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Twelve-inch single

The twelve-inch single (often written as 12-inch or 12) is a type of vinyl (polyvinyl chloride or PVC) gramophone record that has wider groove spacing and shorter playing time with a "single" or a few related sound tracks on each surface, compared to LPs (long play) which have several songs on each side. It is named for its 12 in diameter. This allows for louder levels to be cut on the disc by the mastering engineer, which in turn gives a wider dynamic range, and thus better sound quality. This record type is commonly used in disco and dance music genres, where DJs use them to play in clubs. They are played at either $33 1/3$ or 45 rpm. The conventional 7-inch single usually holds three or four minutes of music at full volume. The 12-inch LP sacrifices volume for extended playing time.

Technical features
Twelve-inch singles typically have much shorter playing time than full-length LPs, and thus require fewer grooves per inch. This extra space permits a broader dynamic range or louder recording level as the grooves' excursions (i.e., the width of the groove waves and distance traveled from side to side by the turntable stylus) can be much greater in amplitude, especially in the bass frequencies important for dance music. Many record companies in the 1970s began producing 12-inch (30 cm) singles at $33 1/3$ rpm, although 45 rpm gives better treble response.

Pre-vinyl period
Gramophone records had been introduced in the latter portion of the 19th century, with several pioneers involved in sound reproduction development such as Thomas Edison and Emilie Berliner. Berliner along with Eldridge R. Johnson merged their efforts within the industry to form the Victor Talking Machine Company in New Jersey, USA and perfected the use of 5 and 7-inch rotating shellac discs for sound replay from 1889, with 10-inch records appearing in 1901. In 1903 12-inch discs were introduced by Victor, on their Deluxe label, these able to play for up to four minutes, so increasing what were non-achievable times of the length of a song or speech on the earlier formats. These first twelve-inch releases were all by the Victor Grand Concert Band, led by Frederick W. Hager. During the 1910s discs became the standard sound reproduction format, although the speeds used could vary between manufacturers until 78 rpm became the norm from around 1925. An album would consist of several of these single discs packaged together. These brittle shellac discs remained a popular medium through the first attempt to introduce vinyl records in 1931, the subsequent move towards microgroove formats from 1948, and would survive until the early 1960s.

Vinyl and microgroove formats
In August 1931, RCA Victor launched the first commercially available vinyl long-playing record, marketed under their Program Transcription series. These revolutionary discs were designed for playback at 331⁄3 rpm (the speed first used on 16‑inch Vitaphone soundtrack discs from 1926) and pressed on ten and twelve-inch diameter flexible Victrolac discs, using a polyvinyl chloride (PVC) compound called vinylite which was licensed from Union Carbide. It was marketed initially as being more robust than shellac, with a duration of up to twenty minutes playing time per side. The first twelve-inch LP (containing only one track per side), was Beethoven's Symphony No. 5 In C Minor by the Philadelphia Orchestra conducted by Leopold Stokowski. RCA's early introduction of a long-play disc was a commercial failure for several reasons including the lack of affordable, reliable consumer playback equipment and consumer wariness during the Great Depression. Because of financial hardships that plagued the recording industry during that period (and RCA's own parched revenues), Victor's long-playing records were discontinued for public sale by early 1933.

However, vinyl continued to be used, notably with broadcasters, on larger 16‑inch radio transcription discs, and later with the V-Disc program that sent records overseas to US troops during World War II to help boost morale. Vinyl as a material for records sold to the public was reintroduced after World War II, first for 78s in 1945 (the Bolshoi Theatre Orchestra and Chorus's Prince Igor a 12‑inch 5-record set from Asch Recordings catalog number M-800), with 12‑inch 331⁄3 rpm LPs (with a narrower "microgroove") and 7‑inch 45 rpm singles being marketed by Columbia Records and RCA Victor respectively in 1948 and 1949. Intriguingly, although there is a scientific basis behind 45 rpm being an ideal speed for sound quality which was touted by RCA, conveniently the 45 rpm number can also be derived by subtracting 33 from 78.

More lengthier tracks could be accommodated on each side of the 12‑inch disc, or shorter but more numerous tracks per side. Indeed, the first 12‑inch vinyl album in 1948, Mendelssohn's Concerto in E Minor featured only three tracks (one 11-minute long suite on side one, and two tracks on side two taking up 14 minutes in total) and was extended play (EP) in appearance, rather than a typical album with multiple tracks each being on average 2–4 minutes such as the first 10‑inch vinyl album, a reissue of The Voice of Frank Sinatra. While one to two short playing songs being sold to the public were more suited for the seven‑inch 45 rpm record, the LP could be anything from 1 track per side, and if in that simplistic configuration it technically could be classed as a twelve‑inch single. Over the ensuing years some works, particularly in the classical and jazz genres, and the relatively few tracks on an occasional album could blur the boundaries of what would become considered a twelve‑inch single, an EP and an album, with the price, catalogue number, any stylistic aims of the performing artist, marketing by the record label, as well as record industry sales charts rules regulating the differences between formats.

Jamaican roots
The gramophone records cut especially for dance-floor DJs came into existence with the advent of recorded Jamaican mento music in the 1950s. By at least 1956 it was already standard practice by Jamaican sound systems owners to give their "selecter" DJs acetate or flexi disc dubs of exclusive mento and Jamaican rhythm and blues recordings before they were issued commercially.

Pre-disco
In March 1970, Cycle/Ampex Records test-marketed a twelve-inch single by jazz-pop guitarist Buddy Fite, featuring "Glad Rag Doll" backed with "For Once in My Life", both from his self-titled debut album issued in 1969. Subtitled 'The world's first 12-inch single!', the experiment aimed to energize the struggling singles market, offering a new option for consumers who had stopped buying traditional singles. The record was pressed at 33 rpm, with identical run times to the seven-inch 45 rpm pressing of the single and album, but with a large runoff area. Several hundred copies were made available for sale for 98 cents each at two Tower Records stores in California.

Shelter Records evidently liked the format enough to use it a few times to promote artists in the US and Australia - they serviced a test pressing of Leon Russell's "It's A Hard Rain's Gonna Fall" backed with "Me And Baby Jane" to radio stations in July 1971, the first track from his Leon Russell and the Shelter People album, with the flip side from his Carney album. Shelter later issued another promotional single "Lowdown in Lodi"/"Me and My Guitar" by Freddie King in 1972 with the tracks taken from his Texas Cannonball LP.

Another early twelve-inch single was released in 1973 by soul/R&B musician/songwriter/producer Jerry Williams, Jr. a.k.a. Swamp Dogg. Twelve-inch promotional copies of "Straight From My Heart" were released on his own Swamp Dogg Presents label (Swamp Dogg Presents #501/SDP-SD01, 33+1⁄3 r.p.m.), with distribution by Jamie/Guyden Distribution Corporation. It was manufactured by Jamie Record Co. of Philadelphia, Pennsylvania. The B-side of the record is blank.

Discotheque era
The disco twelve-inch record came into being from several key developments:

 * using the instrumental or backing track with the vocal version to create an extended version, and later on combining outtakes, adlibs and other unused material;
 * creating a segued mix of different tracks, the length of which was less suited to a 7-inch record;
 * using physically larger and audibly 'hotter' records;
 * progressively increasing the awareness and availability of the format through trade publications, record shops and DJ record pools;

Notably, Tom Moulton, and Scepter Records along with its production chief Mel Cheren (later co-founder of dance label West End Records) were involved in several of these pioneering steps due to their artists and heavy bass-range and strong uptempo song material, forward-thinking company executives and innovative remixing.

First instrumental side
This was released by Scepter to the flip side of the Ultra High Frequency "We're on the Right Track" seven-inch single in 1973. This issuing of the backing track enabled DJ's to use two turntables and mix between the vocal and instrumental versions of popular dance records, seguing them so that dancing patrons could enjoy the overall song for longer. A key reason of the time was that few varispeed turntables existed, so DJ's would only blend into a track with the same drum tempo or BPM, which the instrumental naturally was. This was in comparison to the then occasionally found 'Part 2' B-side of a record, while similar, tended to be the latter half of a lengthy album track, split for seven-inch release, but in many cases this wasn't the full instrumental, so could be more awkward to use. Scepter, from this single onward began to regularly add a non-vocal side, helping to make it an industry standard practice, with several other labels following suit particularly for dance records, and DJs embracing these as a useful tool during their sets.

First extended edit
While not his first production work (his first mixing effort was the northern soul track by the Carstairs "It Really Hurts Me Girl" in 1973), in early 1974, during his quest to adapt songs beyond the radio-friendly three-minute mark for his mixtapes, soon-to-be famed disco mixer Tom Moulton went to record labels for material. At Scepter Records, Cheren recalls playing Tom a previously released Scepter single by singer Don Downing called "Dream World". He had an extra copy of the master tape and let Moulton take it home to experiment. When Moulton brought it back a few days later, Cheren writes, "We were amazed: a so-so record was suddenly snappy, upbeat, and ten times better". But the biggest surprise, Cheren continues, was something "so radical I could hardly believe my ears". Moulton had stretched the original track, not even three minutes long, to almost double its time, and in the process debuted what would become known as the disco break. This innovation would eventually be issued on the song's re-release on a 7-inch 45 in July 1974, and earn Scepter a Billboard Trendsetter of the Year award in December 1974 for 'being the first label to make specialized mixes for discotheques'.

First segued mixes
The first genre-themed set was created by Spring Records, as a commercial compilation of various artists licensed from a number of labels called Disco Par-r-r-ty, released in October 1974. Tom Moulton created one for the A side of the Gloria Gaynor Never Can Say Goodbye album, in January 1975 on MGM Records. As a development from his mixtapes, three songs (Honey Bee/Never Can Say Goodbye/Reach Out, I'll Be There) were not presented as separate pieces of music, but as an uninterrupted, 18-minute-plus, side-long composition optimised for dancing. There were previous albums that had side-long tracks or suites, and medleys which were usually cover versions or re-recordings, but here was a new piece of music composed out of already recorded pieces of music (albeit containing two cover songs), making it the first "DJ mix" committed to vinyl, with Moulton admitting it was conceived as a tool for dancefloors. A followup was the Motown compilation Disc-O-Tech series released mid May 1975, which put together some of the label's most danceable hits onto a number of albums. Disc-O-Tech #2 however, specifically focused on blending a number of their disco releases into a non-stop medley.

