Significance, established in 2004, is a bimonthly print and digital magazine published by Oxford University Press on behalf of the Royal Statistical Society (RSS), the Statistical Society of Australia (SSA) and the American Statistical Association (ASA). It publishes articles on topics of statistical interest presented at a level suited for a general audience. Articles are reviewed by an editorial board of statistics experts drawn from the three societies. The founding editor-in-chief was Helen Joyce. The current editor is Anna Britten. Significance replaced the RSS's journal, The Statistician. The magazine also has a website.

In addition to ordinary articles in the magazine, additional "virtual issues" (collections of articles on a particular subject area) are made available online. In November 2010 the magazine launched its website. Having been launched as a quarterly magazine, Significance changed to a bimonthly frequency in 2011. In 2020, the regular column 'Dr Fisher's Casebook' was renamed 'the secret statistician', a review prompted by concern about Fisher's views on eugenics (and supported by the lack of salience of the homage to Dr Finlay's Casebook).

Members of either the RSS or the ASA receive the magazine as part of their membership. In January 2015, the RSS and ASA decided to make the magazine issues available to the public free of charge a year after their publication.

References

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 Magazine page at publisher's website

2004 establishments in the United Kingdom
American Statistical Association
Bi-monthly magazines published in the United Kingdom
Science and technology magazines published in the United Kingdom
English-language magazines
Magazines established in 2004
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Significance (magazine)


Richard Henry Lee Chichester (April 18, 1870 – February 2, 1930) was an American judge from Virginia.

Chichester was born in Fairfax County, Virginia. His father was for many years a judge of the county courts of Fairfax and Alexandria, and his grandfather was Judge Richard C. L. Moncure, a judge and president of the Supreme Court of Appeals of Virginia. Chichester received his early education in private schools in Fairfax County and at St. John's Military Academy in Alexandria. His higher and legal education was received at the University of Virginia. At twenty-one, he was admitted to the bar and began practice in Stafford County, Virginia where he was shortly elected Commonwealth's Attorney. He was elected judge of Stafford and King George Counties in 1898, serving until 1904 when the county courts were abolished. Resuming practice at that time, he entered a law partnership with his brother in Fredericksburg until his appointment, in 1910, as judge of the Fifteenth Judicial Circuit. In 1924, he was appointed a judge of the Special Court of Appeals where he served until June 1, 1925. Upon the death of Judge Joseph L. Kelly in June 1925, Chichester was elected to the Supreme Court of Appeals of Virginia. He remained on the court until his death.

References

Justices of the Supreme Court of Virginia
Virginia state court judges
Virginia lawyers
1870 births
1930 deaths
Moncure family
People from Fairfax County, Virginia
People from Stafford County, Virginia

Richard Henry Lee Chichester


Kweku Baako Jnr is a Ghanaian journalist and the editor-in-chief of the New Crusading Guide newspaper. He is a frequent panelist on radio shows in Ghana, including Peace FM's weekday morning show and Newsfile, a current affairs programme on Joy FM (Ghana). In 1999 he was awarded the Journalist of the Year by the Ghana Journalists Association. He was polled as the 39th most influential Ghanaian in 2014 by ETV.

References

Living people
Ghanaian journalists
Year of birth missing (living people)

Kweku Baako Jnr


The Dingling were an ancient people who appears in Chinese historiography in the context of the 1st century BCE.

The Dingling are considered to have been an early Turkic-speaking people. They were also proposed to be the ancestors of Tungusic speakers among the later Shiwei people, or are related to Na-Dené and Yeniseian speakers.

Modern archaeologists have identified the Dingling as belonging to the eastern Scythian horizon, namely the Tagar culture.

Etymology
The ethnonym 'Dingling' is regarded by modern scholars in the Western world as being interchangeable with the ethnonym 'Tiele', who are believed to be the descendants of the Dingling. Chinese historiographers believed that 'Tiele' was a mistaken transription, related them to the ancient Red Di (狄翟), and recorded various names like Dili (狄历), Gaoche (高車) or Chile (敕勒).

Several modern scholars, including Peter B. Golden, now believe that all of these ethnonyms described by the Chinese all derive from Altaic exonyms describing wheeled vehicles, with 'Dingling' perhaps being an earlier rendering of a Tuoba word (*tegreg), meaning "wagon".

Peter Golden also wrote that "Gaoche" or "high carts" may be a translation of "Dingling et al.". Edwin Pulleyblank writes that "High Cart" is just one of several variations of exonyms that ultimately reflect the original Turkic meaning of 'Dingling', which is possibly derived from *Tägräg, meaning "circle, hoop".

Origin and migration

The Weilüe mentioned three Dingling groups:
one group south of Majing (馬脛, literally: "Horse-Shank[ed people]"), north of Kangju, and west of Wusun;
another south of the North Sea, identified as Lake Baikal;
and another north of Xiongnu and neighbouring the Qushi (屈射), Hunyun (渾窳), Gekun (隔昆), and Xinli (薪犁), all of whom had once been conquered by the Xiongnu.

Murphy (2003) proposes that the Dingling's country had been in the Minusinsk Basin on the Yenisey river, thus close to the location of the Dingling group who neighbored the Kangju, Wusun, and Horse-Shanked people. Dingling gradually moved southward to Mongolia and northern China. They were a huge independent horde for centuries, but were later defeated and temporarily became subject of the Xiongnu Empire, and thus presumably related to the invaders known as Huns in the west. One group, known as the West Dingling, remained in an area that would become Kazakhstan, while others – expelled from Mongolia by the Rouran – settled in the Tarim Basin during the 5th century and took control of Turpan.

The Dingling had a warlike society, formed by traders, hunters, fishers, and gatherers, living a semi-nomadic life in the southern Siberian mountain taiga region from Lake Baikal to northern Mongolia.  Some ancient sources claims that Di or Zhai (翟) was adopted as the group name because the Zhai family had been the ruling house for centuries.

Other sources claim that they might have been correlated with the Guifang, a northern tribe that appears in the oracle bone inscriptions from Yinxu.

According to the History of the Gaoche of Wei Shou (6th century), the origin of the Dingling can be traced to the Chidi (赤狄) (lit. Red Di), who lived in northern China during the Spring and Autumn period. The Mozi mentions a total of eight related Di groups, of whom only "Red Di" (赤狄, Chidi), the "White Di" (白狄, Baidi), and "Tall Di" (長狄, Changdi) are known.

To the north of the Xiongnu empire and Dingling territories, at the headwaters of the Yenisei around Tannu Uriankhai, lived the Gekun (鬲昆), also known as the Yenisei Kirghiz in later records. Further to the west near the Irtysh river lived the Hujie (呼揭). Other tribes living  of the Xiongnu, such as the Hunyu (浑庾), Qushe (屈射), and Xinli (薪犁), were only mentioned once in Chinese records, and their exact location is unknown.

During the 2nd century BCE, the Dingling became subjects of Modu Chanyu along with 26 other tribes, including the Yuezhi and Wusun.

Dingling and Xiongnu

The Dingling were first subjugated by the Xiongnu, but the latter gradually weakened. In 71 BCE, after numerous conflicts between the Chinese and the Xiongnu, the Dingling, led by Zhai Jin, with help from neighboring tribes, took the opportunity to revolt. From 63 to 60 BCE, during a split within the Xiongnu ruling clan of Luanti (挛鞮), the Dingling attacked the Xiongnu, together with the Wusun from the west, supported by the Chinese from the south and the Wuhuan from the southeast.

In 51 BCE, the Dingling, together with the Hujie and Gekun, were defeated by the Xiongnu under Zhizhi Chanyu, on his way to Kangju. Over the next century there may have been more uprisings, but the only recorded one was in the year 85, when together with the Xianbei they made their final attack on the Xiongnu, and Dingling regained its power under Zhai Ying. After that, under the Dingling pressure, the remaining of northern Xiongnu and the Tuoba formed the confederacy by Xianbei chief Tanshihuai (檀石槐). After his death in 181, the Xianbei moved south and the Dingling took their place on the steppe.

Some groups of Dingling, called the West Dingling by the ancient Chinese, started to migrate into western Asia, but settled in Kangju (康居), modern day Kazakhstan and Uzbekistan. There is no specific source to tell where exactly they settled, but some claim Lake Zaysan (宰桑 or 斋桑).

Assimilation
Between the short-lived Xianbei confederacy in 181 and the foundation of the Rouran Qaghanate in 402, there was a long period without a tribal confederacy on the steppe. During this period, a part of the Dingling were assimilated to the northern Xiongnu by permanently settling further to the south. Another group, documented as about 450,000, moved southeast and merged into the Xianbei.

Some groups of Dingling settled in China during Wang Mang's reign. According to the Weilüe, another group of Dingling escaped to the western steppe in Kazakhstan, which has been called the West Dingling. Around the 3rd century, Dinglings living in China began to adopt family names such as Zhai or Di (翟), Xianyu (鲜于), Luo (洛) and Yan (严). These Dingling became part of the southern Xiongnu tribes known as Chile (赤勒) during the 3rd century, from which the name Chile (敕勒) originated.

