band's final concert ("Stop Breathin'", "Conduit for Sale" and "Here") are presented at the end of the documentary. Also on the DVD is a hidden easter egg clip from the same show, wherein Malkmus talks about how the handcuffs attached to his microphone stand "represent what it's like being in a band all these years." A bonus disc with a complete concert in Seattle, Washington, from the early part of the Terror Twilight tour was included on the second disc, as well as several songs from their penultimate show. 2004 saw the publication of Perfect Sound Forever: The Story of Pavement, a biography on the band written by Rob Jovanovic. Reviews for the book were mixed, with some saying that it contained much of the same information as the Slow Century DVD and expanded very little on it, while others called it a "fond retrospection". Appearances The band appeared in a 1997 episode of Space Ghost Coast to Coast, titled "Pavement", where they were mistaken for The Beatles and played two songs done exclusively for the show. These songs were later included on the deluxe re-issue of Brighten the Corners. Band members Final line-up *Stephen Malkmus – lead vocals, guitar (1989–1999; 2010; 2021) *Scott “Spiral Stairs” Kannberg – guitar, backing and lead vocals (1989–1999; 2010; 2021) *Mark Ibold – bass guitar (1991–1999; 2010; 2021) *Bob Nastanovich – percussion, synthesizers, vocals (1991–1999; 2010; 2021) *Steve West – drums (1993–1999; 2010; 2021) Former members *Gary Young – drums (1989, 1989–1993; 2010) *Jason Turner – drums (1989) Timeline ImageSize = width:1000 height:auto barincrement:25 PlotArea = left:120 bottom:80 top:15 right:0 Alignbars = justify DateFormat = dd/mm/yyyy Period = from:01/01/1989 till:31/12/2022 TimeAxis = orientation:horizontal format:yyyy Legend = orientation:horizontal position:top ScaleMajor = increment:3 start:1989 ScaleMinor = increment:1 start:1989 Colors = id:vocals value:red legend:Vocals id:guitar value:green legend:Guitar id:Keyboards value:purple legend:Keyboards id:bass value:blue legend:Bass id:drums value:orange legend:Drums id:Percussion value:claret legend:Percussion id:studio value:black legend:Studio_album id:bars value:gray(0.9) BackgroundColors = bars:bars Legend = position:bottom columns:2 BarData = bar:Malkmus text:"Stephen Malkmus" bar:Kannberg text:"Scott Kannberg" bar:Ibold text:"Mark Ibold" bar:Young text:"Gary Young" bar:Turner text:"Jason Turner" bar:West text:"Steve West" bar:Nastanovich text:"Bob Nastanovich" PlotData= width:11 textcolor:black align:left anchor:from shift:(10,-4) bar:Malkmus from:01/01/1989 till:20/11/1999 color:vocals bar:Malkmus from:01/01/1989 till:20/11/1999 color:guitar width:3 bar:Malkmus from:01/01/2010 till:31/12/2010 color:vocals bar:Malkmus from:01/01/2010 till:31/12/2010 color:guitar width:3 bar:Malkmus from:31/05/2021 till:05/06/2021 color:vocals bar:Malkmus from:31/05/2021 till:05/06/2021 color:guitar width:3 bar:Kannberg from:01/01/1989 till:20/11/1999 color:guitar bar:Kannberg from:01/01/1989 till:20/11/1999 color:vocals width:3 bar:Kannberg from:01/01/2010 till:31/12/2010 color:guitar bar:Kannberg from:01/01/2010 till:31/12/2010 color:vocals width:3 bar:Kannberg from:31/05/2021 till:05/06/2021 color:guitar bar:Kannberg from:31/05/2021 till:05/06/2021 color:vocals width:3 bar:Young from:01/01/1989 till:01/06/1989 color:drums bar:Young from:01/09/1989 till:01/06/1993 color:drums bar:Young from:24/06/2010 till:25/06/2010 color:drums bar:Turner from:01/06/1989 till:01/09/1989 color:drums bar:Ibold from:01/01/1991 till:20/11/1999 color:bass bar:Ibold from:01/01/2010 till:31/12/2010 color:bass bar:Ibold from:31/05/2021 till:05/06/2021 color:bass bar:Nastanovich from:01/01/1991 till:20/11/1999 color:percussion bar:Nastanovich from:01/01/1991 till:20/11/1999 color:vocals width:3 bar:Nastanovich from:01/01/1991 till:20/11/1999 color:keyboards width:7 bar:Nastanovich from:01/01/2010 till:31/12/2010 color:percussion bar:Nastanovich from:01/01/2010 till:31/12/2010 color:vocals width:3 bar:Nastanovich from:01/01/2010 