Early acetates 10-inch and 12-inch
The first large-format single made specifically for discotheque DJs was a ten-inch acetate used by a mix engineer (José Rodríguez) in need of a Friday-night test copy for a remix created by Tom Moulton in 1974. The song was "I'll Be Holding On" by Al Downing, brother of Don Downing. As no 7-inch (18 cm) acetates could be found, a 10-inch (25 cm) blank was used. Upon completion, Moulton found that such a large disc with only a couple of inches worth of grooves on it made him feel silly wasting all that space. He asked Rodríguez to re-cut it so that the grooves looked more spread out and ran to the normal center of the disc. Rodriguez told him that for it to be viable, the level would have to be increased considerably. Because of the wider spacing of the grooves, not only was a louder sound possible but also a wider overall dynamic range (distinction between loud and soft) as well. This was immediately noticed by them to give a more favorable, 'hotter' sound which would appeal to discothèque play. It also meant that these extended versions being created by Moulton could be given to fellow DJs and tested within a nightclub environment to see how well it worked the dancefloor, with adjustments subsequently made to the remix.

Moulton's position as the premiere mixer and "fix it man" for pop singles ensured that this fortunate accident would instantly become industry practice. This would perhaps have been a natural evolution: as dance tracks became much longer than had been the average for a pop song, and as the DJ in the club wanted sufficient dynamic range, the format would likely have enlarged from the seven-inch single eventually. Ironically, Moulton's mix of Downing would be eventually released by Chess Records for sale to the general public, but only on a standard-issue 7-inch record for sale in October 1974. Further 10-inch acetates featuring the extended versions would be created by Moulton and Rodriguez from late 1974 such as Moment of Truth's "Your Love" on Roulette in October and the aforementioned Don Downing "Dream World". "I'll be Holding On" would eventually be cut onto a twelve-inch acetate, as chronicled by its usage on the music compilation album A Tom Moulton Mix.

An acetate twelve-inch test pressing single hailed as being a first by Moulton was South Shore Commission "Free Man". Again, it was a disco mix prepared by Moulton, pressed by Rodríguez, with only a handful (around 10) cut and handed out to local DJs. These would have plain white labels, or very sparse printing or typed text. In many cases there would be no logos, and many contained handwritten text only. Test pressings being tried out at discos were reported on in Moulton's weekly column in Billboard during early April 1975. Moulton's effort was eventually issued for commercial release by Scepter in June 1975, but on seven-inch vinyl as a 5:35 min mix. Moulton subsequently created a lengthier 7:15 min mix for the album which appeared in late October.

Another record remixed by Moulton was the Philly Devotions "I Just Can't Make It Without You" on Columbia. Mentioned in his Billboard column in mid April as being 'out soon', a known 10-inch acetate had a date of 8 May 1975, with twelve-inch acetates and promos also appearing at some stage. It eventually was issued commercially in August on seven-inch 45 only. A sometimes mentioned candidate among these first acetates is Moment of Truth "So Much For Love", but this effort was commercially released a year later, the band having been signed to Salsoul Records in June 1976, and the song only appearing in disco charts at the end of July of that year.

Promotion through media and DJs
Many of the above disco era timelines were driven by the DJ necessity to give a better nightclub dancefloor experience to patrons, and as the scene grew, it began to be chronicled in trade press publications such as Billboard and Record World. Tom Moulton began to write for the former from 26 October 1974 in the Disco Action column (which changed name as time went on), while Vince Aletti wrote Disco File in the latter from November 1974. The DJs would increasingly be expected to report back, much like with radio, with what songs worked on their dancefloors to the record company and mixers such as Moulton and others so that a strategic decision would be made whether to further fine tune or remix the music to enhance the reaction, typically a new edit would be repeatedly created, pressed on acetates and supplied until a good response was had from nightclubs, so creating a buzz which would drive the eventual commercial sales. To better feed those reactions back record pools were established, the first in New York in June 1975 for better distribution of pre-release records to bona fide DJs as record labels began to appreciate their role in breaking and selling records. A further development began to see the labels tailor the records specifically towards DJs as a result, with for example Scepter Records publicly announcing via both publications that they were to start servicing '12-inch 45s' to DJs in pools and their national promotional mailouts from June 1975. promoting the format earlier in distributor roadshows they hosted in late May, citing a key benefit of high volume levels being maintained with lengthier tracks. Billboard magazine reported that Atlantic Records were the first major label to issue 12-inch 33 rpm vinyl promos to DJs in July, under the watchful eye of a DJ as their promotions director Doug Riddick, who took up the post in May 1975.

Early vinyl test pressings 12-inch
An early test pressing was Bobby Moore "(Call Me Your) Anything Man", on Scepter. Twelve-inch acetates for this single were pressed in April 1975, and was subsequently produced as twelve-inch vinyl promotional singles with typed labels in June. Moulton, now a regular in-house remixer for the label, was again given label credit for the remix. This was released commercially on 7-inch in May 1975.

First wide scale promotional 12-inch
A very early disco prototype vinyl 12-inch pressing was a Midland International promo distributed by RCA - this was a Carol Douglas one sided EP with 4 tracks from The Carol Douglas Album. This was issued in mid February 1975 and was subtitled 'Specially Prepared For Disco Use', but it held same length versions of the selected album tracks.

The first wide-scale record company promotional twelve-inch single according to Moulton (considering his then position with Billboard at the time as disco product reviewer, and that most of the very limited 12-inch records up to this point involved his own remixes), was Frankie Valli "Swearin' To God", issued by Private Stock Records in June 1975 with a 10:32 min running time. Bob Crewe, co-writer and producer, personally presented 10-inch test pressings to DJs in April 1975 after high pre-release demand. It was then issued commercially as a 7-inch at the end of April/start of May 1975.

Barrabas "Mellow Blow" became the first 33 rpm Atlantic promo to be released in July,  but eventually commercially only on an 7" in September. Warner-Spector's Calhoon "(Do You Wanna) Dance, Dance, Dance" had a 10-inch acetate from May, given out as a one sided 10-inch vinyl promo in the same month, and as a 12-inch promo in July 1975.

At first, these special remixed or extended versions were only available as promotional copies to DJs, either given directly to them or issued by record pools who obtained these from record companies' A&R departments. Examples of these promos, released from spring/summer 1975, include:

Motivation for public sale
As time went on, a growing number of record labels became aware of the 12-inch format as a useful promotional tool, the benefits it gave for sound fidelity, and started to issue product in response. However, into early 1976 none considered them at first as suitable for sale to the general public. Companies came to appreciate the place of the nightclub and how they helped to break a record, but still considered an extended remix to ultimately facilitate sales of the original 7-inch single version or the artist's album, and not as a sale item in its own right. Pop orientated labels began to use the format to promote commercial artists with dance elements to their music, but not necessarily lengthening their tracks, concentrating instead on its novel aspects. The costs for the format were also still prohibitive; one label reported 12-inch singles cost more than it did to press an album.

However, demand was being driven by record shops, particularly those serving disco product, with feedback from buyers. It was particularly noticed that many of the previously released promotional 12-inch singles were attracting a premium in the resellers marketplace, with them in some cases changing hands for the price of an album. Along with research done by labels to get a feel of the interest for the proposed format, a push was finally made by a self-proclaimed 'pioneer' label of the disco business.

Releases
The first song found on a twelve-inch single commercially issued for public purchase from the disco era onwards was "Ten Percent" by Double Exposure on Salsoul Records in mid May 1976. This was due to the label co-owner Ken Cayre's decision to release the Walter Gibbons remix for dancers who would hear that version in nightclubs but not be satisfied with the shorter versions sold on the 7-inch vinyl or the album. The seven-inch edit had been released a month earlier but sales of this were slow. With renewed interest and high sales weeks after the release, Salsoul were presented with Billboard awards at the 1976 Disco Forum event for disco disk sales to consumers and best record label as a result of the push into the marketplace with the new format. The issue did not contain the 3 min 5 secs 7-inch edit or the 6 mins 51 mins album version; the A side contained the Gibbons 9 min 43 secs remix, while Cayre himself created an extended 7 min 31 secs mix for the B side.

The second twelve-inch release was the double artist single Jesse Green "Nice And Slow" / Sweet Music "I Get Lifted" on Scepter/Wand in mid June 1976. Third was a disc containing two George Benson sides "Summertime 2001" / "Theme From Good King Bad" in late June 1976 on CTI Records. Salsoul's second 12-inch record for retail was by Moment of Truth "So Much For Love" / "Helplessly" released in early July 1976. Amherst Records released a double headed single by the Chicago Gangsters "Gangster Love" / "Feel Like Making Love" in early July 1976 on their Gold Plate subsidiary. A clutch of releases including Jakki "Sun... Sun... Sun" and Four Below Zero "My Baby's Got E.S.P.", amongst others on Pyramid/Roulette Records was sold from mid July 1976.

Australia
Two twelve-inch promotional vinyl issues for rock/folk Shelter Records artists were serviced to radio stations in April 1972 by local distributor Tempo Records:

France
Early titles first appeared in 1976 on the Pathe Marconi EMI and Disques Vogue imprints, these included pop releases such as Paul McCartney's first ever extended single internationally, credited as Wings "Let 'Em In"

Jamaica 'discomixes'
Although as previously mentioned regarding the Jamaican dub influence on the use of 10-inch acetates, the 12-inch single was only born once imports of US disco singles were established from 1976 onwards. The issued twelve-inch-single trend spread to Jamaica quickly, where hundreds of reggae singles were pressed in this format, and commercially issued as "discomixes" to catch on the disco hype. An early 12-inch pressing, reputedly the first was by the Jayes "Truly" in 1977 on the Channel One label, which was run by the Hoo Kim brothers from their Channel One Studios.