During the Sixteen Kingdoms period, the West Dingling Khan Zhai Bin (翟斌) lead his hordes, migrate from Kazakhstan into Central China, served under the Former Qin, after series of plotting, Zhai Bin was betrayed by Former Qin, to avoid Qin nobles further attempts, he revolted against the Former Qin Dynasty. Murong Chui (慕容垂), the Xianbei leader under Former Qin court, got appointed as the high command of Former Qin army, was expected to take down the revolt, but convinced by Zhai Bin, joined his mutiny to against Former Qin. Their mutiny were also joined by several other Xianbei tribes which formed the Anti-Qin leagues, with the suggestion by Zhai Bin, Murong Chui was elected to be the leader of the leagues. Near end of the same year, Murong Chui styled himself King of Yan (燕王), left Zhai Bin the new leader of the league and a dilemma of the war, later Murong Chui broke the alliance with the leagues, murdered Zhai Bin and his three sons in an ambush. His nephew Zhai Zhen (翟真) inherited the horde, was elected be the new Leader of the leagues, seeking for revenge, but later assassinated by his military advisor Xianyu Qi (鲜于乞), Xian did not escape far, were caught by the Dingling soldiers and got executed, the leagues elected Zhai Zhen's cousin Zhai Cheng (翟成) as the new Leader, but later also been assassinated by Yan spy, then Zhai Liao (翟辽), became the new leader of Dingling horde, with the support from the Leagues, he founded the Wei state, a DingLing Dynasty in China in modern Henan Province.

About one-quarter of the Tuoba clans show similar names as found among the later Gaoche and Tiele tribes. Among them, the Hegu (紇骨) and Yizhan (乙旃) clans kept their high status.

Between the 4th and 7th centuries, the name "Dingling" slowly disappeared from Chinese records, coinciding with the rise of the Uyghur Khaganate.

Cultural and linguistic theories
Several theories have been proposed about the relationship between the Dingling and both ancient and living cultures, based on linguistic, historical and archaeological evidence.

Turkic hypothesis
The Dingling are considered to have been an early Turkic-speaking people.

Weilüe records the Dingling word for the artic fox (vulpes lagopus) as 昆子 kūnzǐ (Middle Chinese (ZS): *kuən-t͡sɨX < Early Middle Chinese: *kwən-tsɨ'/tsi' < Eastern Han Chinese: *kûn-tsəʔ), which is proposed to be from Proto-Turkic *qïrsaq ~ *karsak.

Tungusic hypothesis
Chinese historians linked the Tungusic speakers among the later Shiwei people to the Dingling, considering them as descendants of the Dingling owing to linguistic similarities.

Dené-Yeniseian hypothesis
In Zur jenissejisch-indianischen Urverwandtschaft (Concerning Yeniseian-Indian Primal Relationship), the German scholar, Heinrich Werner developed a new language family which he termed Baikal–Siberic. By extension, he groups together the Yeniseian peoples (Arin, Assan, Yugh, Ket, Kott, and Pumpokol), the Na-Dene Indians, and the Dingling of Chinese chronicles to Proto-Dingling. The linguistic comparison of Na-Dene and Yeniseian shows that the quantity and character of the correspondences points to a possible common origin. According to Russian linguistic experts, they likely spoke a polysynthetic or synthetic language with an active form of morphosyntactic alignment, exhibiting a linguistically and culturally unified community.

The name Dingling resembles both:
 the Yeniseian word *dzheng people >
 the Na-Dene word *ling or  people, i.e. as manifested in the name of the Tlingit (properly  son of man, child of the people).

Although the Dené–Yeniseian language family is now a widely known proposal, his inclusion of the Dingling is not widely accepted.

Physical appearance
There is some evidence that the Dingling looked similar to European people, based on their identification with the Tagar culture of the Altai region in Siberia. In the 20th century, several historians proposed that the Tagar people were characterized by a high frequency of light hair and light eyes, and that the associated Dingling were blond-haired. Genetic testing of fossils from the Tagar culture has confirmed the theory that they were often blue eyed and light-haired. Twenty-first century scholars continue to describe the Dingling in a similar manner. Adrienne Mayor repeated N. Ishjants' description (1994) of the Dingling as "red-haired, blue-eyed giants" while M.V. Dorina called the Dingling "European-looking."

The Chinese sources do not differentiate the Dingling's appearance from the Han Chinese. Chinese histories unanimously depict the Dingling as the ancestors of the Tiele, whose physical appearance is also not described, but seem to have included non-Turkic speaking peoples. The Alans, an Iranic people, are included among them, as well as the Bayegu, who had a somewhat different language than the Tiele according to the New Book of Tang. The New Book also relates that the Kyrgyz intermixed with the Dingling. The Book of Sui states that the Tiele had similar customs to the Göktürks but different marriage and burial traditions.

The Classic of Mountains and Seas described the Dingling as human beings with horses' legs and hooves and excellent at running. However, this description is mythological in nature. A similar description is also echoed in a Wusun account, recorded in the Weilüe (compiled 239-265 CE), which describes the men of Majing ("Horse Shanks"), located north of the Dingling, as possessing horse legs and hooves.

See also
 List of indigenous peoples of Russia
 Xunyu
 Zhai Wei

Notes

References

Further reading
 Duan, Lianqin (1988). Dingling, Gaoju and Tiele. Shanghai: Shanghai People's Press, 1988.
Hill, John E. (2004). "The Peoples of the West" from the Weilüe, Section 15. (Draft version). Downloadable from: .
 Li, Jihe (2003). A Research on Migration of Northwestern Minorities Between pre-Qin to Sui and Tang. Beijing: Nationalities Press.
 Lu, Simian (1996). A History of Ethnic Groups in China. Beijing: Oriental Press.
 Shen, Youliang (1998). A Research on Northern Ethnic Groups and Regimes. Beijing: Central Nationalities University Press.
 Suribadalaha (1986). New Studies of the Origins of the Mongols. Beijing: Nationalities Press.
 Trever, Camilla (1932). Excavations in Northern Mongolia (1924-1925). Leningrad: J. Fedorov Printing House.
 Xue, Zongzheng (1992). A History of Turks. Beijing: Chinese Social Sciences Press.

Ancient peoples of China
Ethnic groups in Chinese history
Ancient Central Asia
History of Siberia
History of the Turkic peoples
Sixteen Kingdoms

Dingling


James Economou (born December 6, 1988) is a former American football offensive lineman. He is one of Southern Connecticut State University players who entered the AFL in 2010 as a 2009 All-American.

College career
Economou attended Southern Connecticut State University in New Haven, Connecticut. There he was a standout offensive lineman who was named an All-American, All-Region and All-Conference player. He was a starter at guard, helping lead the offense to rank second in the league in rushing and passing yards per game, and first in scoring and total offense.

Professional career

Pre-draft
Prior to the 2011 NFL Draft, Economou was projected to be undrafted by NFLDraftScout.com. He was rated as the 129th-best offensive lineman in the draft. He was not invited to the NFL Scouting Combine, he posted the following numbers during his pro-day workouts at Fordham University:

Milwaukee Mustangs
After going undrafted in the 2011 NFL Draft, Economou signed with the Milwaukee Mustangs of the Arena Football League.

Cleveland Gladiators
Economou was with the Cleveland Gladiators in 2012.

Iowa Barnstormers
Economou has re-signed with the Iowa Barnstormers for the 2013 season.

References

American football offensive linemen
1988 births
Living people
Iowa Barnstormers players
Milwaukee Mustangs (2009–2012) players
Cleveland Gladiators players
People from Coram, New York

James Economou


The csikós is the mounted horse-herdsman of Hungary. The csikós tradition is closely associated with the Hungarian puszta, in recent times particularly in the environs of Debrecen and Hortobágy National Park. It is also closely linked to the Nonius breed of horse, for which one of the two principal breeding centres is the Máta Stud, some 3 kilometres from Hortobágy.

Images

See also
Betyárs
Hajduk (soldiers)

References

Animal husbandry occupations
Horse-related professions and professionals

Csikós


The Della Rovere or Saint Jerome Chapel, otherwise the Chapel of the Nativity () is the first side chapel in the south aisle of the Basilica of Santa Maria del Popolo in Rome. It was dedicated to the Virgin and Saint Jerome and decorated with the paintings of Pinturicchio and his pupils. It is one of the best preserved monuments of quattrocento art in Rome.

History
The chapel was bought by Cardinal Domenico della Rovere in 1477 after the reconstruction of the church by his relative, Pope Sixtus IV. The interior of the chapel with the tomb of Cardinal Cristoforo della Rovere (†1478), the marble altar and the pictorial decoration was executed in the following years. The frescoes are attributed to Pinturicchio and his school. According to the traditional dating he worked in the chapel from 1488 to 1490, at the same time when he created the frescoes of Palazzo Della Rovere (now Palazzo dei Penitenzieri) for the cardinal.

Other studies, for example those of Strinati (1995), have proposed an earlier date, before the Bufalini Chapel in Santa Maria in Aracoeli, oscillating between the 1470s and 1482, supported by the dedicatory epigraph of the cardinal that does not report his title of archbishop of Turin which he only got in that year.