till:31/12/2010 color:keyboards width:7 bar:Nastanovich from:31/05/2021 till:05/06/2021 color:percussion bar:Nastanovich from:31/05/2021 till:05/06/2021 color:vocals width:3 bar:Nastanovich from:31/05/2021 till:05/06/2021 color:keyboards width:7 bar:West from:01/06/1993 till:20/11/1999 color:drums bar:West from:01/01/2010 till:31/12/2010 color:drums bar:West from:31/05/2021 till:05/06/2021 color:drums LineData = layer:back color:studio at:20/04/1992 at:14/02/1994 at:11/04/1995 at:11/02/1997 at:09/06/1999 Discography ;Studio albums * Slanted and Enchanted (1992) * Crooked Rain, Crooked Rain (1994) * Wowee Zowee (1995) * Brighten the Corners (1997) * Terror Twilight (1999) References Bibliography *Jovanovic, Rob (2004). Perfect Sound Forever: The Story of Pavement. (Boston) Justin, Charles & Co. . External links * Pavement at Matador Records * Pavement at Domino Recording Company Podcast of Pavement Live at the Great American Music Hall * Stephen Malkmus on Reissue of Pavement's 'Brighten the Corners' * Podcast: Pavement Reunion Tour Concert, Sydney, Australia Arts & Crafts Productions artists Art pop musicians Domino Recording Company artists Drag City (record label) artists Flying Nun Records artists Indie rock musical groups from California Matador Records artists Musical groups established in 1989 Musical groups disestablished in 1999 Musical groups reestablished in 2010 Musical groups disestablished in 2010 Lo-fi music groups Pony Canyon artists "

❤️ Concentric spheres 🐮

"The cosmological model of concentric (or homocentric) spheres, developed by Eudoxus, Callippus, and Aristotle, employed celestial spheres all centered on the Earth. In this respect, it differed from the epicyclic and eccentric models with multiple centers, which were used by Ptolemy and other mathematical astronomers until the time of Copernicus. Origins of the concept of concentric spheres Eudoxus of Cnidus was the first astronomer to develop the concept of concentric spheres. He was originally a student at Plato's academy and is believed to have been influenced by the cosmological speculations of Plato and Pythagoras."Eudoxus of Cnidus." Complete Dictionary of Scientific Biography. Vol. 4. Detroit: Charles Scribner's Sons, 2008. 465–467. Gale Virtual Reference Library. Web. 2 June 2014. He came up with the idea of homocentric spheres in order to explain the perceived inconsistent motions of the planets and to develop a uniform model for accurately calculating the movement of celestial objects. None of his books have survived to the modern day and everything we know about his cosmological theories comes from the works of Aristotle and Simplicius. According to these works, Eudoxus’ model had twenty-seven homocentric spheres with each sphere explaining a type of observable motion for each celestial object. Eudoxus assigns one sphere for the fixed stars which is supposed to explain their daily movement. He assigns three spheres to both the sun and the moon with the first sphere moving in the same manner as the sphere of the fixed stars. The second sphere explains the movement of the sun and the moon on the ecliptic plane. The third sphere was supposed to move on a “latitudinally inclined” circle and explain the latitudinal motion of the sun and the moon in the cosmos. Four spheres were assigned to Mercury, Mars, Venus, Jupiter, and Saturn which were the only known planets at that time. The first and second spheres of the planets moved exactly like the first two spheres of the sun and the moon. According to Simplicius, the third and fourth sphere of the planets were supposed to move in a way that created a curve known as a hippopede. The hippopede was a way to try and explain the retrograde motions of planets. Many historians of science, such as Michael J. Crowe, have argued that Eudoxus did not consider his system of concentric spheres to be a real representation of the universe but thought it was merely a mathematical model for calculating planetary motion. Later additions to Eudoxus' model