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Constructible polygon

In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not. There are infinitely many constructible polygons; some regular polygons are easy to construct with compass and straightedge. The ancient Greek mathematicians knew how to construct a regular polygon with 3, 4, or 5 sides, they knew how to construct a regular polygon with double the number of sides of a given regular polygon; this led to the question being posed: is it possible to construct all regular polygons with compass and straightedge? If not, which n-gons are constructible and which are not? Carl Friedrich Gauss proved the constructibility of the regular 17-gon in 1796. Five years he developed the theory of Gaussian periods in his Disquisitiones Arithmeticae; this theory allowed him to formulate a sufficient condition for the constructibility of regular polygons. Gauss stated without proof that this condition was necessary, but never published his proof.

A full proof of necessity was given by Pierre Wantzel in 1837. The result is known as the Gauss–Wantzel theorem: A regular n-gon can be constructed with compass and straightedge if and only if n is the product of a power of 2 and any number of distinct Fermat primes. In order to reduce a geometric problem to a problem of pure number theory, the proof uses the fact that a regular n-gon is constructible if and only if the cosine, cos ⁡, is a constructible number—that is, can be written in terms of the four basic arithmetic operations and the extraction of square roots. Equivalently, a regular n-gon is constructible if any root of the nth cyclotomic polynomial is constructible. Restating the Gauss-Wantzel theorem: A regular n-gon is constructible with straightedge and compass if and only if n = 2kp1p2...pt where k and t are non-negative integers, the pi's are distinct Fermat primes. The five known Fermat primes are: F0 = 3, F1 = 5, F2 = 17, F3 = 257, F4 = 65537. Since there are 31 combinations of anywhere from one to five Fermat primes, there are 31 known constructible polygons with an odd number of sides.

The next twenty-eight Fermat numbers, F5 through F32, are known to be composite. Thus a regular n-gon is constructible if n = 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96, 102, 120, 128, 136, 160, 170, 192, 204, 240, 255, 256, 257, 272, 320, 340, 384, 408, 480, 510, 512, 514, 544, 640, 680, 768, 771, 816, 960, 1020, 1024, 1028, 1088, 1280, 1285, 1360, 1536, 1542, 1632, 1920, 2040, 2048...while a regular n-gon is not constructible with compass and straightedge if n = 7, 9, 11, 13, 14, 18, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 33, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97, 98, 99, 100, 101, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 123, 124, 125, 126, 127.... Since there are 5 known Fermat primes, we know of 31 numbers that are products of distinct Fermat primes, hence 31 constructible odd-sided regular polygons.

These are 3, 5, 15, 17, 51, 85, 255, 257, 771, 1285, 3855, 4369, 13107, 21845, 65535, 65537, 196611, 327685, 983055, 1114129, 3342387, 5570645, 16711935, 16843009, 50529027, 84215045, 252645135, 286331153, 858993459, 1431655765, 4294967295. As John Conway commented in The Book of Numbers, these numbers, when written in binary, are equal to the first 32 rows of the modulo-2 Pascal's triangle, minus the top row, which corresponds to a monogon; this pattern breaks down after this, as the next Fermat number is composite, so the following rows do not correspond to constructible polygons. It is unknown whether any more Fermat primes exist, it is therefore unknown how many odd-sided constructible regular polygons exist. In general, if there are q Fermat primes there are 2q−1 odd-sided regular constructible polygons. In the light of work on Galois theory, the principles of these proofs have been clarified, it is straightforward to show from analytic geometry that constructible lengths must come from base lengths by the solution of some sequence of quadratic equations.

In terms of field theory, such lengths must be contained in a field extension generated by a tower of quadratic extensions. It follows that a field generated by constructions will always have degree over the base field, a power of two. In the specific case of a regular n-gon, the question reduces to the question of constructing a length cos 2π/n,which is a trigonometric number and hence an algebraic number; this number lies in the n-th cyclotomic field — and in fact in its real subfield, a real field and a rational vector space of dimension ½φ,where φ is Euler's totient function. Wantzel's result comes down to a calculation showing that φ is a power of 2 in the cases specified; as for the construction of Gauss, when the Galois group is 2-group it follows that it has a sequence of subgroups of orders

June 2015 Nationalist Movement Party election campaign

The Nationalist Movement Party election campaign of June 2015 was the official election campaign of the Nationalist Movement Party for the June 2015 general election in Turkey. The campaign was led by Devlet Bahçeli, contesting his fifth successive general election since being elected party leader in 1997. In comparison to the other political parties, the MHP began their campaign late, announcing their manifesto on 3 May 2015; the party's campaign focused on electoral rallies, which were conducted in 60 different Provinces of Turkey within one month and their pledge to end the solution process with the Kurdish rebels. The MHP's rhetoric was dominated by negative campaigning against the AKP, whom Bahçeli accused of corruption and financing terrorism. Since 2013, the MHP was observed to have increased its support, taking many votes from disaffected nationalist voters who had deserted the AKP due to the solution process; the party held its 11th Ordinary Congress on 21 March 2015 as scheduled, which had resulted in the unanimous election of Bahçeli as leader of the party unopposed.