The chapel changed little during the centuries but suffered an important alteration when the tomb of Giovanni da Castro was moved here from its original place on the counterfaçade. This happened during the general reconstruction of the basilica by Pope Alexander VII in 1655–61. The original painted decoration of the fifth wall is hidden behind the large monument.

Description
The chapel is hexagonal with a sexpartite ribbed vault. The side walls are articulated by painted Corinthian pilasters decorated with grotesques, resting on a monochrome base. The ribs and the splays of the two arched windows are decorated with similar grotesques. The main altar-piece, The Adoration of the Child with St. Jerome is an exquisite autograph work by Pinturicchio. The marble slabs of the parapet are decorated with Cardinal Della Rovere's coat-of-arms held by two putti. The entrance arch is supported by travertine half-columns which are similar to other original half-columns of the basilica. They have Corinthianesque capitals with a palmette motif between the scrolls and they were painted gold (these are the only capitals in the basilica without a later plaster coat).

The polychrome grotesques on yellow-gold background are of the highest quality and were enhanced by a cleaning in the 1990s. The rich repertoire of figures contain masks, swans, camels, shells, musical instruments etc. They were painted by a pictor doctus with an ease comparable to the artists of the imperial era. They are therefore ascribed to the hand of the master who put into practice a bold experiment, one of the most successful of its kind.

La Malfa claims that the grotesques on the pilasters were clearly inspired by those painted in the Domus Aurea (the so-called Room of Achilles on Skyros) and the wall decoration was executed no later than 1479, that is before the building of the tomb of Cristoforo Della Rovere. In this case they could be among the earliest of their kind in Rome.

Vault and lunettes
The vault was decorated with delicate all'antica motifs before being covered with a blue carpet and gold stars due to its bad state. There are five frescoes in the lunettes from the Life of St. Jerome, detached and transferred to canvas in the 18th century. The scenes are:

St Jerome debating the perpetual virginity of Mary with the heretic Helvidius
St Jerome in the desert
St Jerome extracting a thorn from a lion's paw
St Jerome in his study with St Augustine
St Jerome's death

The latter was hidden behind the monument of Giovanni de Castro, placed there in the 17th century, and was discovered only recently: perhaps this is the portrait of Cardinal Della Rovere mentioned by Giorgio Vasari.

The lunette paintings are worn and only their general compositional structure could be appreciated which according to Strinati was influenced by the Roman painters of the era: Piermatteo d'Amelia and Antoniazzo Romano. Some deny the authenticity of the lunettes attributing them to the assistant Tiberio d'Assisi. The paintings were originally richly gilded as evidenced by the presence of red wax on the edges of the garments that served as a support for the leaves of precious metal.

The Adoration of the Child

The Adoration of the Child with St. Jerome is located above the main altar, framed by a carved and gilded marble arch. It was always referred to as an autograph work by Pinturicchio whose master workmanship was confirmed during the restoration. In front of the hut of the Nativity, partially ruined and its walls built of different materials (symbolizing the Jewish and pagan religions which predated the rise of Christianity), and with a truss roof seen from underneath, the Holy Family is seen together with St. Jerome and the shepherds adoring the Child. Jesus is resting on a bundle of wheat, reference to the bread of the Eucharist. To the right the ox and the donkey are penned behind a fence of woven twigs while Joseph is represented in a typical dormant posture alluding to his role as a mere guardian of Mary and Jesus without active participation in the procreation. The rich background fades in the distance according to the rules of perspective which makes things appear distantly blurred in bluish colours through the effect of haze; it is populated by a city on the banks of a lake and a series of fantastic rocky spurs, creating an atmospheric setting for the procession of the Magi and the angelic announcement to the shepherds, located at the top center. In the middle an extremely slender tree acts as a pivot for the background and separates the two parts, re-creating equilibrium.

The most beautiful pictorial details are the heads of Mary and the Child; the vivacity of looks and gestures recalls the Adoration of Antoniazzo Romano.

The dedication plaque for the chapel is set at the centre of the base between two Della Rovere coat-of-arms with the cardinal's hat:

DOMINICVS RVVERE CARD[INALI]S S[ANCTIS] CLEMENTIS CAPPELLA MARIAE VIRG[INIS] GENETRICI DEI AC DIVO HIERONIMO DICAVIT.

Domenico della Rovere, the cardinal of San Clemente dedicated this chapel to the Virgin Mary, mother of God, and St. Jerome

Sculptures
The tomb of Cardinal Cristoforo della Rovere, the Bishop of Tarentaise (died in 1478) is one of the best 15th century sculpted funeral monuments in Rome. It is the work of Andrea Bregno but the relief of the Virgin and the Child in the lunette is attributed to Mino da Fiesole. According to the inscription the tomb was prepared by Domenico della Rovere for his brother and himself (c. 1479). The funeral monument of Jorge da Costa in the Costa Chapel shows a strong similarity to this tomb.

The other tomb in the chapel is the sepulchre of Cardinal Juan de Castro who died in 1506. Francesco Sangallo's 16th century work is sometimes compared unfavourably to the superb Della Rovere monument: "With its restless flaunting angels, its pretentious ornament, its self-conscious Virtues, its total lack of quiet repose and thoughtfulness, has in it all the elements of the decay which was to overtake Italian sculpture before the century went out."

The tombstone of the Venuta family is set in the pavement. The slab is decorated with the family coat-of-arms (two maces crossed), and has a Renaissance frame with floral engravings.

Gallery

Notes

Bibliography
Cristina Acidini, Pintoricchio, in Pittori del Rinascimento, Scala, Firenze 2004.
Claudia La Malfa, The Chapel of San Girolamo in Santa Maria del Popolo in Rome, Journal of the Warburg and Courtauld Institutes, 2000, LXIII, pp. 259–270
Gerald S. Davies: Renascence. The Sculptured Tombs of the Fifteenth Century in Rome, E. P. Dutton and Company, New York, 1916

Della Rovere
Church frescos in Rome
Paintings by Pinturicchio
Renaissance paintings
Della Rovere family
Rome R. IV Campo Marzio

Della Rovere Chapel (Santa Maria del Popolo)


The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. Indeed,  the reassembly process involves only moving the pieces around and rotating them without changing their shape. However, the pieces themselves are not "solids" in the usual sense, but infinite scatterings of points.  The reconstruction can work with as few as five pieces.

An alternative form of the theorem states that given any two "reasonable" solid objects (such as a small ball and a huge ball), the cut pieces of either one can be reassembled into the other. This is often stated informally as "a pea can be chopped up and reassembled into the Sun" and called the "pea and the Sun paradox".

The theorem is called a paradox because it contradicts basic geometric intuition. "Doubling the ball" by dividing it into parts and moving them around by rotations and translations, without any stretching, bending, or adding new points, seems to be impossible, since all these operations ought, intuitively speaking, to preserve the volume. The intuition that such operations preserve volumes is not mathematically absurd and it is even included in the formal definition of volumes. However, this is not applicable here because in this case it is impossible to define the volumes of the considered subsets. Reassembling them reproduces a set that has a volume, which happens to be different from the volume at the start.

Unlike most theorems in geometry, the mathematical proof of this result depends on the choice of axioms for set theory in a critical way. It can be proven using the axiom of choice, which allows for the construction of non-measurable sets, i.e., collections of points that do not have a volume in the ordinary sense, and whose construction requires an uncountable number of choices.

It was shown in 2005 that the pieces in the decomposition can be chosen in such a way that they can be moved continuously into place without running into one another.

As proved independently by Leroy and Simpson, the Banach–Tarski paradox does not violate volumes if one works with locales rather than topological spaces. In this abstract setting, it is possible to have subspaces without point but still nonempty. The parts of the paradoxical decomposition do intersect a lot in the sense of locales, so much that some of these intersections should be given a positive mass. Allowing for this hidden mass to be taken into account, the theory of locales permits all subsets (and even all sublocales) of the Euclidean space to be satisfactorily measured.

Banach and Tarski publication
In a paper published in 1924, Stefan Banach and Alfred Tarski gave a construction of such a paradoxical decomposition, based on earlier work by Giuseppe Vitali concerning the unit interval and on the paradoxical decompositions of the sphere by Felix Hausdorff, and discussed a number of related questions concerning decompositions of subsets of Euclidean spaces in various dimensions. They proved the following more general statement, the strong form of the Banach–Tarski paradox:

 Given any two bounded subsets  and  of a Euclidean space in at least three dimensions, both of which have a nonempty interior, there are partitions of  and  into a finite number of disjoint subsets, ,  (for some integer k), such that for each (integer)  between  and , the sets  and  are congruent.

Now let  be the original ball and  be the union of two translated copies of the original ball. Then the proposition means that you can divide the original ball  into a certain number of pieces and then rotate and translate these pieces in such a way that the result is the whole set , which contains two copies of .