United Kingdom
The Jamaican reggae and US disco trend also hit London, reggae being popular along with uptempo forms of music such as Motown and northern soul, the seven-inch record being the primary medium in the early 1970s for this material, with the UK following up a little later than the US with 12-inch singles. The reasons were different, the UK jocks did not have the same need to extend records like the US pioneers who wanted longer records for the dancefloor. Although the use of larger temporary singles (primarily 10-inch) started from the Jamaican influence and before (such as the pre-Beatles band the Quarrymen with the one-off "In Spite of All the Danger" in 1958), acetates were also used by the record labels to quality control the eventual product, and not for servicing single songs or exclusive remixes, and then not in the 12-inch format. The usage of the 12-inch vinyl as a medium followed the US promos introduction but was initially seen as a marketing tool to help promote an artist more uniquely. Another emphasis with the new format was the louder sound and better audio quality afforded to the release. It therefore was not exclusively used for disco songs but included pop artists, however it eventually came into its own in the later 1970s with the lengthened versions of US disco songs being promoted in the UK.

Promo UK 12-inch releases
Atlantic Records was an early front runner with two 12-inch promo singles: Ben E. King "Supernatural Thing" backed with Osiris "Warsaw Concerto", along with Herbie Mann "Hijack" b/w Jimmy Castor Bunch "The Bertha Butt Boogie", both at 33 rpm and issued in approximately June 1975 (based on the catalogue numbers used), but rumoured to be as late as October. Robert Palmer "Which of Us Is the Fool" was released by Island Records also in October 1975. Virgin started a line of 12-inch promos in November 1975, with the first being Ruan O'Lochlainn "Another Street Gang". Disco singles started to appear in earnest months later, Brass Construction "Changin'" was promoted around March 1976 by United Artists. A later 12-inch promo issue was a double sider the Moments "Nine Times" / the Rimshots "Do What You Feel" on All Platinum Records via Phonogram in the middle of April 1976, however both were released commercially and individually on seven-inch 45s only, in April 1976. Candi Staton followed with "Young Hearts Run Free" in the middle of May from Warner Bros. These early issues usually containing the original 7-inch edit, It took a little later for lengthened versions to begin appearing, with 1970s UK club DJ Greg Wilson recalling promotional 12-inch product being mailed out from August 1976, Lalo Schifrin "Jaws" being his first one, which was in extended form. This was followed by disco acts such as James Wells, the Originals, Ultrafunk, Mass Production, Deodato and the Undisputed Truth, however some of these were not UK pressed vinyl but US promos sent over to the UK and distributed through club promotions businesses and record company A&R departments.

First UK 12-inch retail singles
The first commercially released twelve-inch vinyl was Ernie Bush "Breakaway" / Banzaii "Chinese Kung Fu" both as Tom Moulton mixes, along with another disc containing the Armada Orchestra "For the Love of Money" / Ultrafunk "Sting Your Jaws (Part 1)". Bush and the latter two acts had Gerry Shury production involvement, and these two releases were issued by John Abbey's Contempo Records from 8 October 1976, these songs having been previously released in either 7-inch format or as album tracks. Abbey likely had the nod from Scepter Records about the use of the twelve-inch single format, as both had released all these titles on their labels and Contempo had cross-licensed the tracks with Moulton mixes. This was closely followed with a single containing re-issues of the Who's "Substitute" with "I'm A Boy" / "Pictures of Lily" on the flip, all originally from 1966 and 1967 by Polydor Records on 22 October 1976.

Core sales period
Notable 1970s and 1980s releases making use of the new length opportunities of the format included Donna Summer's "Love To Love You Baby" (16 min 50 seconds), "I Feel Love" (15:45), and Sugarhill Gang's "Rapper's Delight" (15:00). The broad visual spacing of the grooves on the twelve-inch records made it easy for the DJ in locating the approximate area of the "breaks" on the disc's surface in dim club light (without having to listen while dropping and re-dropping the stylus to find the right point). A quick study of any DJs favorite discs will reveal mild wear in the "break points" on the discs' surfaces that can clearly be seen by the naked eye, which further eases the "cueing" task (a club DJs tone-arm cartridge will be heavily weighted and mild wear will seldom spoil the sound quality). Many DJ-only remix services, such as Ultimix and Hot Tracks, issued sets with deliberately visualised groove separations (i.e., the record was cut with narrow and wider spacings that could be seen on the surface, marking the mix points on the often multi-song discs). Motown were one of the first to "eye cue" their 12-inch disco discs, giving DJs the track's BPM and info on the exact length of the various sections of the song - one of the earliest examples of a record company recognising how important the DJ was to become by making their product more user-friendly.

Following the lead of the US club DJs, using 12-inch extended versions in the UK as a mixing tool was advocated particularly by James Hamilton of the Record Mirror music weekly paper, with him notably indicating the approximate BPM of late 1970s disco tracks onwards. Increasingly in the 1980s, many pop and even rock artists released twelve-inch singles that included longer, extended, or remixed versions of the actual track being promoted by the single. These versions were frequently labeled with the parenthetical designation "12-inch version", "12-inch mix", "extended remix", "dance mix", or "club mix", before its usage becoming commonplace in the early/mid 1980s developing alongside turntabalism in use for the electro and hip-hop genres, but was regularised with the advent of the late 1980s house music scene. Later musical styles took advantage of this new format and recording levels on vinyl twelve-inch "maxi-singles" have steadily increased, culminating in the extremely loud (or "hot") cuts of drum and bass records of the 1990s and early 2000s.

Many record labels produced mainly twelve-inch singles (in addition to albums) during the 1980s, lots being mostly regular A and B-sides, not remixes. Certain labels such as Factory Records, only ever released a handful of seven-inch singles. One of Factory's resident artists, alternative rock/dance quartet New Order, produced the biggest-selling twelve-inch record ever in the United Kingdom, "Blue Monday", selling about 800,000 copies on the format and over a million copies in total (not counting later remixes). It was somewhat helped by the fact that Factory did not release a seven-inch version of the single until 1988, five years after the single was originally released as a twelve-inch-only release. Besides, the seven-inch version that was released was not the original 1983 version released on twelve-inch, but a re-recording called "Blue Monday 1988".

Decline and ongoing interest
Singles have followed wider vinyl format sales, withstanding competition from the 1960s and 70s reel-to-reel tape, the 8-track cartridge, and the compact cassette formats. The widespread popularity of Sony's Walkman was a key factor that contributed to vinyl's lessening usage in the 1980s. In 1988, the compact disc surpassed the gramophone record in unit sales. Vinyl records experienced a sudden decline in popularity between 1988 and 1991, when the major label distributors restricted their return policies, which retailers had been relying on to maintain and swap out stocks of relatively unpopular titles. Record companies also deleted many vinyl titles from production and distribution, or simply did not make 12-inch singles for many pop artists, further undermining the availability of the format and leading to the closure of pressing plants. This rapid decline in the availability of records accelerated the format's decline in popularity, and is seen by some as a deliberate ploy to make consumers switch to CDs which were at the time more profitable for the record companies,   and more latterly, streaming.

12-inch singles have continued for sales of dance acts or for dance remixes of commercial artists as there was a continued high regard of the format from DJs into the 2000s and 2010s. A growing number of DJs eventually began to use CDJs for their convenience, and later along with a crossover period where turntables could be combined with laptops and used with encoded 12 inch discs and DJ software, which could manipulate MP3 or WAV music files but still allow for a turntablism experience. DJ controller all-in-one decks have in later times become the norm which take up less space than a pair of turntables, reducing DJs dependence on the physical format even further. There is however, a dedicated DJ sub-community that maintain their usage of the format, with retro styled 'vinyl only' nights being a unique selling point. Also, there are some new titles being pressed on the format and available at physical record shops alongside an ongoing cultural vinyl revival, although many sales take place online. There is also a notable second-hand trade business on online sale and auction marketplaces for collectors, of which some titles are still in demand and can be of some value.

title: Twelve-inch single


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Reflection (mathematics)

In mathematics, a reflection (also spelled reflexion) is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as a set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection. For example the mirror image of the small Latin letter p for a reflection with respect to a vertical axis (a vertical reflection) would look like q. Its image by reflection in a horizontal axis (a horizontal reflection) would look like b. A reflection is an involution: when applied twice in succession, every point returns to its original location, and every geometrical object is restored to its original state.

The term reflection is sometimes used for a larger class of mappings from a Euclidean space to itself, namely the non-identity isometries that are involutions. Such isometries have a set of fixed points (the "mirror") that is an affine subspace, but is possibly smaller than a hyperplane. For instance a reflection through a point is an involutive isometry with just one fixed point; the image of the letter p under it would look like a d. This operation is also known as a central inversion, and exhibits Euclidean space as a symmetric space. In a Euclidean vector space, the reflection in the point situated at the origin is the same as vector negation. Other examples include reflections in a line in three-dimensional space. Typically, however, unqualified use of the term "reflection" means reflection in a hyperplane.

Some mathematicians use "flip" as a synonym for "reflection".

Construction

In a plane (or, respectively, 3-dimensional) geometry, to find the reflection of a point drop a perpendicular from the point to the line (plane) used for reflection, and extend it the same distance on the other side. To find the reflection of a figure, reflect each point in the figure.

To reflect point $P$ through the line $AB$ using compass and straightedge, proceed as follows (see figure):

 * Step 1 (red): construct a circle with center at $P$ and some fixed radius $r$ to create points $A′$ and $B′$ on the line $AB$, which will be equidistant from $P$.
 * Step 2 (green): construct circles centered at $A′$ and $B′$ having radius $r$. $P$ and $Q$ will be the points of intersection of these two circles.

Point $Q$ is then the reflection of point $P$ through line $AB$.

Properties
The matrix for a reflection is orthogonal with determinant −1 and eigenvalues −1, 1, 1, ..., 1. The product of two such matrices is a special orthogonal matrix that represents a rotation. Every rotation is the result of reflecting in an even number of reflections in hyperplanes through the origin, and every improper rotation is the result of reflecting in an odd number. Thus reflections generate the orthogonal group, and this result is known as the Cartan–Dieudonné theorem.

