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Schläfli symbol

In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more than three dimensions and discovered all their convex regular polytopes, including the six that occur in four dimensions; the Schläfli symbol is a recursive description, starting with for a p-sided regular polygon, convex. For example, is an equilateral triangle, is a square, a convex regular pentagon and so on. Regular star polygons are not convex, their Schläfli symbols contain irreducible fractions p/q, where p is the number of vertices, q is their turning number. Equivalently, is created from the vertices of, connected every q. For example, is a pentagram. A regular polyhedron that has q regular p-sided polygon faces around each vertex is represented by. For example, the cube is represented by. A regular 4-dimensional polytope, with r regular polyhedral cells around each edge is represented by.

For example, a tesseract, has 3 cubes, around an edge. In general, a regular polytope has z facets around every peak, where a peak is a vertex in a polyhedron, an edge in a 4-polytope, a face in a 5-polytope, a cell in a 6-polytope, an -face in an n-polytope. A regular polytope has a regular vertex figure; the vertex figure of a regular polytope is. Regular polytopes can have star polygon elements, like the pentagram, with symbol, represented by the vertices of a pentagon but connected alternately; the Schläfli symbol can represent a finite convex polyhedron, an infinite tessellation of Euclidean space, or an infinite tessellation of hyperbolic space, depending on the angle defect of the construction. A positive angle defect allows the vertex figure to fold into a higher dimension and loops back into itself as a polytope. A zero angle defect tessellates space of the same dimension as the facets. A negative angle defect can be constructed in hyperbolic space. A facet or a vertex figure is assumed to be a finite polytope, but can sometimes itself be considered a tessellation.

A regular polytope has a dual polytope, represented by the Schläfli symbol elements in reverse order. A self-dual regular polytope will have a symmetric Schläfli symbol. In addition to describing Euclidean polytopes, Schlafli symbols can be used to describe spherical polytopes or spherical honeycombs. Schläfli's work was unknown in his lifetime, his notation for describing polytopes was rediscovered independently by several others. In particular, Thorold Gosset rediscovered the Schlafli symbol which he wrote as | p | q | r |... | z | rather than with brackets and commas as Schlafli did. Gosset's form has greater symmetry, so the number of dimensions is the number of vertical bars, the symbol includes the sub-symbols for facet and vertex figure. Gosset regarded | p as an operator, which can be applied to | q |... | z | to produce a polytope with p-gonal faces whose vertex figure is | q |... | z |. Schläfli symbols are related to reflection symmetry groups, which correspond to the finite Coxeter groups and are specified with the same indices, but square brackets instead.

Such groups are named by the regular polytopes they generate. For example, is the Coxeter group for reflective tetrahedral symmetry, is reflective octahedral symmetry, is reflective icosahedral symmetry; the Schläfli symbol of a regular polygon with p edges is. For example, a regular pentagon is represented by. For star polygons, the constructive notation is used, where p is the number of vertices and q - 1 is the number of vertexes skipped when drawing each edge of the star. For example, represents the pentagram; the Schläfli symbol of a regular polyhedron is if its faces are p-gons, each vertex is surrounded by q faces. For example, is the regular dodecahedron, it has pentagonal faces, 3 pentagons around each vertex. See the 5 convex Platonic solids, the 4 nonconvex Kepler-Poinsot polyhedra. Topologically, a regular 2-dimensional tessellation may be regarded as similar to a polyhedron, but such that the angular defect is zero. Thus, Schläfli symbols may be defined for regular tessellations of Euclidean or hyperbolic space in a similar way as for polyhedra.