The strong form of the Banach–Tarski paradox is false in dimensions one and two, but Banach and Tarski showed that an analogous statement remains true if countably many subsets are allowed. The difference between dimensions 1 and 2 on the one hand, and 3 and higher on the other hand, is due to the richer structure of the group  of Euclidean motions in 3 dimensions. For  the group is solvable, but for   it contains a free group with two generators. John von Neumann studied the properties of the group of equivalences that make a paradoxical decomposition possible, and introduced the notion of amenable groups. He also found a form of the paradox in the plane which uses area-preserving affine transformations in place of the usual congruences.

Tarski proved that amenable groups are precisely those for which no paradoxical decompositions exist. Since only free subgroups are needed in the Banach–Tarski paradox, this led to the long-standing von Neumann conjecture, which was disproved in 1980.

Formal treatment
The Banach–Tarski paradox states that a ball in the ordinary Euclidean space can be doubled using only the operations of partitioning into subsets, replacing a set with a congruent set, and reassembling. Its mathematical structure is greatly elucidated by emphasizing the role played by the group of Euclidean motions and introducing the notions of equidecomposable sets and a paradoxical set. Suppose that  is a group acting on a set . In the most important special case,  is an -dimensional Euclidean space (for integral n), and  consists of all isometries of , i.e. the transformations of  into itself that preserve the distances, usually denoted . Two geometric figures that can be transformed into each other are called congruent, and this terminology will be extended to the general -action. Two subsets  and  of  are called -equidecomposable, or equidecomposable with respect to , if  and  can be partitioned into the same finite number of respectively -congruent pieces. This defines an equivalence relation among all subsets of . Formally, if there exist non-empty sets ,  such that

and there exist elements  such that

then it can be said that  and  are -equidecomposable using  pieces. If a set  has two disjoint subsets  and  such that  and , as well as  and , are  -equidecomposable, then  is called paradoxical.

Using this terminology, the Banach–Tarski paradox can be reformulated as follows:

 A three-dimensional Euclidean ball is equidecomposable with two copies of itself.

In fact, there is a sharp result in this case, due to Raphael M. Robinson: doubling the ball can be accomplished with five pieces, and fewer than five pieces will not suffice.

The strong version of the paradox claims:

 Any two bounded subsets of 3-dimensional Euclidean space with non-empty interiors are equidecomposable.

While apparently more general, this statement is derived in a simple way from the doubling of a ball by using a generalization of the Bernstein–Schroeder theorem due to Banach that implies that if  is equidecomposable with a subset of  and  is equidecomposable with a subset of , then  and  are equidecomposable.

The Banach–Tarski paradox can be put in context by pointing out that for two sets in the strong form of the paradox, there is always a bijective function that can map the points in one shape into the other in a one-to-one fashion. In the language of Georg Cantor's set theory, these two sets have equal cardinality. Thus, if one enlarges the group to allow arbitrary bijections of , then all sets with non-empty interior become congruent. Likewise, one ball can be made into a larger or smaller ball by stretching, or in other words, by applying similarity transformations. Hence, if the group  is large enough, -equidecomposable sets may be found whose "size"s vary. Moreover, since a countable set can be made into two copies of itself, one might expect that using countably many pieces could somehow do the trick.

On the other hand, in the Banach–Tarski paradox, the number of pieces is finite and the allowed equivalences are Euclidean congruences, which preserve the volumes. Yet, somehow, they end up doubling the volume of the ball! While this is certainly surprising, some of the pieces used in the paradoxical decomposition are non-measurable sets, so the notion of volume (more precisely, Lebesgue measure) is not defined for them, and the partitioning cannot be accomplished in a practical way. In fact, the Banach–Tarski paradox demonstrates that it is impossible to find a finitely-additive measure (or a Banach measure) defined on all subsets of a Euclidean space of three (and greater) dimensions that is invariant with respect to Euclidean motions and takes the value one on a unit cube. In his later work, Tarski showed that, conversely, non-existence of paradoxical decompositions of this type implies the existence of a finitely-additive invariant measure.

The heart of the proof of the "doubling the ball" form of the paradox presented below is the remarkable fact that by a Euclidean isometry (and renaming of elements), one can divide a certain set (essentially, the surface of a unit sphere) into four parts, then rotate one of them to become itself plus two of the other parts. This follows rather easily from a -paradoxical decomposition of , the free group with two generators. Banach and Tarski's proof relied on an analogous fact discovered by Hausdorff some years earlier: the surface of a unit sphere in space is a disjoint union of three sets  and a countable set  such that, on the one hand,  are pairwise congruent, and on the other hand,  is congruent with the union of  and . This is often called the Hausdorff paradox.

Connection with earlier work and the role of the axiom of choice

Banach and Tarski explicitly acknowledge Giuseppe Vitali's 1905 construction of the set bearing his name, Hausdorff's paradox (1914), and an earlier (1923) paper of Banach as the precursors to their work. Vitali's and Hausdorff's constructions depend on Zermelo's axiom of choice ("AC"), which is also crucial to the Banach–Tarski paper, both for proving their paradox and for the proof of another result:

 Two Euclidean polygons, one of which strictly contains the other, are not equidecomposable.

They remark:

 Le rôle que joue cet axiome dans nos raisonnements nous semble mériter l'attention
 (The role this axiom plays in our reasoning seems to us to deserve attention)

They point out that while the second result fully agrees with geometric intuition, its proof uses AC in an even more substantial way than the proof of the paradox. Thus Banach and Tarski imply that AC should not be rejected solely because it produces a paradoxical decomposition, for such an argument also undermines proofs of geometrically intuitive statements.

However, in 1949, A. P. Morse showed that the statement about Euclidean polygons can be proved in ZF set theory and thus does not require the axiom of choice. In 1964, Paul Cohen proved that the axiom of choice is independent from ZF – that is, it cannot be proved from ZF. A weaker version of an axiom of choice is the axiom of dependent choice, DC, and it has been shown that DC is  sufficient for proving the Banach–Tarski paradox, that is,

 The Banach–Tarski paradox is not a theorem of ZF, nor of ZF+DC.

Large amounts of mathematics use AC. As Stan Wagon points out at the end of his monograph, the Banach–Tarski paradox has been more significant for its role in pure mathematics than for foundational questions:  it motivated a fruitful new direction for research, the amenability of groups, which has nothing to do with the foundational questions.

In 1991, using then-recent results by Matthew Foreman and Friedrich Wehrung, Janusz Pawlikowski proved that the Banach–Tarski paradox follows from ZF plus the Hahn–Banach theorem. The Hahn–Banach theorem does not rely on the full axiom of choice but can be proved using a weaker version of AC called the ultrafilter lemma. So Pawlikowski proved that the set theory needed to prove the Banach–Tarski paradox, while stronger than ZF, is weaker than full ZFC.

A sketch of the proof

Here a proof is sketched which is similar but not identical to that given by Banach and Tarski. Essentially, the paradoxical decomposition of the ball is achieved in four steps:

 Find a paradoxical decomposition of the free group in two generators.
 Find a group of rotations in 3-d space isomorphic to the free group in two generators.
 Use the paradoxical decomposition of that group and the axiom of choice to produce a paradoxical decomposition of the hollow unit sphere.
 Extend this decomposition of the sphere to a decomposition of the solid unit ball.

These steps are discussed in more detail below.

Step 1

The free group with two generators a and b consists of all finite strings that can be formed from the four symbols a, a−1, b and b−1 such that no a appears directly next to an a−1 and no b appears directly next to a b−1. Two such strings can be concatenated and converted into a string of this type by repeatedly replacing the "forbidden" substrings with the empty string. For instance: abab−1a−1 concatenated with abab−1a yields abab−1a−1abab−1a, which contains the substring a−1a, and so gets reduced to abab−1bab−1a, which contains the substring b−1b, which gets reduced to . One can check that the set of those strings with this operation forms a group with identity element the empty string e. This group may be called F2.

The group  can be "paradoxically decomposed" as follows: Let S(a) be the subset of  consisting of all strings that start with a, and define S(a−1), S(b) and S(b−1) similarly. Clearly,

but also

and

where the notation aS(a−1) means take all the strings in S(a−1) and concatenate them on the left with a.

This is at the core of the proof. For example, there may be a string  in the set  which, because of the rule that  must not appear next to , reduces to the string .  Similarly,  contains all the strings that start with  (for example, the string  which reduces to ). In this way,  contains all the strings that start with ,  and , as well as the empty string .

Group F2 has been cut into four pieces (plus the singleton {e}), then two of them "shifted" by multiplying with a or b, then "reassembled" as two pieces to make one copy of  and the other two to make another copy of . That is exactly what is intended to do to the ball.

Step 2
In order to find a free group of rotations of 3D space, i.e. that behaves just like (or "is isomorphic to") the free group F2, two orthogonal axes are taken (e.g. the x and z axes). Then, A is taken to be a rotation of  about the x axis, and B to be a rotation of  about the z axis (there are many other suitable pairs of irrational multiples of π that could be used here as well).

The group of rotations generated by A and B will be called H.
Let  be an element of H that starts with a positive rotation about the z axis, that is, an element of the form  with . It can be shown by induction that  maps the point  to , for some . Analyzing  and  modulo 3, one can show that . The same argument repeated (by symmetry of the problem) is valid when  starts with a negative rotation about the z axis, or a rotation about the x axis. This shows that if  is given by a non-trivial word in A and B, then . Therefore, the group H is a free group, isomorphic to F2.