Similarly the Euclidean group, which consists of all isometries of Euclidean space, is generated by reflections in affine hyperplanes. In general, a group generated by reflections in affine hyperplanes is known as a reflection group. The finite groups generated in this way are examples of Coxeter groups.

Reflection across a line in the plane
Reflection across an arbitrary line through the origin in two dimensions can be described by the following formula
 * $$\operatorname{Ref}_l(v) = 2\frac{v \cdot l}{l \cdot l}l - v,$$

where $$v$$ denotes the vector being reflected, $$l$$ denotes any vector in the line across which the reflection is performed, and $$v\cdot l$$ denotes the dot product of $$v$$ with $$l$$. Note the formula above can also be written as
 * $$\operatorname{Ref}_l(v) = 2\operatorname{Proj}_l(v) - v,$$

saying that a reflection of $$v$$ across $$l$$ is equal to 2 times the projection of $$v$$ on $$l$$, minus the vector $$v$$. Reflections in a line have the eigenvalues of 1, and −1.

Reflection through a hyperplane in n dimensions
Given a vector $$v$$ in Euclidean space $$\mathbb R^n$$, the formula for the reflection in the hyperplane through the origin, orthogonal to $$a$$, is given by

 * $$\operatorname{Ref}_a(v) = v - 2\frac{v\cdot a}{a\cdot a}a,$$

where $$v\cdot a$$ denotes the dot product of $$v$$ with $$a$$. Note that the second term in the above equation is just twice the vector projection of $$v$$ onto $$a$$. One can easily check that
 * $Ref_{a}(v) = −v$, if $$v$$ is parallel to $$a$$, and
 * $Ref_{a}(v) = v$, if $$v$$ is perpendicular to $Q$.

Using the geometric product, the formula is

 * $$\operatorname{Ref}_a(v) = -\frac{a v a}{a^2} .$$

Since these reflections are isometries of Euclidean space fixing the origin they may be represented by orthogonal matrices. The orthogonal matrix corresponding to the above reflection is the matrix

 * $$R = I-2\frac{aa^T}{a^Ta},$$

where $$I$$ denotes the $$n \times n$$ identity matrix and $$a^T$$ is the transpose of a. Its entries are

 * $$R_{ij} = \delta_{ij} - 2\frac{a_i a_j}{ \left\| a \right\| ^2 },$$

where $δ_{ij}$ is the Kronecker delta.

The formula for the reflection in the affine hyperplane $$v\cdot a=c$$ not through the origin is
 * $$\operatorname{Ref}_{a,c}(v) = v - 2\frac{v \cdot a - c}{a\cdot a}a.$$

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Reflection (mathematics)


text:
Young Communist League USA

The Young Communist League USA (YCLUSA) is a communist youth organization in the United States. The stated aim of the League is the development of its members into Communists, through studying Socialism and through active participation in the struggles of the American working class. The YCL recognizes the Communist Party USA as the party for socialism in the United States and operates as the Party's youth wing. Although the name of the group changed a number of times during its existence, its origins trace back to 1920, shortly after the establishment of the first communist parties in the United States.

Although independent, in its final years the organization came under direct control of the CPUSA. After a backlash by members towards the suspension of elections and ideological shifts towards the right, membership plummeted. On November 14, 2015, the CPUSA's National Committee voted to suspend funding to the Young Communist League and the organization was subsequently dissolved.

In 2019, at the 31st party convention, the organization was re-established.

The underground period
As early as 1920, a skeleton of a "Young People's Communist League" was in existence. This minuscule, largely paper organization sent a fraternal delegate to the 2nd Convention of the United Communist Party, held at Kingston, New York from December 24, 1920, to January 2, 1921. A report was delivered by this delegate on the youth situation in America and the convention at this time first decided to establish a serious youth section, to be called the Young Communist League. The resolution passed by the convention pledged the UCP would provide its youth section assistance by helping to produce and distribute its literature, by helping to gain control of existing units of the Independent YPSL and organizing them into communist groups, by helping to organize new units, by providing it financial assistance, by lending it speakers and teachers, and by allotting it space in the official party periodicals.

This did not mean that there was no national convention of the organization. The founding convention of the YCL was held early in May 1922, apparently in Bethel, Connecticut. It was a small and low key gathering, including just fourteen delegates from four of the Communist Party's twelve national districts. The gathering heard a report from Max Bedacht of the adult party dealing with the discussions and decisions of the 3rd World Congress of the Communist International and its February 1922 special conference. The convention adopted a constitution and a program for the YCL, as well as a resolution delineating the relationship of the youth league with the adult party. A governing National Executive Committee of five members was elected. The initiation fee to join the YCL was 50 cents and dues were 25 cents per month, receipted with stamps issued by the National Office. The basic unit of organization was the "group" consisting, ideally, of from five to ten members and meeting at least every other week. Groups elected their own captains to coordinate their activities with the center. Multiple groups were parts of a "section" of up to five groups; multiple sections were part of a "sub-district," which was in turn a subdivision of the regular geographic "districts" of the Communist Party.

Establishment of the "overground" organization

As a YWL leader, Nat Ganley (then still using his birth name Nat Kaplan) summarized the difference between his communist group and others by stating: "Let us remember that is it mainly on this point that we differ from the old form of child organization – the worker's Sunday schools.  We are not only preparing the child for future participation in class struggle–we are leading the child in the class struggle now!"

The founding convention of the YWL was held in Brooklyn, New York from May 13 to 15, 1922, held appropriately enough at Finnish Socialist Hall. Oliver Carlson delivered the keynote speech to the thirty regular and five fraternal delegates. Carlson claimed a presence for the nascent YWL in forty-six cities and a membership of "at the very least," 2,200.

The basic unit of organization of the YWL was the "branch," consisting of at least five but no more than one hundred and fifty members. Two or more branches in a single large city were to form a "City Central Committee" to coordinate their activities, and all units were to be part of the regular array of districts used by the adult party. The initiation fee was 25 cents and dues 25 cents per month, with all initiation fees and 10 cents of every month's dues going to support the National Office.

The depression decade and after
On October 17, 1943, the YCL convened in national convention in New York City, passed a resolution dissolving itself, and immediately reconvened under a new organizational name, the American Youth for Democracy (AYD). This predated a similar move transforming the adult Communist Party, USA into the "Communist Political Association" by seven months. The change of names proved to be strictly semantic, as all important positions within the "new" AYD were held by former members of the YCL. National Executive Secretary of the AYD at the time of its October 1943 launch was Carl Ross, the former head of the YCL for a period of more than five years.

The CPUSA reestablished a youth organization in 1949 as the Labor Youth League, which dissolved in the dissension following the Hungarian Revolution of 1956 and the 20th Congress of the CPSU. In 1965, after a period of mainly local activity, the DuBois Clubs were formed and later renamed the Young Workers Liberation League before reaffirming the original name Young Communist League in 1984.

Re-establishment
In 2019, a resolution at the 31st National Convention of the CPUSA was passed which called to re-establish the Young Communist League.

title: Young Communist League USA


text:
Alogia

In psychology, alogia (from Greek ἀ-, "without", and λόγος, "speech" + New Latin -ia) is poor thinking inferred from speech and language usage. There may be a general lack of additional, unprompted content seen in normal speech, so replies to questions may be brief and concrete, with less spontaneous speech. This is termed poverty of speech or laconic speech. The amount of speech may be normal but conveys little information because it is vague, empty, stereotyped, overconcrete, overabstract, or repetitive. This is termed poverty of content or poverty of content of speech. Under Scale for the Assessment of Negative Symptoms used in clinical research, thought blocking is considered a part of alogia, and so is increased latency in response.

This condition is associated with schizophrenia, dementia, severe depression, and autism. As a symptom, it is commonly seen in patients with schizophrenia and schizotypal personality disorder, and is traditionally considered a negative symptom. It can complicate psychotherapy severely because of the considerable difficulty in holding a fluent conversation.

The alternative meaning of alogia is inability to speak because of dysfunction in the central nervous system, found in mental deficiency and dementia. In this sense, the word is synonymous with aphasia, and in less severe form, it is sometimes called dyslogia.

Characteristics
Alogia may be on a continuum with normal behaviors. People without mental illness may have it occasionally including when fatigued or disinhibited, when writers use language creatively, when people in certain disciplines—such as politicians, administrators, philosophers, ministers, and scientists—use language pedantically. Hence, deciding if an individual has alogia depends on contextual clues. Is the person in control? Can the person moderate the effect if asked to be specific or concise? Is it better with another topic? Are there other significant symptoms?

Alogia is characterized by a lack of speech, often caused by a disruption in the thought process. Usually, an injury to the left side of the brain may cause alogia to appear in an individual. While in conversation, alogic patients will reply very sparsely and their answers to questions will lack spontaneous content; sometimes, they will even fail to answer at all. Their responses will be brief, generally only appearing as a response to a question or prompt.

Apart from the lack of content in a reply, the manner in which the person delivers the reply is affected as well. Patients affected by alogia will often slur their responses, and not pronounce the consonants as clearly as usual. The few words spoken usually trail off into a whisper, or are just ended by the second syllable. Studies have shown a correlation between alogic ratings in individuals and the amount and duration of pauses in their speech when responding to a series of questions posed by the researcher. The inability to speak stems from a deeper mental inability that causes alogic patients to have difficulty grasping the right words mentally, as well as formulating their thoughts. A study investigating alogiacs and their results on the category fluency task showed that people with schizophrenia who exhibit alogia display a more disorganized semantic memory than controls. While both groups produced the same number of words, the words produced by people with schizophrenia were much more disorderly and the results of cluster analysis revealed bizarre coherence in the alogiac group.

If the condition is assessed using a language other than the individual's primary language, the medical professional needs to make sure that the problem is not from language barriers.

This condition is associated with schizophrenia, dementia, and severe depression.