The analogy holds for higher dimensions. For example, the hexagonal tiling is represented by; the Schläfli symbol of a regular 4-polytope is of the form. Its faces are regular p-gons, the cells are regular polyhedra of type, the vertex figures are regular polyhedra of type, the edge figures are regular r-gons. See the six convex regular and 10 regular star 4-polytopes. For example, the 120-cell is represented by, it is made of dodecahedron cells, has 3 cells around each edge. There is one regular tessellation of Euclidean 3-space: the cubic honeycomb, with a Schläfli symbol of, made of cubic cells and 4 cubes around each edge. There are 4 regular compact hyperbolic tessellations including, the hyperbolic small dodecahedral honeycomb, which fills space with dodecahedron cells. For higher-dimensional regular polytopes, the Schläfli symbol is defined recursively as if the facets have Schläfli symbol and the vertex figures have Schläfli symbol. A vertex figure of a facet of a polytope and a facet of a vertex figure of the same polytop

Richard Ashcroft

Richard Paul Ashcroft is an English singer and songwriter. He was the lead singer and occasional rhythm guitarist of the alternative rock band the Verve from their formation in 1990 until their original split in 1999. Songs he wrote for the band include "Bitter Sweet Symphony", "Lucky Man", the UK number one "The Drugs Don't Work", he became a successful solo artist. The Verve reformed in 2007 but again broke up by summer 2009. Ashcroft founded a new band, RPA & The United Nations of Sound, released a new album on 19 July 2010. On 22 February 2016, Ashcroft announced his fourth solo album, These People, released on 20 May 2016. In May 2019, Ashcroft received the Ivor Novello Award for Outstanding Contribution to British Music from the British Academy of Songwriters and Authors. Chris Martin of Coldplay has described Ashcroft as "the best singer in the world". Ashcroft was the only son of hairdresser Louise Ashcroft, his middle name, Paul, is the name of a paternal uncle. When Ashcroft was 11, his father died of a brain haemorrhage.

Ashcroft soon "fell under the influence of his stepfather" to the Rosicrucians. Ashcroft attended Up Holland High School in West Lancashire, near Manchester, along with future bandmates Simon Jones, Peter Salisbury and Simon Tong, attended nearby Winstanley College, where he met Nick McCabe, his teachers referred to him as "the cancer of the class", though one member of staff recalled him being "incredibly intelligent". Ashcroft was an avid football player. For some time, Ashcroft wanted to be a professional football player, idolising George Best, but as he grew older he lost interest in this, turning to music instead. Ashcroft formed The Verve in 1990 with McCabe and Salisbury; the band signed to Hut Records and became well known for their appetite for both psychedelic music and drugs. They became a part of the Britpop movement; the band split in 1995, around this time Ashcroft wrote a collection of songs he intended to release as his first solo album. However, by 1997 he had changed his mind and asked McCabe to return, reforming the Verve and releasing the successful album Urban Hymns.

Ashcroft was at the forefront of the band's popularity, receiving an Ivor Novello Award for his songwriting and being referred to by the press as "the unmistakable face of the Number One rock band in England". However, the pressures of touring and the tensions within the band led to McCabe's departure in mid-1998 and the announcement of the band's break-up in April 1999. In early 2007, Ashcroft made peace with McCabe and Jones and the Verve's reunion was announced in June; the band played gigs that year and continued touring in 2008, headlining at several festivals around the world. A new album, was released in August. In August 2008, the Verve broke up for the third time, though it was not announced until the following year. Ashcroft's first solo single, "A Song for the Lovers", peaked at No. 3 in the UK charts in April 2000. It was followed by the single "Money to Burn" which reached the UK Top 20 at No. 17. The album, Alone with Everybody, was released in June, reaching number 1 and receiving platinum status in the UK.

Album reviews were positive. In September, a third single was released – "C'mon People" – entering the charts at No. 21. Richard does not publish the lyrics to his songs in the inlay cards of his albums or singles as he feels they are personal to him. Ashcroft began work on his second album Human Conditions in 2002; the lead single, "Check the Meaning", was released in early October, peaked at No. 11 on the UK Singles Chart. The album was released that month and reached No. 3 in the UK Album Chart. Reception to the album was positive. Review aggregating website Metacritic reports a normalised score of 61% based on 15 reviews. In response to negative reaction to the album, Coldplay's Chris Martin – a fan of both Ashcroft and the Verve – defended the album's merits which "made an impression" on Ashcroft; the appreciation shown would result in a support slot for Ashcroft, serving as the opening act for Coldplay during a European tour. The album's second single, "Science of Silence", was released the following January and charted at No. 14 in the UK.