The party's Central Executive Committee due to oversee the party's election campaign was elected. Splits between key party officials, such as those between Bahçeli, Meral Akşener and Sinan Oğan attracted media attention. In the election, the party came third with 16.29% of the vote and came joint third with the Peoples' Democratic Party with 80 seats in Parliament. During the course of the hung parliament, the party opted to steer clear of joining any coalition government, instead calling for an early election; this was called for 1 November 2015. The 11th Nationalist Movement Party Ordinary Congress was held on 21 March 2015; the Congress resulted in the re-election of Devlet Bahçeli as party leader, with rivals failing to obtain enough signatures to contest the leadership election. Held 78 days before the June 2015 general election, the congress determined the MHP's election team and slogans; the main slogan used in the congress was'Walk with us, Turkey!'. Bahçeli's speech focussed on criticising the Solution process initiated by the Turkish government with Kurdish separatists, thereby indicating the main campaign issues and policies in the preparation for the general election.

The MHP has implemented one of the lowest application fees of any party for potential candidates. Between 12 and 18 March, all male and disabled citizens wishing to apply had to pay a fee of ₺2,000; the MHP had charged ₺2,500 in the 2011 general election. In response to plans to build a hotel in a greenfield site at İstinye in İstanbul, the MHP organised a seed-planting protest in the area In January. Actor Mehmet Aslan declared his potential candidacy during the event; the MHP has been predicted to build on its success in the 2014 local elections and improve its vote share, with support coming from former AKP nationalist voters who are disillusioned with the Kurdish peace process. The MHP leader Devlet Bahçeli claimed that there was a low possibility that such voters would return to the AKP, accusing the AKP of continuously changing ideologies from the Islamist National View to the more moderate Conservative democracy; the MHP has targeted the AKP in their campaign, accusing the Supreme Electoral Council of serving no-one apart from Recep Tayyip Erdoğan and vowing to bring violations of the law to justice in the future.

Such promises include the sending of Erdoğan to the Court of Cassation and turning the new presidential palace into a museum about Atatürk and the Republic. The MHP began its electoral rallies on 6 May in the north of the country, with Bahçeli making stops at local cafes during his campaign to directly meet with voters. Bahçeli endorsed the new legal process that had begun against the Gülen Movement, though criticised the AKP due to their previous relations with the Movement. In May, the AKP appeared to have started a smear campaign against Meral Akşener, one of the MHP's most prominent politicians, claiming to be in possession of a video recording of her with private content. Akşener subsequently began legal proceedings against two pro-government journalists whom she accused of starting the defamation campaign against her. One of these journalists, Latif Erdoğan, claimed that the Gülen Movement had obtained the recording to blackmail Akşener, though both Akşener and Gülen's lawyer denied the claims and accused them of being baseless.

The MHP's manifesto pledged to extend education to nine years and to end the Solution process with Kurdish militants. In their constitutional reform proposals, the party promised to keep a constitutional reference to the Turkish ethnicity while not allowing other minorities to obtain constitutional recognition. Pressing for a manufacturing based economy, the MHP has pledged to investigate every individual involved in the 2013 government corruption scandal and to establish an anti-corruption commission to lead the investigation; the MHP has promised a strong, international response to terrorism and to raise the minimum wage to ₺1,400. The MHP spent over ₺40 million of the ₺77 million it obtained from the Treasury on its election campaign, holding a rally in Rize for the first time in 20 years. In the space of one month, the MHP conducted election rallies in 60 provinces, with one clashing with a HDP rally in Mersin. Numerous breakouts of violence between MHP and HDP supporters were documented in their respective election events.

One of the MHP's election songs was an adaptation of the popular Turkish folk music song Ankara'nın Bağları. Justice and Development Party election campaign, June 2015 Republican People's Party electi

1995 FIBA Americas Championship for Women

The 1995 FIBA Americas Championship for Women, was the third FIBA Americas Championship for Women regional basketball championship held by FIBA Americas, which served as Americas qualifier for the 1996 Summer Olympics, granting berths to the champion and runner-up. It was held in Canada between June 22 and June 29, 1995. Five national teams entered the event under the auspices of FIBA Americas, the sport's regional governing body; the city of Hamilton hosted the tournament. Canada won their first title after defeating hosts Cuba in the final; the preliminary stage consisted of a single round-robin group. The top four teams advanced to the knockout semifinals, where the winners qualified directly to the 1996 Summer Olympics; the winners from the semifinals competed for the championship. There was no third place game. 1995 American Olympic Qualifying Tournament for Women, FIBA.com. Retrieved January 22, 2015

Bodoland People's Front

The Bodoland People's Front is a state political party in Assam, India. The party has its headquarters in Kokrajhar Town and is in power in the autonomous region of Bodoland, it is an ally of Bharatiya Janata Party whose alliance known as NDA formed the Government of Assam in 2016, it is a member of the regional alliance of North East Democratic Alliance. In the 2009 general election its candidate, Sansuma Khunggur Bwiswmuthiary was elected to the 15th Lok Sabha from Kokrajhar constituency. Biswajit Daimary of this party was elected to the Rajya Sabha in May 2008. BPF had 10 members in the 12th Assam Legislative Assembly, it is a constituent of the current Bharatiya Janata Party led ruling coalition in Assam. In the 2011 Assam Assembly election, BPF won 12 seats. In 2011, with 12 elected lawmakers, the BPF supported the Congress government because of the efforts of Himanta Biswa Sarma, the former minister in the Tarun Gogoi government who joined the Bharatiya Janata Party. With Assam heading for elections in 2016, Bharatiya Janata Party made its first alliance with the Bodoland People's Front in lower Assam.