Example
The following table shows an example of "poverty of speech" which shows replies to questions that are brief and concrete, with a reduction in spontaneous speech:

The following example of "poverty of content of speech" is a response from a patient when asked why he was in a hospital. Speech is vague, conveys little information, but is not grossly incoherent and the amount of speech is not reduced. "I often contemplate—it is a general stance of the world—it is a tendency which varies from time to time—it defines things more than others—it is in the nature of habit—this is what I would like to say to explain everything."

Causes
Alogia can be brought on by frontostriatal dysfunction which causes degradation of the semantic store, the center located in the temporal lobe that processes meaning in language. A subgroup of chronic schizophrenia patients in a word generation experiment generated fewer words than the unaffected subjects and had limited lexicons, evidence of the weakening of the semantic store. Another study found that when given the task of naming items in a category, schizophrenia patients displayed a great struggle but improved significantly when experimenters employed a second stimulus to guide behavior unconsciously. This conclusion was similar to results produced from patients with Huntington's and Parkinson's disease, ailments which also involve frontostriatal dysfunction.

Treatment
Medical studies conclude that certain adjunctive drugs effectively palliate the negative symptoms of schizophrenia, mainly alogia. In one study, Maprotiline produced the greatest reduction in alogia symptoms with severity reduction in 50% of patients (out of 10). Of the negative symptoms of schizophrenia, alogia had the second best responsiveness to the drugs, surpassed only by attention deficiency. D-amphetamine is another drug that has been tested on people with schizophrenia and found success in alleviating negative symptoms. This treatment, however, has not been developed greatly as it seems to have adverse effects on other aspects of schizophrenia such as increasing the severity of positive symptoms.

Relation to schizophrenia
Although alogia is found as a symptom in a variety of health disorders, it is most commonly found as a negative symptom of schizophrenia.

Previous studies and analyses conclude that at least three factors are needed to cover both the positive and negative symptoms of schizophrenia; the three are: psychotic, disorganization, and negative symptom factors. Studies suggest that an inappropriate affect is strongly associated with bizarre behavior and positive formal thought disorder on a disorganization factor; attention impairment correlates significantly with psychotic, disorganization, and negative symptom factors. Alogia contains both positive and negative symptoms, with the poverty of content of speech as the disorganization factor, and poverty of speech, response latency, and thought blocking as the negative symptom factors.

Alogia is a major diagnostic sign of schizophrenia, when organic mental disorders have been excluded.

In schizophrenia, negative symptoms including flattening of affect, avolition, and alogia are responsible for the considerable morbidity of the disease compared with other psychotic disorders. Negative symptoms are common in the prodromal and residual phases of the disease and can be severe. During the first year, negative symptoms can progress, especially alogia, which may start off from a relatively low rate. Within 2 years, up to 25% of patients will have significant negative symptoms. Psychotic symptoms tend to diminish as the individuals age, but negative symptoms tend to persist. Prominent negative symptoms at disease onset, including alogia, are good predictors of worse outcomes.

Negative symptoms can arise in the presence of other psychiatric symptoms. Positive symptoms are a common cause of apathy, social withdrawal, and alogia. Secondary causes of negative symptoms, such as depression and demoralization, often remit within a year, which helps distinguishing them from primary negative symptoms. Symptoms that don't diminish over a year with medications should be reconsidered as possible primary negative symptoms.

title: Alogia


text:
Mount Holly Cemetery

Mount Holly Cemetery is a historic cemetery located in the Quapaw Quarter area of downtown Little Rock in the U.S. state of Arkansas, and is the burial place for numerous Arkansans of note. It was listed on the National Register of Historic Places in 1970 and has been nicknamed "The Westminster Abbey of Arkansas".

"Tales of the Crypt"
Every year in October several drama students from Parkview Arts and Science Magnet High School are each given a person buried in the cemetery to research. They then prepare short monologues or dialogues, complete with period costumes, to be performed in front of the researched person's grave. Audiences are led through the cemetery from grave to grave by guides with candles. The event is called "Tales of the Crypt". Although it takes place around the same time as the American holiday Halloween, the event is meant to be historic rather than spooky.

2016 vandalism
The cemetery experienced heavy vandalism in the overnight hours of April 20, 2016. Numerous headstones were toppled and smashed, including the well-known statues of a mourner next to statues of two little girls.

Notable burials
The cemetery is the burial place for ten former Arkansas governors, six United States senators, 14 Arkansas Supreme Court justices, 21 Little Rock mayors, numerous Arkansas literary figures, Confederate generals, and other worthies. There are also several slaves who are buried there, marked by modest gravestones.

 * Samuel Adams (1805–1850), Arkansas Governor
 * Dale Alford (1916–2000), U.S. Representative (1959–63) and ophthalmologist
 * Chester Ashley (1791–1848), US Senator
 * W. Jasper Blackburn (1820–1899), US Congressman
 * Solon Borland (1811–1864), physician and US Senator
 * Elias Nelson Conway (1812–1892), Arkansas Governor

 * Jeff Davis (1862–1913), Arkansas Governor
 * David Owen Dodd (1846–1864), boy martyr of the Confederacy
 * James Philip Eagle (1837–1904), Arkansas Governor
 * James Fleming Fagan (1828–1893), Confederate Major General
 * Sandford C. Faulkner (1803–1874), the original 'Arkansas Traveller'
 * John Gould Fletcher (1886–1950), Pulitzer Prize–winning poet, with his wife, noted children's book author Charlie May Simon (1897–1977)
 * Thomas Fletcher (1817–1880), acting governor of Arkansas (1862)
 * William Savin Fulton (1795–1844), U.S. senator from Arkansas (1836–1844)
 * Augustus Hill Garland (1876–1907), Arkansas Governor
 * John N. Heiskell (1872–1972), US Senator and editor of the Arkansas Gazette
 * Simon Pollard Hughes Jr. (1830–1906), Arkansas Governor
 * George Izard (1776–1828), governor of Arkansas Territory from 1825 to 1828
 * Robert Ward Johnson (1814–1879), C.S. senator from Arkansas (1862–1865)
 * George R. Mann (1856–1939), architect
 * William Read Miller (1823–1887) Arkansas Governor

 * Allison Nelson (1822–1862), politician and Confederate Brigadier General
 * Thomas Willoughby Newton (1804–1853) US Congressman

 * Elizabeth Quatie Brown Ross, wife of John Ross (Cherokee chief) (1791–1839), Cherokee figure
 * Albert Rust (1818–1870), US Congressman and Confederate Brigadier General (cenotaph)
 * Ambrose H. Sevier (1801–1848), U.S. senator from Arkansas (1836–1848)
 * Bernard Smith (1776–1835), US Congressman
 * David D. Terry (1881–1963), US Congressman
 * Cephas Washburn (1793–1860), missionary
 * Edward Washburn (1831–1860), artist
 * Frank D. White (1933–2003), governor of Arkansas from 1981 to 1983
 * William W. Wilshire (1830–1888), U.S. representative from 1873 to 1877; chief justice of the Arkansas Supreme Court from 1868 to 1871
 * William E. Woodruff (1795–1885), founder of the Arkansas Gazette

title: Mount Holly Cemetery


text:
Norwegian Student Choral Society

The Norwegian Student Choral Society (Den norske Studentersangforening) is a Norwegian male voice choir, founded in 1845 by Johan Diederich Behrens. It is the second oldest choir in Norway and the official choir of the University of Oslo.

Premiering a number of well known pieces, including Norway's national anthem "Ja, vi elsker dette landet", the choir is most commonly known for their yearly outdoor concert at University Square on 17 May, the Norwegian Constitution Day. The concert is broadcast live on NRK Radio.

The choirs' first official performance was at Henrik Wergeland's funeral.

Commonly referred to as "DnS" by members and friends, the choir holds regular concerts, both independently and in collaboration with other artists, choirs and orchestras.

Present day choir
As of 2016, the choir has around 60 active singers, who are conducted by Marit Tøndel Bodsberg.

It consists of the so-called "Concert Choir", containing the most active members, and a number of older members, with their own conductor, who go by the name "DnX".

DnS have competed in a number of international competitions. In the World Choir Games of 2006, in Xiamen, China, it won first place in the male voice choir category and third place in sacred choral music

Conductors

 * Halfdan Kjerulf (1815 – 1868) 1845 – 1849
 * (1820 – 1890) 1849 – 1888
 * Olaus Andreas Grøndahl (1847 – 1923) 1888 – 1912
 * Ansgar Guldberg (1868 – 1924) 1912 – 1916
 * Per Winge (1858 – 1935) 1916 – 1919
 * Emil Nielsen (1877 – 1942) 1919 – 1929
 * Sigurd Torkildsen (1897 – 1990) 1929 – 1961
 * Sverre Bruland (1923) 1961 – 1973
 * Torstein Grythe (1918 – 2009) 1973 – 1976
 * Øyvind Otterstad (1944 – 1986) 1976 – 1980
 * Sverre Bruland (1923) 1980 – 1982
 * Carl Høgset (1941) 1982 – 1985
 * Trond Dahlen (1955) 1986 – 1991
 * (1960) 1991 – 2003
 * (1960) 2003-2008
 * Ketil J.B. Belsaas (1969) 2007-2008
 * Steffen Kammler (1965) 2009 – 2011
 * Carl Høgset (1941) 2012 – 2015
 * Marit Tøndel Bodsberg (1981) 2016 –

Johan Diederich Behrens
Founder of The Norwegian Student Choral Society was Johan Diederich Behrens; and the "Johan Behrens Quartet", a male voice ensemble founded in 1843, formed the core of the first choir. The initial reason for the formation of the choir was a wish, in the summer of 1845, to participate at the Nordic student meeting in Lund and Copenhagen. The students contacted Behrens a short time before their departure to Copenhagen. Behrens, at first, declined to assist in the formation of choir due to the limited amount of time they had left to rehearse. He was eventually persuaded and accepted on the condition that he had the final word on which songs were rehearsed well enough for performance. On July 10, 1845, the society's laws were signed. However, Behrens, wanting to focus on his studies, did not wish to continue as conductor; and passed on the job to Halfdan Kjerulf.