On 26 March, Ashcroft made his first live appearance of 2003 at London's Royal Albert Hall as part of the third annual Teenage Cancer Trust charity fundraising event, before "Buy It in Bottles", the third and final single to be taken from the album, was released on 7 April, charting at No. 26. Aside from a limited number of appearances in 2003, Ashcroft was absent from the music business for about two years, he explained this in 2006, stating that "veryone got it into their heads over the last few years that I was in my ivory tower like Lennon, baking bread all day. The fact of the matter was that I was bringing up kids". Ashcroft began playing gigs again in mid-2005, and, on 2 July at the Live 8 concert in Hyde Park, Coldplay invited him to perform with them during their set, they performed the Verve's hit "Bitter Sweet Symphony", after having rehearsed the song in Crystal Palace. Ashcroft's performance of the song was introduced by Chris Martin as "the best song written, here's the best singer in the world", helping to create renewed interest in Ashcroft.

At Christmas 2005, a documentary entitled Live 8: A Bitter Sweet Symphony was aired on the BBC reliving moments of the day featuring a portion of Ashcroft's performance as the show's opening soundtrack. After the disintegration of Hut Records in

Egis Group

Egis Group is a French engineering group involved in the areas of infrastructure and transport systems, planning and environment. Egis is involved in the business of setting up projects and operations for roads and airports. Egis, a subsidiary of the French “Caisse des Dépôts” and “Iosis Partenaires” on a 75%-25% basis, is a consulting and engineering group working in the fields of construction for transport, urban development, industry, the environment and energy; the group is involved in project financing, tolling and airport operations. The new group results from the merger, on 1 January 2011, of Egis, a leader in infrastructure engineering and Iosis, a French leader in construction and civil nuclear engineering. Egis provides engineering services in the field of urban development, from buildings to transport infrastructure: hospitals, tertiary buildings, museums, airports, tramways. Egis Group has interests in many areas, offering a broad range of services, as follows: Transport Roads Urban & Rail Transport Maritime & Inland Waterways Transport Airports & Air Navigation Systems & “Soft” Transport Building & Urban Development Offices Health, Culture, Teaching Sports Facilities, Leisure Activities Commercial centers, Air Terminals Urban Development, New Cities, Eco-districts Industry, Energy & Civil Engineering Tunnels, Structures Nuclear Civil Engineering Geotechnics, Dams Mines, Industry Major Science Projects Water & Environment Drinking Water Supply and Sewerage Systems Hydraulics Industrial Process Water Environment Waste & Polluted Soils Egis Group, through Egis Projects Canada, operates the Golden Ears Bridge in British Columbia: an All Electronic Toll bridge in Metro Vancouver.

Egis Group were selected, as part of a consortium with CS America named Secure Interagency Flow, to build and run a pilot interoperability hub for ATI: the Alliance for Toll Interoperability. Egis Group

John Polanyi Collegiate Institute

John Polanyi Collegiate Institute Sir Sandford Fleming Academy and Sir Sandford Fleming Secondary School is a public high school housed in the former Bathurst Heights Secondary School building located in the North York district of Toronto, serving the area of Lawrence Avenue West and Allen Road. Prior to 1998, the school was part of the North York Board of Education, it is a semestered school offering a "full range of university and apprenticeship programs." The school was named after the Scottish-Canadian inventor of time zones Sandford Fleming. Sir Sandford Fleming Secondary School was constructed and opened in September 1964; the building shares the same design as Newtonbrook Secondary School. The teams name was known as the Chargers. On September 6, 2011, the school was renamed to John Polanyi Collegiate Institute with the Fleming staff and students moved into the former Bathurst Heights Secondary School on Lawrence and Allen. Bathurst Heights Secondary School opened on September 4, 1951 as North York's second high school and opened on May 29, 1952.