The two rotations behave just like the elements a and b in the group F2: there is now a paradoxical decomposition of H.

This step cannot be performed in two dimensions since it involves rotations in three dimensions. If two rotations are taken about the same axis, the resulting group is the abelian circle group and does not have the property required in step 1.

An alternate arithmetic proof of the existence of free groups in some special orthogonal groups using integral quaternions leads to paradoxical decompositions of the rotation group.

Step 3
The unit sphere S2 is partitioned into orbits by the action of our group H: two points belong to the same orbit if and only if there is a rotation in H which moves the first point into the second. (Note that the orbit of a point is a dense set in S2.) The axiom of choice can be used to pick exactly one point from every orbit; collect these points into a set M. The action of H on a given orbit is free and transitive and so
each orbit can be identified with H. In other words, every point in S2 can be reached in exactly one way by applying the proper rotation from H to the proper element from M. Because of this, the paradoxical decomposition of H yields a paradoxical decomposition of S2 into four pieces A1, A2, A3, A4 as follows:

where we define

and likewise for the other sets, and where we define

(The five "paradoxical" parts of F2 were not used directly, as they would leave M as an extra piece after doubling, owing to the presence of the singleton {e}.)

The (majority of the) sphere has now been divided into four sets (each one dense on the sphere), and when two of these are rotated, the result is double of what was had before:

Step 4
Finally, connect every point on S2 with a half-open segment to the origin; the paradoxical decomposition of S2 then yields a paradoxical decomposition of the solid unit ball minus the point at the ball's center. (This center point needs a bit more care; see below.)

N.B. This sketch glosses over some details. One has to be careful about the set of points on the sphere which happen to lie on the axis of some rotation in H. However, there are only countably many such points, and like the case of the point at the center of the ball, it is possible to patch the proof to account for them all. (See below.)

Some details, fleshed out

In Step 3, the sphere was partitioned into orbits of our group H. To streamline the proof, the discussion of points that are fixed by some rotation was omitted; since the paradoxical decomposition of F2 relies on shifting certain subsets, the fact that some points are fixed might cause some trouble. Since any rotation of S2 (other than the null rotation) has exactly two fixed points, and since H, which is isomorphic to F2, is countable, there are countably many points of S2 that are fixed by some rotation in H. Denote this set of fixed points as D. Step 3 proves that S2 − D admits a paradoxical decomposition.

What remains to be shown is the Claim: S2 − D is equidecomposable with S2.

Proof. Let λ be some line through the origin that does not intersect any point in D. This is possible since D is countable. Let J be the set of angles, α, such that for some natural number n, and some P in D, r(nα)P is also in D, where r(nα) is a rotation about λ of nα. Then J is countable. So there exists an angle θ not in J. Let ρ be the rotation about λ by θ. Then ρ acts on S2 with no fixed points in D, i.e., ρn(D) is disjoint from D, and for natural m<n, ρn(D) is disjoint from ρm(D). Let E be the disjoint union of ρn(D) over n = 0, 1, 2, ... . Then S2 = E ∪ (S2 − E) ~ ρ(E) ∪ (S2 − E) = (E − D) ∪ (S2 − E) = S2 − D, where ~ denotes "is equidecomposable to".

For step 4, it has already been shown that the ball minus a point admits a paradoxical decomposition; it remains to be shown that the ball minus a point is equidecomposable with the ball. Consider a circle within the ball, containing the point at the center of the ball. Using an argument like that used to prove the Claim, one can see that the full circle is equidecomposable with the circle minus the point at the ball's center. (Basically, a countable set of points on the circle can be rotated to give itself plus one more point.) Note that this involves the rotation about a point other than the origin, so the Banach–Tarski paradox involves isometries of Euclidean 3-space rather than just SO(3).

Use is made of the fact that if A ~ B and B ~ C, then A ~ C. The decomposition of A into C can be done using number of pieces equal to the product of the numbers needed for taking A into B and for taking B into C.

The proof sketched above requires 2 × 4 × 2 + 8 = 24 pieces - a factor of 2 to remove fixed points, a factor 4 from step 1, a factor 2 to recreate fixed points, and 8 for the center point of the second ball. But in step 1 when moving {e} and all strings of the form an into S(a−1), do this to all orbits except one. Move {e} of this last orbit to the center point of the second ball. This brings the total down to 16 + 1 pieces. With more algebra, one can also decompose fixed orbits into 4 sets as in step 1. This gives 5 pieces and is the best possible.

Obtaining infinitely many balls from one
Using the Banach–Tarski paradox, it is possible to obtain k copies of a ball in the Euclidean n-space from one, for any integers n ≥ 3 and k ≥ 1, i.e. a ball can be cut into k pieces so that each of them is equidecomposable to a ball of the same size as the original. Using the fact that the free group F2 of rank 2 admits a free subgroup of countably infinite rank, a similar proof yields that the unit sphere Sn−1 can be partitioned into countably infinitely many pieces, each of which is equidecomposable (with two pieces) to the Sn−1 using rotations. By using analytic properties of the rotation group SO(n), which is a connected analytic Lie group, one can further prove that the sphere Sn−1 can be partitioned into as many pieces as there are real numbers (that is,  pieces), so that each piece is equidecomposable with two pieces to Sn−1 using rotations. These results then extend to the unit ball deprived of the origin. A 2010 article by Valeriy Churkin gives a new proof of the continuous version of the Banach–Tarski paradox.

Von Neumann paradox in the Euclidean plane

In the Euclidean plane, two figures that are equidecomposable with respect to the group of Euclidean motions are necessarily of the same area, and therefore, a paradoxical decomposition of a square or disk of Banach–Tarski type that uses only Euclidean congruences is impossible. A conceptual explanation of the distinction between the planar and higher-dimensional cases was given by John von Neumann: unlike the group SO(3) of rotations in three dimensions, the group E(2) of Euclidean motions of the plane is solvable, which implies the existence of a finitely-additive measure on E(2) and R2 which is invariant under translations and rotations, and rules out paradoxical decompositions of non-negligible sets. Von Neumann then posed the following question: can such a paradoxical decomposition be constructed if one allows a larger group of equivalences?

It is clear that if one permits similarities, any two squares in the plane become equivalent even without further subdivision. This motivates restricting one's attention to the group SA2 of area-preserving affine transformations. Since the area is preserved, any paradoxical decomposition of a square with respect to this group would be counterintuitive for the same reasons as the Banach–Tarski decomposition of a ball. In fact, the group SA2 contains as a subgroup the special linear group SL(2,R), which in its turn contains the free group F2 with two generators as a subgroup. This makes it plausible that the proof of Banach–Tarski paradox can be imitated in the plane. The main difficulty here lies in the fact that the unit square is not invariant under the action of the linear group SL(2, R), hence one cannot simply transfer a paradoxical decomposition from the group to the square, as in the third step of the above proof of the Banach–Tarski paradox. Moreover, the fixed points of the group present difficulties (for example, the origin is fixed under all linear transformations). This is why von Neumann used the larger group SA2 including the translations, and he constructed a paradoxical decomposition of the unit square with respect to the enlarged group (in 1929). Applying the Banach–Tarski method, the paradox for the square can be strengthened as follows:

 Any two bounded subsets of the Euclidean plane with non-empty interiors are equidecomposable with respect to the area-preserving affine maps.

As von Neumann notes:

"Infolgedessen gibt es bereits in der Ebene kein nichtnegatives additives Maß (wo das Einheitsquadrat das Maß 1 hat), das gegenüber allen Abbildungen von A2 invariant wäre."

"In accordance with this, already in the plane there is no non-negative additive measure (for which the unit square has a measure of 1), which is invariant with respect to all transformations belonging to A2 [the group of area-preserving affine transformations]."

To explain further, the question of whether a finitely additive measure (that is preserved under certain transformations) exists or not depends on what transformations are allowed. The Banach measure of sets in the plane, which is preserved by translations and rotations, is not preserved by non-isometric transformations even when they do preserve the area of polygons. The points of the plane (other than the origin) can be divided into two dense sets which may be called A and B. If the A points of a given polygon are transformed by a certain area-preserving transformation and the B points by another, both sets can become subsets of the A points in two new polygons. The new polygons have the same area as the old polygon, but the two transformed sets cannot have the same measure as before (since they contain only part of the A points), and therefore there is no measure that "works".

The class of groups isolated by von Neumann in the course of study of Banach–Tarski phenomenon turned out to be very important for many areas of Mathematics: these are amenable groups, or groups with an invariant mean, and include all finite and all solvable groups. Generally speaking, paradoxical decompositions arise when the group used for equivalences in the definition of equidecomposability is not amenable.