The announcement was made on 19 January when Prime Minister Narendra Modi visits Kokrajhar headquarters of the autonomous Bodoland Territorial Council and announces a possible economic package of Rs 1000cr or the Bodoland Territorial Council area. It was declared that Out of 126 Assembly constituencies, Bharatiya Janata Party would contest in 84 constituencies and leave 42 others to its four allies - Asom Gana Parishad, BPF, Rabha Jatiya Aikya Manch and Tiwa Jatiya Aikya Manch. In the 2016 Assembly Elections, the BPF won 12 seats and the alliance was established after Bharatiya Janata Party led National Democratic Alliance which includes Asom Gana Parishad formed its first government in Assam with Pramila Rani Brahma and Rihon Daimary sworn in as ministers in the Sarbananda Sonowal ministry. During the cabinet expansion Shri Chandan Brahma was inducted as Cabinet minister for Tourism and other departments. In May 2016, after the Bharatiya Janata Party led National Democratic Alliance formed its first government in Assam, formed a new alliance called the North-East Democratic Alliance with Himanta Biswa Sarma as its convener.

The Chief Ministers of the north eastern states of Sikkim and Nagaland too belong to this alliance. Bodo Liberation Tigers Force Bodo People's Progressive Front

Little tern

The little tern is a seabird of the family Laridae. It was placed into the genus Sterna, which now is restricted to the large white terns; the genus name is a diminutive of Sterna, "tern". The specific albifrons is from Latin albus, "white", "frons", forehead; the former North American and Red Sea S. a. saundersi subspecies are now considered to be separate species, the least tern and Saunders's tern. This bird breeds on the coasts and inland waterways of Asia, it is migratory, wintering in the subtropical and tropical oceans as far south as South Africa and Australia. There are three subspecies, the nominate albifrons occurring in Europe to North Africa and western Asia; the little tern breeds in colonies on islands. It lays two to four eggs on the ground. Like all white terns, it will attack intruders. Like most other white terns, the little tern feeds by plunge-diving for fish from saline environments; the offering of fish by the male to the female is part of the courtship display. This is 21 -- 25 cm long with a 41 -- 47 cm wingspan.

It is not to be confused with other species, apart from fairy tern and Saunders's tern, because of its size and white forehead in breeding plumage. Its thin sharp bill is yellow with a black tip and its legs are yellow. In winter, the forehead is more extensively white, the legs duller; the call is a distinctive creaking noise. The little tern was described by the German naturalist Peter Simon Pallas in 1764 and given the binomial name Sterna albifrons. At the beginning of the 19th century the little tern was a common bird of European shores and wetlands, but in the 20th century populations of coastal areas decreased because of habitat loss and human disturbance; the loss of inland populations has been more severe, since due to dams, river regulation and sediment extraction it has lost most of its former habitats. The Little Tern population has declined or become extinct in many European countries, former breeding places on large rivers like the Danube and Rhine ceased. Nowadays, only few river systems in Europe possess suitable habitats.

The status of the little tern on the rivers Tagus and lower Danube is uncertain. The Drava population is one of the most threatened. Old fashioned water management practices, including river regulation and sediment extraction, endanger the remaining pairs. Only 15 pairs still breed on extensive sand or gravel banks along the border between Hungary and Croatia; the WWF and its partners are involved in working for the protection of this bird and this unique European river ecosystem. The little tern is one of the species to which the Agreement on the Conservation of African-Eurasian Migratory Waterbirds applies. BTO Little tern fact sheet Little tern - Species text in The Atlas of Southern African Birds Sterna albifrons in the Flickr: Field Guide Birds of the World BBC Norfolk news item Little tern

2002 Euro Beach Soccer Cup

The 2002 Euro Beach Soccer Cup was the fourth Euro Beach Soccer Cup, one of Europe's two major beach soccer championships at the time, held in February 2002, in Barcelona, Spain. Portugal won the championship, claiming their second successive title and third overall, with hosts Spain finishing second. France beat Italy in the third place play off to finish fourth respectively. Eight teams participated in the tournament who played in a straightforward knockout tournament, starting with the quarter finals, with extra matches deciding the nations who finished in fifth, sixth and eighth place. England France Germany Republic of Ireland Italy Portugal Spain Switzerland The following matches took place between the losing nations in the quarter finals to determine the final standings of the nations finishing in fifth to eighth place; the semi finals took place on the same day of the semi finals of the main tournament and the play offs took place on the day of the final