Chronology
1845 "Studentertog" to Copenhagen results in formation of DnS. The society is founded on 10 July 1845. (21 November is later officiated as jubilee day)

1849 "Aftenunderholdning" in Christiania Theater. "Brudefærden i Hardanger" performed for the first time. Johan D. Behrens takes over from Halfdan Kjerulf as conductor

1861 Oslo University 50th anniversary. Cantata by Welhaven/Arnold.

1863 Fifth public song festival in Bergen. Celebration of student unions 50th birthday.

1865 DnS' 20th jubilee. Bjørnson, Welhaven, Vinje, Ole Bull and Halfdan Kjerulf amongst the guests. Bjørnson holds keynote speech. "Studenter-sangforeningens Vise" by Bjørnson and "Fjellguten i songarlaget av studentar i Christiania" by Vinje composed for the event.

1868 Spring concert: "Aftenrøster" by Johan Svendsen performed for the first time. Halfdan Kjerulf laid to rest.

1869 Nordic student meeting in Christiania.

1870 DnS' 25th jubilee. Bjørnson, Grieg, Jonas Lie, Andreas Munch and Theodor Kjerulf amongst the guests.

1872 Concert with Bazaar to raise funds for the restoration of Trondheim cathedral. "Landkænning" performed for the first time.

1874 Halfdan Kjerulf monument unveiled at "Halfdan Kjerulfs Plass".

1875 Karl-Johan monument unveiled: Cantata by Jørgen Moe/F.A. Reissiger. (Elite choir "Johanitterne" founded.)

1878 Song tour to Paris (together with "Allmänna Sången", Uppsala). Concerts in Drammen, Arendal, Christianssand, Paris, Le Havre and Christiania.

1881 Wergeland monument unveiled: Cantata by Jonas Lie/Johan Svendsen.

1888 Johan D. Behrens steps down as conductor. O.A. Grøndahl appointed. First Nordic music festival in Copenhagen.

1889 Concert tour to Paris. Performances in Drammen, Arendal, Christianssand, Paris and Le Havre.

1891 Student union and Student Choral Society's party and parade in honour of Edvard Grieg.

1893 Performance for Camilla Collett on her 80th birthday. Parade for Jonas Lie.

1895 DnS' 50th jubilee. "The Order of DnS" established.

1905 First tour to USA.

1906 Rikard Nordraaks monolith in Berlin unveiled, with participation by Bjørnstjerne Bjørnson and the Student Choral Society.

1908 Denmark tour with concerts in Copenhagen, Roskilde, Aarhus, Odense and Nyborg.

1911 Oslo University 100th anniversary. The new "University Hall" is inaugurated. University cantata "Lyset" is performed for the first time.

1912 O.A. Grøndahl slutter steps down as conductor. Ansgar Guldberg appointed.

1916 Ansgar Guldberg steps down as conductor. Per Winge takes over. "Guldbergs Academic Choir" founded.

1922 Sørland tour with concerts in Sanderfjord, Skien, Larvik, Arendal and Kristiansand.

1923 Northern Norway tour with concerts in Lillehammer, Trondheim, Namsos, Mosjøen, Bodø, Melbu, Harstad, Tromsø, Hammerfest and Sandnessjøen.

1924 Scandinavian Student Meet in Stockholm. Trondheim Student Choral Society visit Kristiania for the first time

1928 West Coast tour with concerts in Bergen, Haugesund, Stavanger, Flekkefjord and Kristiansand.

1929 Emil Nielsen steps down as conductor. Sigurd Torkildsen takes over. Birthday concert for Johan Halvorsens 60th.

1939 Second tour to USA.

1945 Performance at National theatre. Song tour to Denmark. 100 year jubilee.

1948 Tour to Møre, with konserts in Gjøvik, Hamar, Kristiansand, Molde, Ålesund and Åndalsnes.

1949 Finland tour.

1960 Northern Norway tour with concerts in konserter i Tromsø, Harstad, Svolvær and Bodø.

1961 Sigurd Torkildsen steps down as conductor, Sverre Bruland appointed.

1962 Iceland tour with concerts in Reykjavik and Akureyri.

1964 Finland tour with concerts in Åbo og Helsingfors.

1966 Czechoslovakia tour

1968 West Germany tour with concerts in Kiel and Hamburg.

1972 Unification Anniversary. Performances in Hafrsfjord, Ullandhaug, Haraldshaugen and Skudeneshavn.

1973 Sverre Bruland steps down as conductor, Torstein Grythe steps in. 8th Nordic student song meet in Århus.

1976 Torstein Grythe steps down as conductor, Øyvind Otterstad appointed.

1978 "Academic Choir Meet" in Trondheim.

1980 Øyvind Otterstad steps down as conductor, Sverre Bruland temporarily appointed.

1982 Performance for His Majesty King Olav V, in honour of his 25th jubilee. Sverre Bruland steps down as conductor, Carl Høgset steps in.

1984 Italy tour. "Academic Choir Meet" in Oslo.

1985 Italy tour. Participants in international choir competition, Arezzo. 3rd place. Carl Høgset steps down as conductor, Trond Dahlen steps in.

1986 "Academic Choir Meet" in Trondheim.

1991 Trond Dahlen slutter steps down as conductor. Tone Bianca Dahl appointed.

1993 Tour to England and Wales. Concerts in London, Canterbury and Sandwich. Participants in "Llangollen International Musical Eisteddfod". 3rd place in male choir category, and 3rd place in folk music category. "Nordic Student Choir Meet" in Trondheim.

1994 Tour to Greece. Concert in Piraeus. Participation in Athens international choir competition. 1st place in equal voices category.

1995 150th anniversary. "Natten skrider förbi" by Knut Nystedt performed for first time at anniversary concert. Third tour to USA. CD "DnS i 150!" released.

1996 Tour to Spain. Participation in the international choir games, Tolosa. Two 3rd places.

1997 Tour to Germany, with an invitation to the "Ostsee Choir Festival" in Flensburg

1999 Participation in the national choir games "Syng For Oss" in Bergen. 1st place in A-class for male choir. "Kullervo-symfoni" by Jean Sibelius performed together with the Oslo Philomonic and choir. CD "Transeamus usque Bethlehem" released. Nordic Academic Choir Meet in Tallinn.

2000 Tour to Wales and Ireland, with concert in St. Patricks cathedral, Dublin. Participation in the "Llangollen International Musical Eisteddfodd". 3rd place in male choir category.

2001 DnS moves to new premises in Professor Dahls gate 8. The CD "En mannshowinistisk aften" is released.

2003 Tone Bianca Dahl steps down as conductor, Thomas Caplin appointed new conductor.

2004 DnS wins male choir category, and takes 3rd place for church music in the international choir games, Helsingborg.

2005 CD/DVD "Norske Perspektiver" is released on 7 June. Collaborative project with "Collegium Vocale" from Hamar.

2006 DnS takes first place in Male Choir category at The International Choir Olympics in Xiamen, China.

2007 DnS takes first place in Sacred Choral Music category, and 2nd place in category for world music at the 2nd international choir games in Bergen. Ketil J.B. Belsaas steps in temporarily for Thomas Caplin.

2008 Thomas Caplin steps down as conductor, Ketil J.B. Belsaas appointed on temporary basis.

2009 Steffen Kammler appointed as conductor.

2012 Carl Høgset is appointed as conductor.

2015 The CD "Sangen har lysning" is released on September 18.

2016 Marit Tøndel Bodsberg is appointed as conductor.

title: Norwegian Student Choral Society


text:
Arnulf of Chocques

Arnulf of Chocques (died 1118) was a leading member of the clergy during the First Crusade, being made Latin Patriarch of Jerusalem in 1099 and again from 1112 to 1118. Sometimes referred to as Arnulf of Rœulx, presumably after the village of Rœulx some 70km from his home village of Chocques, he was given the nickname Malecorne, meaning badly tonsured.

Biography
Arnulf was the illegitimate son of a Flemish priest, and studied under Lanfranc at Caen. In the 1070s he was a tutor to Cecilia, daughter of William I of England. He also taught Ralph of Caen, one of the later chroniclers of the First Crusade. He was also close to Odo of Bayeux, who he accompanied on the Crusade. He was the chaplain of the Norman crusader army led by Robert of Normandy, Cecilia's brother and William's son. He was most likely appointed a papal legate, under the authority of the overall legate Adhemar of Le Puy, and after Adhemar's death in 1098 he shared control of the clergy with fellow legate Peter of Narbonne. Some of the non-Norman knights in the other crusader armies believed he was corrupt, and they apparently sang vulgar songs about him, but most crusaders respected him as an eloquent preacher. He was one of the chief skeptics about Peter Bartholomew's claims to have discovered the Holy Lance in Antioch, and because of Arnulf's opposition Peter volunteered to undergo an ordeal by fire. Arnulf's opposition to Peter brought him into conflict with Raymond of St. Gilles, who believed Peter's story. To help ease the crisis among the crusaders over the issue, and also to lift spirits after Peter's death during the ordeal, Arnulf helped make a statue of Christ which was placed on one of the siege engines during the siege of Jerusalem. After the capture of Jerusalem he discovered the True Cross in the Church of the Holy Sepulchre. This discovery was not as controversial as the discovery of the Lance, although it was just as suspicious. Arnulf may have been trying to make up for the problems he caused disproving the authenticity of the Lance, and the True Cross became the most sacred relic of the Kingdom of Jerusalem.

After Raymond left Jerusalem on 1 August 1099, Arnulf was elected Latin Patriarch of Jerusalem. He was supported by Godfrey of Bouillon, the first ruler of Jerusalem, and in turn he supported Godfrey's decision to make Jerusalem a secular kingdom rather than one ruled by the clergy. He accompanied Godfrey in the Battle of Ascalon, with a relic of the True Cross. Arnulf enforced the Latin rite among the crusaders, banning all others thus further alienating the disaffected Greeks. However, his election was soon subject to doubts concerning its canonicity, as he was not yet a deacon. Before he could be ordained, he was replaced in December by Dagobert of Pisa, whom Pope Paschal II had appointed legate. Arnulf was instead appointed archdeacon of Jerusalem.

