In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, they play an important role in group theory and chemistry; the notation for the dihedral group differs in abstract algebra. In geometry, Dn or Dihn refers to the symmetries of a group of order 2n. In abstract algebra, D2n refers to this same dihedral group; the geometric convention is used in this article. A regular polygon with n sides has 2 n different symmetries: n rotational symmetries and n reflection symmetries. We take n ≥ 3 here; the associated rotations and reflections make up the dihedral group D n. If n is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If n is there are n/2 axes of symmetry connecting the midpoints of opposite sides and n / 2 axes of symmetry connecting opposite vertices. In either case, there are 2 n elements in the symmetry group. Reflecting in one axis of symmetry followed by reflecting in another axis of symmetry produces a rotation through twice the angle between the axes.
The following picture shows the effect of the sixteen elements of D 8 on a stop sign: The first row shows the effect of the eight rotations, the second row shows the effect of the eight reflections, in each case acting on the stop sign with the orientation as shown at the top left. As with any geometric object, the composition of two symmetries of a regular polygon is again a symmetry of this object. With composition of symmetries to produce another as the binary operation, this gives the symmetries of a polygon the algebraic structure of a finite group; the following Cayley table shows the effect of composition in the group D3. R0 denotes the identity. For example, s2s1 = r1, because the reflection s1 followed by the reflection s2 results in a rotation of 120°; the order of elements denoting the composition is right to left, reflecting the convention that the element acts on the expression to its right. The composition operation is not commutative. In general, the group Dn has elements r0... rn−1 and s0... sn−1, with composition given by the following formulae: r i r j = r i + j, r i s j = s i + j, s i r j = s i − j, s i s j = r i − j.
In all cases and subtraction of subscripts are to be performed using modular arithmetic with modulus n. If we center the regular polygon at the origin elements of the dihedral group act as linear transformations of the plane; this lets us represent elements of Dn with composition being matrix multiplication. This is an example of a group representation. For example, the elements of the group D4 can be represented by the following eight matrices: r 0 =, r 1 =, r 2 =, r 3 =, s 0 =, s 1 =, s 2 =
Alexander Muir was a Canadian songwriter, poet and school headmaster. He was the composer of The Maple Leaf Forever, which he wrote in October 1867 to celebrate the Confederation of Canada. In 1833 Muir immigrated to Toronto, from Lesmahagow, where he grew up and he was educated by his father. Muir studied at Queen's College, where he graduated in 1851. Muir taught in the Greater Toronto Area in such places as Scarborough and Toronto, as well as in Newmarket, in suburban areas as Parkdale and Leslieville, where he lived on Laing Avenue. During the early 1870s, Alexander Muir was an elementary school teacher in Newmarket; when the cornerstone of the Christian Church in Newmarket was being laid on June 25, 1874, by the Governor General, Lord Dufferin, Muir brought his school choir to the event to sing his new composition The Maple Leaf Forever, its first public performance. From 1860 to 1870, he was principal of Leslieville School in Toronto, he was principal of Toronto's Alexander Muir/Gladstone Junior and Senior Public School.
Muir was a noted Canadian Orangeman. He served with The Queen's Own Rifles of Canada, fought with them at the Battle of Ridgeway, he wrote The Maple Leaf Forever. Although Muir's musical activities were on an amateur level, they were emphasized along with athletics and patriotism during his teaching career. Muir wrote several songs about Canada during his career, including Canada Forever and Young Canada Was Here, but his most enduring composition was The Maple Leaf Forever written in 1867, the year of Confederation. Muir wrote the poem for a patriotic poetry contest in Montreal, winning second prize, he looked for an existing melody that would fit, a common practice. When he failed to find a suitable tune, Muir wrote the music himself. Alexander Muir Memorial Gardens, a formal garden and park, just south of Yonge Street and Lawrence Avenue in the Lawrence Park neighbourhood of Toronto, is named in his honour. Maple Leaf Forever Park is in the rear of Maple Cottage at 62 Laing Street, near Leslie Street and Queen Street East in Toronto.
Schools which have been named after him are: Alexander Muir/Gladstone Ave Junior and Senior Public School, 108 Gladstone Ave. Toronto Alexmuir Junior Public School, 95 Alexmuir Blvd. Scarborough Alexander Muir Public School, 75 Ford Wilson Blvd. Newmarket Alex Muir Public School, 188 Kohler St. Sault Ste Marie which has now been repurposed and renamed the Urban Aboriginal Alternative High School. Mount Muir in Alberta is named for him. Bibliography Media related to Alexander Muir at Wikimedia Commons The Maple Leaf Forever MP3 The Maple Leaf Forever MIDI File The Maple Leaf Forever Johnson, Edward, 1878-1959
"Let It Rock" is the debut single by American musician Kevin Rudolf. It was produced by Rudolf for his debut album, In the City, features a verse from American rapper Lil Wayne. In the U. S. and Canada, the song reached #5 on the Billboard Hot 100, number #6 on the Pop 100 and #2 on the Canadian Hot 100. The first week, the song debuted at number 45 on the Australian Singles Chart and reached number 3. In the United Kingdom, the song climbed to number 40 on physical release but due to an increase in airplay, due to BBC Radio 1 adding it to their A-list, the song climbed thirty places to number 10 in the song's second week on the chart. "Let It Rock" was number 28 on Rolling Stone's list of the 100 Best Songs of 2008. The single was the official theme song for WWE's 2009 Royal Rumble, it is featured in the video game Madden NFL 11 and Guitar Hero Live. According to in-game files, the song was originally meant to appear on Grand Theft Auto V's Non-Stop Pop FM radio station, but it was not included in the final release of the game.
"Let It Rock" was featured in tourism ads for Atlantic City, New Jersey in 2013. "Let It Rock" was certified triple platinum by the RIAA on May 5, 2009. As of July 2011, the song has sold 4,000,000 digital downloads, making it Lil Wayne's third 4-million-seller. Rudolf explained what "Let It Rock" is about, saying: "'Let It Rock' was written from a place of anger and dissatisfaction with the world. It's a song about hypocrisy, it contains biblical references - and yet people still think it's a party song. Whatever works." Rudolf said about "Let It Rock": "It's a song about the hypocrisy in the world, I'm saying that when I come through, I'm bringing the truth," Rudolf explained. Rudolf said: "I use the Parable of the Prodigal Son, because I want to expose all the fakes out there — in the music industry, in the world, anywhere. A lot of people think it's a party record, but it's not." Digital download"Let It Rock" - 3:56U. S. Digital CD single"Let It Rock" - 3:56 "Let It Rock" - 3:56 "Let It Rock" - 3:50 "Let It Rock" - 3:50UK digital CD single"Let It Rock" - 3:56 "Let It Rock" - 6:35 "Let It Rock" - 6:11 "Let It Rock" - 3:54 "Let It Rock" - 2:48 The music video for "Let It Rock" features Kevin Rudolf and Lil Wayne performing in front of both a crowd and a dark background, sometimes lit with various effects such as strobe and laser lights.
The NBA on TNT features a version of the music video to promote the 2009 NBA All-Star Game, featuring the players competing in the event. The song's chorus is sampled in the Iyaz song "Damn They Hot"; the song is sampled and incorporated into Natasha Bedingfield's song "All I Need" from her album Strip Me, which features Rudolf. Let It Rock on YouTube Lyrics of this song at MetroLyrics