Callippus, a contemporary of Eudoxus, attempted to improve his system by increasing the total number of homocentric spheres. He added two additional spheres for the sun and the moon as well as one additional sphere for Mars, Mercury, and Venus. These additional spheres were supposed to fix some of the calculation problems in Eudoxus’ original system. Callippus’ system was able to better predict the motions of certain celestial objects but his system still had many problems and was not able to account for many astronomical observations. Aristotle developed his own system of concentric spheres in Metaphysics and De Caelo (On the Heavens). He thought that both Eudoxus and Callippus had too few spheres within their models and added more spheres onto Callippus’ system. He added three spheres to Jupiter and Mars as well as four spheres to Venus, Mercury, the sun, and the moon for a total of fifty-five spheres. He later doubted the accuracy of his results and stated that he believed there were either forty seven or forty nine concentric spheres. Historians are unsure about how many spheres Aristotle thought there were in the cosmos with theories ranging from 43 to 55. Unlike Eudoxus, Aristotle believed that his system represented an actual model of the cosmos. See also * Celestial spheres * Geocentric model Notes Further reading Kieffer, John S. "Callippus." Dictionary of Scientific Biography 3:21-22. External links * Working model and complete explanation of the Eudoxus's Spheres * Eudoxos of Knidos (Eudoxus of Cnidus): astronomy and homocentric spheres Henry Mendell, Cal State U, LA * Models of Planetary Motion—Eudoxus, Craig McConnell, Ph.D., Cal State, Fullerton * Aristotle's On the Heavens * Aristotle's metaphysics Ancient Greek astronomy Early scientific cosmologies "

❤️ Rader's FFT algorithm 🐮

"Rader's algorithm (1968),C. M. Rader, "Discrete Fourier transforms when the number of data samples is prime," Proc. IEEE 56, 1107–1108 (1968). named for Charles M. Rader of MIT Lincoln Laboratory, is a fast Fourier transform (FFT) algorithm that computes the discrete Fourier transform (DFT) of prime sizes by re-expressing the DFT as a cyclic convolution (the other algorithm for FFTs of prime sizes, Bluestein's algorithm, also works by rewriting the DFT as a convolution). Since Rader's algorithm only depends upon the periodicity of the DFT kernel, it is directly applicable to any other transform (of prime order) with a similar property, such as a number-theoretic transform or the discrete Hartley transform. The algorithm can be modified to gain a factor of two savings for the case of DFTs of real data, using a slightly modified re- indexing/permutation to obtain two half-size cyclic convolutions of real data;S. Chu and C. Burrus, "A prime factor FTT [sic] algorithm using distributed arithmetic," IEEE Transactions on Acoustics, Speech, and Signal Processing 30 (2), 217-227 (1982). an alternative adaptation for DFTs of real data uses the discrete Hartley transform.Matteo Frigo and Steven G. Johnson, "The Design and Implementation of FFTW3," Proceedings of the IEEE 93 (2), 216–231 (2005). Winograd extended Rader's algorithm to include prime-power DFT sizes p^m,S. Winograd, "On Computing the Discrete Fourier Transform", Proc. National Academy of Sciences USA, 73(4), 1005-1006 (1976).S. Winograd, "On Computing the Discrete Fourier Transform", Mathematics of Computation, 32(141), 175-199 (1978). and today Rader's algorithm is sometimes described as a special case of Winograd's FFT algorithm, also called the multiplicative Fourier transform algorithm (Tolimieri et al., 1997),R. Tolimieri, M. An, and C.Lu, Algorithms for Discrete Fourier Transform and Convolution, Springer- Verlag, 2nd ed., 1997. which applies to an even larger class of sizes. However, for composite sizes such as prime powers, the Cooley–Tukey FFT algorithm is much simpler and more practical to implement, so Rader's algorithm is typically only used for large-prime base cases of Cooley–Tukey's recursive decomposition of the