In 1112 he officially became Patriarch, though many of the other clerics distrusted him and found him unnecessarily harsh. He was especially unpopular with the Orthodox and Syriac Christians when he prohibited non-Latin masses at the Holy Sepulchre. He was accused of various crimes: sexual relations with a Muslim woman, simony, and most importantly condoning the bigamous marriage of King Baldwin I to Adelaide del Vasto while his first wife Arda of Armenia was still alive. He was briefly deposed by a papal legate in 1115, but appealed to Pope Paschal II and was reinstated in 1116, provided that he annul Baldwin and Adelaide's marriage.

He remained Patriarch until his death in 1118.

Arnulf married his niece Emma, also called Emelota, to Eustace Garnier, the Latin seigneur of Caesarea and Sidon. After his death, Emma married Hugh II of Le Puiset.

title: Arnulf of Chocques


text:
Robert of Normandy

Robert of Normandy may refer to:

 * Rollo, baptized as Robert, (c. 860 - c. 932), viking founder and first ruler of Normandy
 * Robert the Magnificent (1000 – 1035), also called the Devil or Robert I, Duke of Normandy, son of Richard II, Duke of Normandy
 * Robert Curthose or Robert II (c. 1051 or 1054–1134), Duke of Normandy, son of William the Conqueror, the first Norman king of England

title: Robert of Normandy


text:
Inversive geometry

In geometry, inversive geometry is the study of inversion, a transformation of the Euclidean plane that maps circles or lines to other circles or lines and that preserves the angles between crossing curves. Many difficult problems in geometry become much more tractable when an inversion is applied. Inversion seems to have been discovered by a number of people contemporaneously, including Steiner (1824), Quetelet (1825), Bellavitis (1836), Stubbs and Ingram (1842–3) and Kelvin (1845).

The concept of inversion can be generalized to higher-dimensional spaces.

Inverse of a point

To invert a number in arithmetic usually means to take its reciprocal. A closely related idea in geometry is that of "inverting" a point. In the plane, the inverse of a point P with respect to a reference circle (Ø) with center O and radius r is a point P', lying on the ray from O through P such that

 * $$OP \cdot OP^{\prime} = r^2.$$

This is called circle inversion or plane inversion. The inversion taking any point P (other than O) to its image P also takes P back to P, so the result of applying the same inversion twice is the identity transformation which makes it a self-inversion (i.e. an involution). To make the inversion a total function that is also defined for O, it is necessary to introduce a point at infinity, a single point placed on all the lines, and extend the inversion, by definition, to interchange the center O and this point at infinity.

It follows from the definition that the inversion of any point inside the reference circle must lie outside it, and vice versa, with the center and the point at infinity changing positions, whilst any point on the circle is unaffected (is invariant under inversion). In summary, for a point inside the circle, the nearer the point to the center, the further away its transformation. While for any point (inside or outside the circle), the nearer the point to the circle, the closer its transformation.

Point outside circle
To construct the inverse P' of a point P outside a circle Ø:
 * Draw the segment from O (center of circle Ø) to P.
 * Let M be the midpoint of OP. (Not shown)
 * Draw the circle c with center M going through P. (Not labeled. It's the blue circle)
 * Let N and N' be the points where Ø and c intersect.
 * Draw segment NN'.
 * P is where OP and NN intersect.

Point inside circle
To construct the inverse P of a point P' inside a circle Ø:
 * Draw ray r from O (center of circle Ø) through P'. (Not labeled, it's the horizontal line)
 * Draw line s through P' perpendicular to r. (Not labeled. It's the vertical line)
 * Let N be one of the points where Ø and s intersect.
 * Draw the segment ON.
 * Draw line t through N perpendicular to ON.
 * P is where ray r and line t intersect.

Dutta's construction
There is a construction of the inverse point to A with respect to a circle P that is independent of whether A is inside or outside P.

Consider a circle P with center O and a point A which may lie inside or outside the circle P.
 * Take the intersection point C of the ray OA with the circle P.
 * Connect the point C with an arbitrary point B on the circle P (different from C and from the point on P antipodal to C)
 * Let h be the reflection of ray BA in line BC. Then h cuts ray OC in a point A. A is the inverse point of A with respect to circle P.

Properties
The inversion of a set of points in the plane with respect to a circle is the set of inverses of these points. The following properties make circle inversion useful.
 * A circle that passes through the center O of the reference circle inverts to a line not passing through O, but parallel to the tangent to the original circle at O, and vice versa; whereas a line passing through O is inverted into itself (but not pointwise invariant).
 * A circle not passing through O inverts to a circle not passing through O. If the circle meets the reference circle, these invariant points of intersection are also on the inverse circle. A circle (or line) is unchanged by inversion if and only if it is orthogonal to the reference circle at the points of intersection.

Additional properties include:
 * If a circle q passes through two distinct points A and A' which are inverses with respect to a circle k, then the circles k and q are orthogonal.
 * If the circles k and q are orthogonal, then a straight line passing through the center O of k and intersecting q, does so at inverse points with respect to k.
 * Given a triangle OAB in which O is the center of a circle k, and points A' and B' inverses of A and B with respect to k, then
 * $$ \angle OAB = \angle OB'A' \ \text{ and }\ \angle OBA = \angle OA'B'.$$

 * The points of intersection of two circles p and q orthogonal to a circle k, are inverses with respect to k.
 * If M and M' are inverse points with respect to a circle k on two curves m and m', also inverses with respect to k, then the tangents to m and m' at the points M and M' are either perpendicular to the straight line MM' or form with this line an isosceles triangle with base MM'.
 * Inversion leaves the measure of angles unaltered, but reverses the orientation of oriented angles.

Examples in two dimensions

 * Inversion of a line is a circle containing the center of inversion; or it is the line itself if it contains the center
 * Inversion of a circle is another circle; or it is a line if the original circle contains the center
 * Inversion of a parabola is a cardioid
 * Inversion of hyperbola is a lemniscate of Bernoulli

Application
For a circle not passing through the center of inversion, the center of the circle being inverted and the center of its image under inversion are collinear with the center of the reference circle. This fact can be used to prove that the Euler line of the intouch triangle of a triangle coincides with its OI line. The proof roughly goes as below:

Invert with respect to the incircle of triangle ABC. The medial triangle of the intouch triangle is inverted into triangle ABC, meaning the circumcenter of the medial triangle, that is, the nine-point center of the intouch triangle, the incenter and circumcenter of triangle ABC are collinear.

Any two non-intersecting circles may be inverted into concentric circles. Then the inversive distance (usually denoted δ) is defined as the natural logarithm of the ratio of the radii of the two concentric circles.

In addition, any two non-intersecting circles may be inverted into congruent circles, using circle of inversion centered at a point on the circle of antisimilitude.

The Peaucellier–Lipkin linkage is a mechanical implementation of inversion in a circle. It provides an exact solution to the important problem of converting between linear and circular motion.

Pole and polar

If point R is the inverse of point P then the lines perpendicular to the line PR through one of the points is the polar of the other point (the pole).

Poles and polars have several useful properties:

 * If a point P lies on a line l, then the pole L of the line l lies on the polar p of point P.
 * If a point P moves along a line l, its polar p rotates about the pole L of the line l.
 * If two tangent lines can be drawn from a pole to the circle, then its polar passes through both tangent points.
 * If a point lies on the circle, its polar is the tangent through this point.
 * If a point P lies on its own polar line, then P is on the circle.
 * Each line has exactly one pole.

In three dimensions

Circle inversion is generalizable to sphere inversion in three dimensions. The inversion of a point P in 3D with respect to a reference sphere centered at a point O with radius R is a point P ' on the ray with direction OP such that $$OP \cdot OP^{\prime} = ||OP|| \cdot ||OP^{\prime}|| = R^2$$. As with the 2D version, a sphere inverts to a sphere, except that if a sphere passes through the center O of the reference sphere, then it inverts to a plane. Any plane passing through O, inverts to a sphere touching at O. A circle, that is, the intersection of a sphere with a secant plane, inverts into a circle, except that if the circle passes through O it inverts into a line. This reduces to the 2D case when the secant plane passes through O, but is a true 3D phenomenon if the secant plane does not pass through O.

Sphere
The simplest surface (besides a plane) is the sphere. The first picture shows a non trivial inversion (the center of the sphere is not the center of inversion) of a sphere together with two orthogonal intersecting pencils of circles.

Cylinder, cone, torus
The inversion of a cylinder, cone, or torus results in a Dupin cyclide.

Spheroid
A spheroid is a surface of revolution and contains a pencil of circles which is mapped onto a pencil of circles (see picture). The inverse image of a spheroid is a surface of degree 4.

Hyperboloid of one sheet
A hyperboloid of one sheet, which is a surface of revolution contains a pencil of circles which is mapped onto a pencil of circles. A hyperboloid of one sheet contains additional two pencils of lines, which are mapped onto pencils of circles. The picture shows one such line (blue) and its inversion.

Stereographic projection as the inversion of a sphere
A stereographic projection usually projects a sphere from a point $$N$$ (north pole) of the sphere onto the tangent plane at the opposite point $$S$$ (south pole). This mapping can be performed by an inversion of the sphere onto its tangent plane. If the sphere (to be projected) has the equation $$x^2+y^2+z^2 = -z$$ (alternately written $$x^2+y^2+(z+\tfrac{1}{2})^2=\tfrac{1}{4}$$; center $$(0,0,-0.5)$$, radius $$0.5$$, green in the picture), then it will be mapped by the inversion at the unit sphere (red) onto the tangent plane at point $$S=(0,0,-1)$$. The lines through the center of inversion (point $$N$$) are mapped onto themselves. They are the projection lines of the stereographic projection.

6-sphere coordinates
The 6-sphere coordinates are a coordinate system for three-dimensional space obtained by inverting the Cartesian coordinates.

Axiomatics and generalization
One of the first to consider foundations of inversive geometry was Mario Pieri in 1911 and 1912. Edward Kasner wrote his thesis on "Invariant theory of the inversion group".