Recent progress
 2000: Von Neumann's paper left open the possibility of a paradoxical decomposition of the interior of the unit square with respect to the linear group SL(2,R) (Wagon, Question 7.4). In 2000, Miklós Laczkovich proved that such a decomposition exists. More precisely, let A be the family of all bounded subsets of the plane with non-empty interior and at a positive distance from the origin, and B the family of all planar sets with the property that a union of finitely many translates under some elements of SL(2, R) contains a punctured neighborhood of the origin. Then all sets in the family A are SL(2, R)-equidecomposable, and likewise for the sets in B. It follows that both families consist of paradoxical sets.
 2003: It had been known for a long time that the full plane was paradoxical with respect to SA2, and that the minimal number of pieces would equal four provided that there exists a locally commutative free subgroup of SA2. In 2003 Kenzi Satô constructed such a subgroup, confirming that four pieces suffice.
 2011: Laczkovich's paper left open the possibility if there exists a free group F of piecewise linear transformations acting on the punctured disk D \{0,0} without fixed points. Grzegorz Tomkowicz constructed such a group, showing that the system of congruences A ≈ B ≈ C ≈ B U C can be realized by means of F and D \{0,0}.
 2017: It has been known for a long time that there exists in the hyperbolic plane H2 a set E that is a third, a fourth and ... and a -th part of H2. The requirement was satisfied by orientation-preserving isometries of H2. Analogous results were obtained by John Frank Adams and Jan Mycielski who showed that the unit sphere S2 contains a set E that is a half, a third, a fourth and ... and a -th part of S2. Grzegorz Tomkowicz showed that Adams and Mycielski construction can be generalized to obtain a set E of H2 with the same properties as in S2.
 2017: Von Neumann's paradox concerns the Euclidean plane, but there are also other classical spaces where the paradoxes are possible. For example, one can ask if there is a Banach–Tarski paradox in the hyperbolic plane H2. This was shown by Jan Mycielski and Grzegorz Tomkowicz. Tomkowicz proved also that most of the classical paradoxes are an easy consequence of a graph theoretical result and the fact that the groups in question are rich enough.
 2018: In 1984, Jan Mycielski and Stan Wagon  constructed a paradoxical decomposition of the hyperbolic plane H2 that uses Borel sets. The paradox depends on the existence of a properly discontinuous subgroup of the group of isometries of H2. Similar paradox is obtained by Grzegorz Tomkowicz  who constructed a free properly discontinuous subgroup G of the affine group SA(3,Z). The existence of such a group implies the existence of a subset E of Z3 such that for any finite F of Z3 there exists an element  of  such that , where  denotes the symmetric difference of  and .
 2019: Banach–Tarski paradox uses finitely many pieces in the duplication. In the case of countably many pieces, any two sets with non-empty interiors are equidecomposable using translations. But allowing only Lebesgue measurable pieces one obtains: If A and B are subsets of Rn with non-empty interiors, then they have equal Lebesgue measures if and only if they are countably equidecomposable using Lebesgue measurable pieces. Jan Mycielski and Grzegorz Tomkowicz  extended this result to finite dimensional Lie groups and second countable locally compact topological groups that are totally disconnected or have countably many connected components.

See also

Notes

References

 Edward Kasner & James Newman (1940) Mathematics and the Imagination, pp 205–7, Simon & Schuster.

External links

 The Banach-Tarski Paradox by Stan Wagon (Macalester College), the Wolfram Demonstrations Project.
 Irregular Webcomic! #2339 by David Morgan-Mar provides a non-technical explanation of the paradox. It includes a step-by-step demonstration of how to create two spheres from one.
  gives an overview on the fundamental basics of the paradox.
 Banach-Tarski and the Paradox of Infinite Cloning

Eponymous paradoxes
Group theory
Measure theory
Mathematical paradoxes
Theorems in the foundations of mathematics
Geometric dissection
1924 introductions
Paradoxes of infinity

Banach–Tarski paradox


In Norse mythology, Mist (Old Norse "cloud" or mist)  is a valkyrie. Mist appears in valkyrie list in the Poetic Edda poem Grímnismál and both of the Nafnaþulur valkyrie lists. No further information is provided about her. Rudolf Simek says that her name, Mist, is likely related to Old Norse mistr, meaning "cloud, mist," and that this "reminds us of the way in which valkyries can ride through the air and over water," such as in the Poetic Edda poems Helgakviða Hjörvarðssonar and Helgakviða Hundingsbana II.

Notes

References

 Orchard, Andy (1997). Dictionary of Norse Myth and Legend. Cassell.
 Simek, Rudolf (2007) translated by Angela Hall. Dictionary of Northern Mythology. D.S. Brewer

Valkyries

Mist (valkyrie)


OpenCola, Opencola, or open cola may refer to:
 Open-source cola, cola with an openly available recipe.
 Opencola, the free-software P2P company started by Grad Conn, Cory Doctorow, and John Henson.
 OpenCola (drink), a brand of open-source cola produced by the free-software company Opencola.

OpenCola


Robert Bellarmine Carl Katter (born 22 May 1945) is an Australian politician who has been a member of the House of Representatives since 1993. He was previously active in Queensland state politics from 1974 to 1992. Katter was a member of the National Party until 2001, when he left to sit as an independent. He formed his own party, Katter's Australian Party, in 2011.

Katter was born in Cloncurry, Queensland. His father, Bob Katter Sr., was also a politician. Katter was elected to the Queensland Legislative Assembly at the 1974 state election, representing the seat of Flinders. He was elevated to cabinet in 1983, under Joh Bjelke-Petersen, and was a government minister until the National Party's defeat at the 1989 state election.

Katter left state politics in 1992, and the following year was elected to federal parliament standing in the Division of Kennedy (his father's old seat). He resigned from the National Party in the lead-up to the 2001 federal election, and has since been re-elected four more times as an independent and another four times for his own party. Katter is known for his social conservatism. His son, Robbie Katter, is a state MP in Queensland, the third generation of the family to be a member of parliament.

Early life, education and career

Katter was born in Cloncurry, Queensland, the son of Robert Cummin Katter, the member for Kennedy from 1966 to 1990, and his wife, Mabel. His paternal grandparents went to Cloncurry in a stage coach around 1900. Katter's paternal grandfather was a Lebanese migrant, who owned clothing stores throughout north Queensland. His father, Bob Katter Sr., was an Australian politician who was in the House of Representatives from 1966 to 1990, representing the National Party (originally named the Country Party).

Growing up in Cloncurry, Katter's family owned a clothing shop and managed a local cinema. He was one of only six at his school who finished year 12.

Katter attended the University of Queensland, where he studied law, but later dropped out without graduating. While at university, Katter was President of the University of Queensland Law Society and St Leo's College. As a university student, Katter pelted the Beatles with rotten eggs during their 1964 tour of Australia, declaring in a later meeting with band that he undertook this as "an intellectual reaction against Beatlemania". He also served in the Citizens Military Forces.

Returning to Cloncurry, he worked in his family's businesses, and as a "labourer in the Mt Isa Mines".

State politics (1974–1992)
Katter's father was a member of the Australian Labor Party until 1957, when he left during the Labor split of that year. He later joined the Country Party, now the National Party. The younger Katter was a Country Party member of the Legislative Assembly of Queensland from 1974 to 1992, representing Flinders in north Queensland. He was Minister for Northern Development and Aboriginal and Islander Affairs from 1983 to 1987, Minister for Northern Development, Community Services and Ethnic Affairs from 1987 to 1989, Minister for Community Services and Ethnic Affairs in 1989, Minister for Mines and Energy in 1989, and Minister for Northern and Regional Development for a brief time in 1989 until the Nationals were defeated in that year's election.

While in the Queensland Parliament, Katter junior was a strong supporter of Premier Joh Bjelke-Petersen, though he remained in cabinet under Mike Ahern, ultimately resigning from Cabinet along with Russell Cooper. He was on the backbench. Then appointed again to Cabinet in the traditional number two position of Mines & Energy. This was under the Bjelke-Petersen's factions restoration to power.

Federal politics (1993–present)
Katter did not run for re-election to state Parliament in 1992. He ran as the National candidate in his father's former seat of Kennedy at the 1993 federal election, facing his father's successor, Labor's Rob Hulls. Despite name recognition, Katter trailed Hulls for most of the night. On the eighth count, a Liberal candidate's preferences flowed overwhelmingly to Katter, allowing him to defeat Hulls by 4,000 votes. He would not face another contest nearly that close for two decades.

In 1994, Katter advocated against federal privacy laws that bypassed Tasmania's anti-gay laws, claiming the government was "helping the spread of AIDS" and legitimizing "homosexual behavior". He also believed the laws jeopardized states' rights in Australia.

Katter was re-elected with a large swing in 1996, and was re-elected almost as easily in 1998. However, when he transferred to federal politics, he found himself increasingly out of sympathy with the federal Liberal and National parties on economic and social issues, with Katter being opposed to neoliberalism and social liberalism. In 2001, he resigned from the National Party and easily retained his seat as an independent at the general elections of 2001, 2004, 2007 and 2010, each time ending up with a percentage vote in the high sixties after preferences were distributed.

In the aftermath of the 2010 hung federal election, Katter offered a range of views on the way forward for government. Two other former National Party MPs, both independents from rural electorates, Tony Windsor, Rob Oakeshott decided to support a Labor government. Katter presented his 20 points document and asked the major parties to respond before deciding which party he would support. As a result, he broke with Windsor and Oakeshott and supported the Liberal/National Coalition for Government. On 7 September 2010, Katter announced his support for a Liberal/National Party coalition minority government.