DFT. Algorithm Visual representation of DFT matrix in Rader's FFT algorithm, The array consists of colored clocks represent DFT matrix of size 11. By permuting rows and columns using a sequence generated by the primitive root of 11 (which is 2) except the 1st row and column, the original DFT matrix becomes a circulant matrix. Multiplying a circulant matrix to a data sequence is equivalent to the cyclic convolution. This relation is an example of the fact that multiplicative group is cyclic: (\mathbb Z/p\mathbb Z)^\times \cong C_{p-1}. : X_k = \sum_{n=0}^{N-1} x_n e^{-\frac{2\pi i}{N} nk } \qquad k = 0,\dots,N-1. If N is a prime number, then the set of non-zero indices n = 1,...,N-1 forms a group under multiplication modulo N. One consequence of the number theory of such groups is that there exists a generator of the group (sometimes called a primitive root, which can be found quickly by exhaustive search or slightly better algorithmsDonald E. Knuth, The Art of Computer Programming, vol. 2: Seminumerical Algorithms, 3rd edition, section 4.5.4, p. 391 (Addison–Wesley, 1998).), an integer g such that n = gq (mod N) for any non-zero index n and for a unique q in 0,...,N-2 (forming a bijection from q to non-zero n). Similarly k = g-p (mod N) for any non-zero index k and for a unique p in 0,...,N-2, where the negative exponent denotes the multiplicative inverse of gp modulo N. That means that we can rewrite the DFT using these new indices p and q as: : X_0 = \sum_{n=0}^{N-1} x_n, : X_{g^{-p}} = x_0 + \sum_{q=0}^{N-2} x_{g^q} e^{-\frac{2\pi i}{N} g^{-(p-q)} } \qquad p = 0,\dots,N-2. (Recall that xn and Xk are implicitly periodic in N, and also that e2πi=1. Thus, all indices and exponents are taken modulo N as required by the group arithmetic.) The final summation, above, is precisely a cyclic convolution of the two sequences aq and bq of length N-1 (q = 0,...,N-2) defined by: :a_q = x_{g^q} :b_q = e^{-\frac{2\pi i}{N} g^{-q} }. =Evaluating the convolution= Since N-1 is composite, this convolution can be performed directly via the convolution theorem and more conventional FFT algorithms. However, that may not be efficient if N-1 itself has large prime factors, requiring recursive use of Rader's algorithm. Instead, one can compute a length-(N-1) cyclic convolution exactly by zero-padding it to a length of at least 2(N-1)-1, say to a power of two, which can then be evaluated in O(N log N) time without the recursive application of Rader's algorithm. This algorithm, then, requires O(N) additions plus O(N log N) time for the convolution. In practice, the O(N) additions can often be performed by absorbing the additions into the convolution: if the convolution is performed by a pair of FFTs, then the sum of xn is given by the DC (0th) output of the FFT of aq plus x0, and x0 can be added to all the outputs by adding it to the DC term of the convolution prior to the inverse FFT. Still, this algorithm requires intrinsically more operations than FFTs of nearby composite sizes, and typically takes 3-10 times as long in practice. If Rader's algorithm is performed by using FFTs of size N-1 to compute the convolution, rather than by zero padding as mentioned above, the efficiency depends strongly upon N and the number of times that Rader's algorithm must be applied recursively. The worst case would be if N-1 were 2N2 where N2 is prime, with N2-1 = 2N3 where N3 is prime, and so on. In such cases, supposing that the chain of primes extended all the way down to some bounded value, the recursive application of Rader's algorithm would actually require O(N2) time. Such Nj are called Sophie Germain primes, and such a sequence of them is called a Cunningham chain of the first kind. The lengths of Cunningham chains, however, are observed to grow more slowly than log2(N), so Rader's algorithm applied in this way is probably not O(N2), though it is possibly worse than O(N log N) for the worst cases. Fortunately, a guarantee of O(N log N) complexity can be achieved by zero padding. References FFT algorithms "