Six additions were made throughout the years and adult education was introduced before its program being abolished in 2000. After years of enrollment declining, Bathurst Heights was closed its doors as a regular operating school on June 23, 2001. In 2004 the school was used in the filming of the Zero Hour episode: Massacre at Columbine High. In the 2000s, the Toronto Catholic District School Board used the Bathurst Heights building to house the students from Brebeuf College School during re-construction and Dante Alighieri Academy Beatrice Campus; the former school's building is now leased out by the Toronto District School Board to several tenants: Swim camp Toronto ESL Learning Centre Rosalie Abella, justice of the Supreme Court of Canada Denham Brown, basketball player Paul Godfrey, municipal politician and newspaper publisher Urjo Kareda, late theatre and music critic David Shiner, municipal politician

Billings County Courthouse

The Billings County Courthouse in Medora, North Dakota was built c.1880 and was remodeled in 1913 with architect John Tester. It was listed on the National Register of Historic Places in 1977. According to its NRHP nomination, the courthouse "symbolizes the orderly administration of justice in a county, one of the least populated in North Dakota; the structure is a physical link with the first years of settlement in the region, its core being formed from a building believed to date from the 1880s."The Billings County Museum has been located in the former county courthouse since 1986. Exhibits include a collection of barbed wire, fencing and automotive tools and weapons, military artifacts, horse riding and Western items, displays of ethnic items from area pioneers. Visitors can view the restored courtroom and jail, a period bunkhouse, general store and kitchen; the museum is operated by the Billings County Historical Society. Billings County Museum - official site

History of East Carolina University

The History of East Carolina University can trace its roots to 1901 when City of Wilson citizens went to Raleigh to petition the N. C. General Assembly for an Eastern North Carolina Normal College; the delegation wanted the Normal College in Wilson. The request was denied because the politicians did not want to take money away from the State Normal College. In February 1905, Elizabeth City sent 11 citizens to persuade the General Assembly to charter a State Normal School for Eastern North Carolina; that delegation failed as well. On March 8, 1907, at the urging of State Senator James Leonidas Fleming, the General Assembly passed an act titled: "AN ACT TO STIMULATE HIGH SCHOOL INSTRUCTION IN THE PUBLIC SCHOOLS OF THE STATE AND TEACHER TRAINING". In this act it said "That there shall be established and maintained at some suitable point in eastern North Carolina a teachers' training school for the training of young white men and women under the corporate name of the East Carolina Teachers' Training School."

$15,000 was appropriated to the fund to build the school, $5,000 annually to fund the school. The Board of Trustees was selected on March 15, 1907 and Thomas Jordan Jarvis was chosen as the Chairman. Eight eastern North Carolina towns and cities placed bids on the location of the school; the towns and cities and what they offered is as follows: Washington - $75,000 and the choice of two sites, 200 acres or 133 acres. When Greenville put a bond to the city and County of Pitt, Haywood Dail promised Thomas Jarvis that Greenville would pass the bond. Dail wanted the ballots to be small and have only "For Bond Issue" and "Against Bond Issue" written on them. Dail went on to chew the "Against Bond Issue" to ensure that the bond will pass; the vote passed by 352. The State Board of Education toured all eight cities in the month of June. On June 10, 1907 the SBE voted on the location of the school. On the first vote Kinston received three votes, Rocky Mount received two votes and Greenville received one vote.

On the second vote, all three locations received two votes. On the last and final vote, Greenville received four votes to Kinston two votes. One of the jobs the Board of Trustees did was to choose the President of the School. Robert Herring Wright was selected on June 11, 1907, he set out to recruit faculty. ECTTS had 10 faculty members. On October 5, 1909, Pattie Simmons Dowell was the first student to enroll to the Training School; when classes started in 1909, ECTTS has six buildings. These buildings included two dormitories: Jarvis and Wilson, Administration Building, Power Plant and Refectory. Tuition was free for all students who pledged to teach for two years, $45.00 for all other students. In total, 104 women and 19 men accompanied the 11 faculty on the first day of classes; the School became a two-year Teacher Training School. The first vote by the students was to choose the school colors. June 11, 1911, 16 students became the first graduating class as East Carolina. In 1912, the Athletic League was formed.

The three sports played were basketball and cross-country walking. This was the beginning of the Athletic Department. A Constitution was written for the first Student Self-Government Association. After growing expediently the School decided to change its mission. Mattie Virginia Cox Thornton was the first student in line at East Carolina University in 1908, she rode a wagon pulled by a mule down dirt roads from Kinston to Greenville to be the first to enroll. Mrs. Thornton went on to become a teacher for many years. After enrollment passed 1,000 students, the Administration decided to change the School