On 5 June 2011, Katter launched a new political party, Katter's Australian Party, which he said would "unashamedly represent agriculture". He made headlines after singing to his party's candidates during a meeting on 17 October 2011, saying it was his "election jingle".

In the 2013 election, however, Katter faced his first serious contest since his initial run for Kennedy in 1993. He had gone into the election holding the seat with a majority of 18 percent, making it the second-safest seat in Australia. However, reportedly due to anger at his decision to back Kevin Rudd (ALP) for Prime Minister following Julia Gillard's (Prime Minister) live cattle export ban (Rudd, within weeks, reopened the live export market), Katter still suffered a primary-vote swing of over 17 points. His name heavily associated with Rudd.  In the end, Katter was re-elected on Labor preferences, suffering a two-party swing of 16 points to the Liberal National party.

In the 2016 election, Katter retained his seat of Kennedy, with an increased swing of 8.93 points toward him.

On 15 August 2017 Katter announced that the Turnbull government could not take his support for granted in the wake of the 2017 Australian parliamentary eligibility crisis, which ensued over concerns that several MPs held dual citizenship and thus may be constitutionally ineligible to be in Parliament. Katter added that if one of the affected MPs, Deputy Prime Minister Barnaby Joyce, lost his seat, the Coalition could not count on his support for confidence and supply.

In November 2018, Katter secured funds for three inland dam-irrigation schemes in North Queensland.

In the 2019 election, Katter was returned to his seat of Kennedy with a swing of 2.9 points towards him, in spite of an unfavourable redistribution of his electorate. In the 2022 election, he was re-elected again, and became the Father of the Australian House of Representatives following the retirement of Kevin Andrews.

Political views

Katter is known as an unabashed social conservative and agrarian socialist. His views on economic matters echo "Old Labor" policy as it was before the Australian Labor Party split of 1955 as he opposes privatisation and economic deregulation and strongly supports traditional Country Party statutory marketing. The sobriquet 'Mad Katter' was coined by his opponents to describe his nationalistic developmentalism. He has a very sporadic attendance record in parliament, and by the end of 2019 had only attended 42% of votes on the floor of parliament, the lowest of any member of parliament.

As of 2020, Katter described himself as belonging to the "hard left," citing his continuing membership of the Construction, Forestry, Maritime, Mining and Energy Union. In a 2022 interview with The Chaser, Katter claimed that he had never pledged allegiance to the Monarch of the United Kingdom when entering parliament.

Abortion
In 2006 Bob Katter voted against a bill which would increase the availability of abortion drugs.

Environment
Katter has opposed enacting climate change legislation to control emissions. He advocates for measures that reduce carbon footprints. Katter has championed the mandating of ethanol fuel content. He has additionally pioneered protests against imported bananas, and is an opponent of the concentration of the Australian supermarket industry amongst Coles and Woolworths.

Gun laws
An opponent of the tougher gun control laws introduced in the wake of the 1996 massacre in Port Arthur, Tasmania, Katter was accused in 2001 of signing a petition promoted by the Citizens Electoral Council (CEC), an organisation that claims the Port Arthur massacre was a conspiracy. He has stated that he always and still believes there was no conspiracy.

Immigration
In 2017, Katter called for a "Trump-like travel ban" in Australia after a New South Welshman was arrested on terrorism charges. That same year, Katter repeated a pledge used by the far-right organisation "Proud Boys", including that he was "a proud western chauvinist". When asked about the incident when it was publicised in 2019, Katter distanced himself from the group, saying "I don't know who this group is or anything about it".

Indigenous Australians
In 1989, while Community Services Minister in the Queensland state government, Katter opposed installing condom vending machines in Aboriginal and Torres Strait Islander communities to reduce the spread of AIDS, describing the plan instead as an attempt at eugenics, or "racist genocide". He is also an opponent of voter identification laws, denouncing the Coalition's proposed introduction of them in 2021 as a racist system that would disenfranchise Aboriginal communities.

LGBT rights
In November 1989, Katter claimed there were almost no homosexuals in North Queensland. He promised to walk backwards from Bourke across his electorate if they represented more than 0.001 percent of the population. Katter also said "mind you, if there are more, then I might take to walking backwards everywhere!"  Katter voted against the , which decriminalised homosexuality in Tasmania. He does not support same-sex marriage. His response to the Australian Marriage Law Postal Survey result was the subject of international attention, as in response he declared that the issue of crocodiles killing people in North Queensland was more pressing than same-sex marriage. Therefore he declared that "I ain't spending any time on it!" on the latter issue. In December 2017, Katter was one of only four members of the House of Representatives to oppose the Marriage Amendment (Definition and Religious Freedoms) Bill 2017 legalising same-sex marriage in Australia.

Personal life
Katter occasionally identifies as being an Aboriginal Australian and has described himself as a blackfella in federal parliament, in interviews, during television appearances and at public events. Katter claims that in his youth he was accepted as a member of the Kalkadoon tribe in the Cloncurry area, otherwise known as the "Curry mob", and said he has long since felt a deep connection with Aboriginal people.

His son Robbie has been a member of the Queensland Legislative Assembly since 2012, representing Mount Isa from 2012 to 2017, and  Traeger since 2017. He represents much of the territory that his father represented in state parliament.

Bibliography

See also

Politics of Queensland

References

External links

 Maiden Speech – Australian House of Representatives (6 May 1993), aph.gov.au

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1945 births
Independent members of the Parliament of Australia
Katter's Australian Party members of the Parliament of Australia
Australian monarchists
National Party of Australia members of the Parliament of Australia
National Party of Australia members of the Parliament of Queensland
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Members of the Australian House of Representatives for Kennedy
People from Cloncurry, Queensland
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Living people
Australian Roman Catholics
Australian nationalists
21st-century Australian politicians
20th-century Australian politicians
Australian people of Lebanese descent

Bob Katter


Norway was represented by Bobbysocks!, with the song "La det swinge", at the 1985 Eurovision Song Contest, which took place on 4 May in Gothenburg, Sweden. "La det swinge" was chosen as the Norwegian entry at the Melodi Grand Prix on 30 March, and went on to bring Norway a famous first Eurovision victory after decades of being the butt of jokes about their status as Eurovision's perennial also-rans. It was admitted that the lyrics of "La det swinge" had been written with the specific intention of avoiding those Norwegian consonant combinations which had been said to sound harsh in song to non-Scandinavian ears.

Bobbysocks! were Elisabeth Andreassen and Hanne Krogh, who both had previous Eurovision experience – Krogh as a 15-year-old for Norway in 1971 and Andreassen for Sweden in 1982. Both would make further appearances for Norway, Krogh in 1991 and Andreassen in 1994 and 1996.

Before Eurovision

Melodi Grand Prix 1985
The MGP was held at the Chateau Neuf in Oslo, hosted by Rita Westvik. The staging was now been set in a modern way for accompanying live music for the first time in the history of national final, while the usage of the orchestra were replaced with a live band (e.g. band drums, electric guitars, keyboard synthesizers and horns) and was conducted by Terje Fjærn. Ten songs took part in the final, with the winner chosen by voting from five regional juries and a panel of "experts" consisting of Tony Visconti, Anne Marie David, Stikkan Andersson and Ronnie Hazlehurst. One of the other participants was former three-time Norwegian representative Anita Skorgan.

At Eurovision
On the night of the final Bobbysocks! performed 13th in the running order, following Italy and preceding the United Kingdom. The voting in 1985 was particularly disparate, with points being spread all across the board to a much greater extent than usual. However at the close of voting "La det swinge" had picked up 123 points, enough for victory by an 18-point margin over runners-up Germany, although the total was and remains the lowest-ever winning score under the 12 points system. "La det swinge" appeared to have sharply divided opinion with the national juries, as it had received an unusually high eight maximum 12 points votes, while the remaining ten juries had awarded it only 27 points between them. The Norwegian jury awarded its 12 points to Sweden.

The congratulations to Bobbysocks! after the victory are fondly remembered for the infamously double-edged remark by Swedish host Lill Lindfors "I must say I'm honestly very happy that this happened, because Norway has been last song so many times that you really deserve it!", prompting much laughter from the audience.

Voting

References

External links
Full national final on nrk.no

1985
Countries in the Eurovision Song Contest 1985
1985
Eurovision
Eurovision

Norway in the Eurovision Song Contest 1985


Ernst Goldenbaum (15 December 1898, Parchim, Mecklenburg-Schwerin – 13 March 1990) was an East German politician.

Biography
Goldenbaum was born in Parchim. During World War I he served in the military and he participated in the German November Revolution. In 1919 he joined the left-wing USPD and a few years later the Communist Party of Germany.

From 1923 to 1925 he was a member of the city council of Parchim and from 1924 to 1932 he was a member of the Landtag of Mecklenburg-Schwerin. From 1932 to 1933 he was the editor of Volkswacht, a communist newspaper. After the Nazis seized power he became a farmer and a member of the German resistance.