More recently the mathematical structure of inversive geometry has been interpreted as an incidence structure where the generalized circles are called "blocks": In incidence geometry, any affine plane together with a single point at infinity forms a Möbius plane, also known as an inversive plane. The point at infinity is added to all the lines. These Möbius planes can be described axiomatically and exist in both finite and infinite versions.

A model for the Möbius plane that comes from the Euclidean plane is the Riemann sphere.

Invariant
The cross-ratio between 4 points $$x, y, z, w$$ is invariant under an inversion. In particular if O is the centre of the inversion and $$r_1$$ and $$r_2$$ are distances to the ends of a line L, then length of the line $$d$$ will become $$d/(r_1 r_2)$$ under an inversion with radius 1. The invariant is:

 * $$ I =\frac{|x-y| |w-z|}{|x-w| |y-z|}$$

Relation to Erlangen program
According to Coxeter, the transformation by inversion in circle was invented by L. I. Magnus in 1831. Since then this mapping has become an avenue to higher mathematics. Through some steps of application of the circle inversion map, a student of transformation geometry soon appreciates the significance of Felix Klein's Erlangen program, an outgrowth of certain models of hyperbolic geometry

Dilation
The combination of two inversions in concentric circles results in a similarity, homothetic transformation, or dilation characterized by the ratio of the circle radii.

 * $$x \mapsto R^2 \frac {x} {|x|^2} = y \mapsto T^2 \frac {y} {|y|^2} = \left( \frac {T} {R} \right)^2 x.$$

Reciprocation
When a point in the plane is interpreted as a complex number $$z=x+iy,$$ with complex conjugate $$\bar{z}=x-iy,$$ then the reciprocal of z is

 * $$\frac{1}{z} = \frac{\bar{z}}{|z|^2}.$$

Consequently, the algebraic form of the inversion in a unit circle is given by $$z \mapsto w$$ where:

 * $$w=\frac{1}{\bar z}=\overline{\left(\frac{1}{z}\right)}$$.

Reciprocation is key in transformation theory as a generator of the Möbius group. The other generators are translation and rotation, both familiar through physical manipulations in the ambient 3-space. Introduction of reciprocation (dependent upon circle inversion) is what produces the peculiar nature of Möbius geometry, which is sometimes identified with inversive geometry (of the Euclidean plane). However, inversive geometry is the larger study since it includes the raw inversion in a circle (not yet made, with conjugation, into reciprocation). Inversive geometry also includes the conjugation mapping. Neither conjugation nor inversion-in-a-circle are in the Möbius group since they are non-conformal (see below). Möbius group elements are analytic functions of the whole plane and so are necessarily conformal.

Transforming circles into circles
Consider, in the complex plane, the circle of radius $$ r$$ around the point $$ a $$

 * $$(z-a)(z-a)^* = r^2 $$

where without loss of generality, $$a \in \R.$$ Using the definition of inversion

 * $$ w = \frac{1}{z^*} $$

it is straightforward to show that $$ w $$ obeys the equation

 * $$w w^*-\frac{a}{(a^2-r^2)}(w+w^{*})+\frac{a^2}{(a^2-r^2)^2}=\frac{r^2}{(a^2-r^2)^2}$$

and hence that $$ w $$ describes the circle of center $\frac{a}{a^2-r^2}$ and radius $ \frac{r}{|a^2-r^2|} .$

When $$a\to r,$$ the circle transforms into the line parallel to the imaginary axis $$ w+w^* = \tfrac{1}{a}. $$

For $$a \not\in \R$$ and $$a a^* \ne r^2 $$ the result for $$ w $$ is

\begin{align} & w w^*-\frac{a w+a^* w^*}{(a^* a-r^2)}+\frac{a a^*}{(a a^*-r^2)^2}=\frac{r^2}{(a a^*-r^2)^2} \\[4pt] \Longleftrightarrow {} & \left(w-\frac{a^*}{a a^*-r^2}\right)\left(w^*-\frac{a}{a^* a-r^2}\right)=\left(\frac{r}{\left|a a^*-r^2\right|}\right)^2 \end{align} $$

showing that the $$ w $$ describes the circle of center $\frac{a}{(a a^*-r^2)}$ and radius $ \frac{r}{\left|a^* a-r^2\right|} $.

When $$a^* a\to r^2,$$ the equation for $$w$$ becomes

\begin{align} & a w+ a^* w^*=1 \Longleftrightarrow 2\operatorname{Re}\{a w\}=1 \Longleftrightarrow \operatorname{Re} \{a\} \operatorname{Re}\{w\} - \operatorname{Im}\{a\}\operatorname{Im}\{w\}=\frac{1}{2} \\[4pt] \Longleftrightarrow {} & \operatorname{Im} \{w\}=\frac{\operatorname{Re}\{a\}}{\operatorname{Im}\{a\}}\cdot \operatorname{Re}\{w\}-\frac{1}{2\cdot\operatorname{Im}\{a\}}. \end{align} $$

Higher geometry
As mentioned above, zero, the origin, requires special consideration in the circle inversion mapping. The approach is to adjoin a point at infinity designated ∞ or 1/0. In the complex number approach, where reciprocation is the apparent operation, this procedure leads to the complex projective line, often called the Riemann sphere. It was subspaces and subgroups of this space and group of mappings that were applied to produce early models of hyperbolic geometry by Beltrami, Cayley, and Klein. Thus inversive geometry includes the ideas originated by Lobachevsky and Bolyai in their plane geometry. Furthermore, Felix Klein was so overcome by this facility of mappings to identify geometrical phenomena that he delivered a manifesto, the Erlangen program, in 1872. Since then many mathematicians reserve the term geometry for a space together with a group of mappings of that space. The significant properties of figures in the geometry are those that are invariant under this group.

For example, Smogorzhevsky develops several theorems of inversive geometry before beginning Lobachevskian geometry.

In higher dimensions
In a real n-dimensional Euclidean space, an inversion in the sphere of radius $r$ centered at the point $$O = (o_1, ..., o_n)$$ is a map of an arbitrary point $$P = (p_1, ..., p_n)$$ found by inverting the length of the displacement vector $$P - O$$ and multiplying by $r^2$:

 * $$\begin{align}

P &\mapsto P' = O + \frac{r^2(P - O)}{\|P - O\|^2}, \\[5mu] p_j &\mapsto p_j' = o_j + \frac{r^2(p_j - o_j)}{\sum_k (p_k - o_k)^2}. \end{align}$$

The transformation by inversion in hyperplanes or hyperspheres in En can be used to generate dilations, translations, or rotations. Indeed, two concentric hyperspheres, used to produce successive inversions, result in a dilation or homothety about the hyperspheres' center.

When two parallel hyperplanes are used to produce successive reflections, the result is a translation. When two hyperplanes intersect in an (n–2)-flat, successive reflections produce a rotation where every point of the (n–2)-flat is a fixed point of each reflection and thus of the composition.

Any combination of reflections, translations, and rotations is called an isometry. Any combination of reflections, dilations, translations, and rotations is a similarity.

All of these are conformal maps, and in fact, where the space has three or more dimensions, the mappings generated by inversion are the only conformal mappings. Liouville's theorem is a classical theorem of conformal geometry.

The addition of a point at infinity to the space obviates the distinction between hyperplane and hypersphere; higher dimensional inversive geometry is frequently studied then in the presumed context of an n-sphere as the base space. The transformations of inversive geometry are often referred to as Möbius transformations. Inversive geometry has been applied to the study of colorings, or partitionings, of an n-sphere.

Anticonformal mapping property
The circle inversion map is anticonformal, which means that at every point it preserves angles and reverses orientation (a map is called conformal if it preserves oriented angles). Algebraically, a map is anticonformal if at every point the Jacobian is a scalar times an orthogonal matrix with negative determinant: in two dimensions the Jacobian must be a scalar times a reflection at every point. This means that if J is the Jacobian, then $$J \cdot J^T = k I$$ and $$\det(J) = -\sqrt{k}.$$ Computing the Jacobian in the case zi = xi/$\|x\|$2, where $\|x\|$2 = x12 + ... + xn2 gives JJT = kI, with k = 1/$\|x\|$4n, and additionally det(J) is negative; hence the inversive map is anticonformal.

In the complex plane, the most obvious circle inversion map (i.e., using the unit circle centered at the origin) is the complex conjugate of the complex inverse map taking z to 1/z. The complex analytic inverse map is conformal and its conjugate, circle inversion, is anticonformal. In this case a homography is conformal while an anti-homography is anticonformal.

Inversive geometry and hyperbolic geometry
The (n − 1)-sphere with equation

 * $$x_1^2 + \cdots + x_n^2 + 2a_1x_1 + \cdots + 2a_nx_n + c = 0$$

will have a positive radius if a12 + ... + an2 is greater than c, and on inversion gives the sphere

 * $$x_1^2 + \cdots + x_n^2 + 2\frac{a_1}{c}x_1 + \cdots + 2\frac{a_n}{c}x_n + \frac{1}{c} = 0.$$

Hence, it will be invariant under inversion if and only if c = 1. But this is the condition of being orthogonal to the unit sphere. Hence we are led to consider the (n − 1)-spheres with equation

 * $$x_1^2 + \cdots + x_n^2 + 2a_1x_1 + \cdots + 2a_nx_n + 1 = 0,$$

which are invariant under inversion, orthogonal to the unit sphere, and have centers outside of the sphere. These together with the subspace hyperplanes separating hemispheres are the hypersurfaces of the Poincaré disc model of hyperbolic geometry.

Since inversion in the unit sphere leaves the spheres orthogonal to it invariant, the inversion maps the points inside the unit sphere to the outside and vice versa. This is therefore true in general of orthogonal spheres, and in particular inversion in one of the spheres orthogonal to the unit sphere maps the unit sphere to itself. It also maps the interior of the unit sphere to itself, with points outside the orthogonal sphere mapping inside, and vice versa; this defines the reflections of the Poincaré disc model if we also include with them the reflections through the diameters separating hemispheres of the unit sphere. These reflections generate the group of isometries of the model, which tells us that the isometries are conformal. Hence, the angle between two curves in the model is the same as the angle between two curves in the hyperbolic space.

title: Inversive geometry