In 1944 he was arrested and he spent the last year of the war in concentration camp Neuengamme. In 1945, he was one of very few who survived the sinking of the SS Cap Arcona which claimed over 4000 lives.

After the war he joined the Socialist Unity Party (SED), but in 1948 he co-founded the communist-sponsored Democratic Farmers' Party of Germany (DBD). The DBD was a close ally of the SED. Until 1982, Goldenbaum was the chairman of the party.

From 1949 to 1990 Goldenbaum was a member of the People's Chamber. From 1949 to 1950 Goldenbaum was East Germany's first Minister of Agriculture and Forestry. Goldenbaum supported the SED's collectivisation in the 1950s and 1960s. From 1950 to 1963 he was the deputy chairman of the People's Chamber. After 1963 he was a member of the Presidium of the People's Chamber.

External links

1898 births
1990 deaths
People from Parchim
People from the Grand Duchy of Mecklenburg-Schwerin
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Ernst Goldenbaum


Citron is a Czech hard rock/heavy metal band founded in Ostrava in 1976. The band's history is divided into two distinct eras: between 1976 and 1981, when their name was spelled with a long "ó", Citrón, and they played mainly blues rock. During this time, the band leader was bassist Petr Michalík. The second era, from 1982 until the present, signalled a change in musical style towards hard rock and heavy metal, with drummer Radim Pařízek as band leader and the group's name slightly altered to Citron. As of 2021, Citron has released twelve studio albums (including three English versions), two EPs, and three official compilation albums.

History

Beginning and end of first era: 1976–1981
Citrón's inception dates back to 1976, when several musicians left the side of František Ringo Čech and founded an accompanying band for singer Aleš Ulm. Among them were guitarist Olda Říha, drummer Tolja Kohout (Katapult), keyboardist Miloš Nop, and bass guitarist Petr Michalík, who gave the group its name. They recorded two singles in this lineup, but Michalík went his separate way shortly after and Citrón became its own entity, though with only one member. Vladimír Kubala soon joined on vocals, followed by guitarist Jiří Schmutz and drummer Milan Tutsch. They began to play blues rock, influenced by such bands as Bad Company, Free, and Led Zeppelin.

The four were already experienced musicians, and they soon established themselves not only on the radio, but also on television. They released their first single, "Tvých sedmnáct/Pantomima reklam", under Supraphon in 1978, and it proved a success. Their next single, "Až se vrátí rokenrol/Diskžokej", came out the following year, and Citrón found themselves at the top of the Czechoslovak rock scene alongside such acts as Katapult and Jiří Schelinger's band.
The same year, one major lineup change occurred: drummer Milan Tutsch left to join Katapult and was replaced by Radim Pařízek, who would later go on to lead the group. Additionally, Miloslav Benýšek joined Citrón on second guitar.
They played a large number of concerts throughout Czechoslovakia, and released two more singles as they worked on their debut album, Obratník Raka, which was recorded in 1981. Two versions of the record were produced, one for the domestic market and one for international audiences. The Czech version was never released, however; only the English one, titled Tropic of Cancer, was published, in 1983, though by that point the band had mostly disintegrated: Michalík and Kubala, and later Benýšek, left the project. Only guitarist Schmutz and drummer Pařízek remained.

New direction: 1982–1987
Radim Pařízek took over the leadership of Citron and brought in seasoned musician Stanislav Hranický (Majestic) on vocals. They were joined by guitarist Jindřich Kvita and bassist Jiří Krejčí. In this lineup, they recorded a few singles in 1983, and moved sonically towards a heavier sound. The same year, Schmutz and Krejčí left the group and were replaced by guitarist Jaroslav Bartoň and bassist Václav Vlasák, both from the band Proto.

Citron's musical style began to change gradually around this time, tending more towards heavy metal. Work began on their upcoming album in 1985, recorded at Prague's Mozarteum. Two versions of the album were created, again. The Czech version, Plni energie, was released in 1986, and Full of Energy followed a year later. The Czech album was certified Gold, and Citron won a Zlatý slavík award, taking second place after the Slovak band Elán. A concert tour was organized, which saw the band travel to West Germany, among other locations. In 1987, vocalist Stanislav Hranický broke his leg after one concert in Bratislava and became unable to perform.

Ladislav Křížek: 1987–1989
The band had to solve the unexpected situation quickly, and they hired vocalist Ladislav Křížek from the Prague band Vitacit. Preparations for their third album, Radegast, began the same year, with the band recording in their own studio. Two singles were released, and demonstrated a definitive shift towards heavy metal. The album came out in the fall of 1987, again in two versions, and the Czech record was certified Gold. The English record came out two years later, under the same title.
A number of large concerts followed, this time not only in Czechoslovakia but also in Poland, the USSR, and Germany. Citron also made numerous television appearances, topped various music charts, and won Song of the Year with their single "Zase dál".
In 1988, they achieved a crowning victory by winning the Zlatý slavík award, becoming Band of the Year. During this time, the band also began collaborating with singer Tanja Kauerová, known by the mononym Tanja, who went on to marry frontman Radim Pařízek. Tanja also sang the successful 1988 duet "Kam jen jdou lásky mé" with Křížek, which was later released on his debut solo album Zlatej Chlapec in 1991.

At the beginning of 1989, Citron began preparing their next studio album, Vypusťte psy!, but disagreements within the band led to Křížek's departure after the release of only one single, "Uragán/Svět patří nám", though the song became an immediate hit. By August, Křížek was gone, followed shortly after by Jaroslav Bartoň, and the two went on to start their own band, Kreyson. Citron found replacements in Slovakia in the form of singer Tibor Šándor and guitarist Pavol Chodelka. In this new lineup, the band completed a number of planned concerts in the fall and continued recording the new album, though Šándor would not end up singing on it.

Lineup changes, new albums: 1990–1993
At the beginning of 1990, Citron discovered singer František "Fany" Michalík and replaced Tibor Šándor. The band went on a concert tour of France. Another Slovak guitarist, Henry Tóth, joined up. With this new lineup, they went back into the studio to rework their upcoming album. Vypusťte psy! was finally released in May 1990. A concert tour to promote it followed, but afterward a period of inactivity ensued, which resulted in further personnel changes. In the spring of 1991, Pavel Chodelka and Henry Tóth left and were replaced by Libor Kozelský. Citron began preparing another record, but in the summer, the band neared disintegration. Jindřich Kvita, Václav Vlasák, and František Michalík chose to leave and form the band Funny. Only Libor Kozelský and Radim Pařízek remained, so again, there was a need to rebuild Citron from the ground up. In the fall of 1991, new musicians were hired: Aleš Hubáček on vocals, Pavel Silber on bass, and Pavel Novotný on guitar, though this latter was soon replaced by Ladislav Krečmer. In this format, the band opened for Ian Gillan in Czechoslovakia and in the spring of 1992, they released the album Sexbomby. Few concerts took place to promote it, however. At the end of 1993, Citron's most successful lineup, from their Plni energie era, met up and began planning a comeback.

Tragedy , Best of, more tragedy: 1994–2013
This period was marked mainly by the band's comeback to large stages, as well as one notable loss. Just as Citron was in the process of producing their first hits compilation in 1997, guitarist Jindra Kvita died. This was a major shock to the band, and put a stop to the album preparations. It wasn't until 1999 that Best of Citron, dedicated to the memory of Jindra Kvita, was released. The album did well commercially and was certified Gold. The band continued to perform as a quartet for a while, but eventually hired guitarist Stanislav Balko, though he didn't stay long and was replaced by the returning Libor Kozelský.

In 2001, Citron reunited with former vocalist Ladislav Křížek. A retrospective album, Síla návratů, consisting of re-recorded versions of old songs, mainly from the Radegast era, was released. A short concert tour followed, after which Křížek departed again. In 2004, the band was invited to perform in the United States, where they held several successful concerts and shot the video for a new single, "Rock, rock, rock". More performances followed, including at festivals, and Citron again returned to the studio to prepare a new album.

In 2008, Citron opened for Judas Priest at the Ostravar Aréna in their hometown. At the beginning of 2009, Libor Kozelský left the band and was replaced by guitarist Jiří Šperl. In May 2010, the long-awaited album Bigbítový pánbů was finally released.

On 7 April 2013, former Citron vocalist Stanislav Hranický died of cancer.

Rebuilding the band, Křížek, and Rebelie: 2013–present
In May 2015, Citron supported Def Leppard on their tour of Czechia, opening for them both in Prague and Ostrava. In September, after much internal squabbling, Citron came close to disintegrating yet again. František Michalík, Jaroslav Bartoň, Jiří Šperl, and Václav Vlasák went on to found the group Limetal, leaving drummer Radim Pařízek as the only remaining member of Citron. He called on Ladislav Křížek to return to the band once more, and they were joined by Djordje Erič and Pavel Hanus on guitar, and Jiří "George" Rain (Kreyson) on bass. The newly reformed band went straight back to work on a new album. Before its release, they launched a concert tour and issued the EP Rebelie Vol. 1, followed a year later by another EP, Rebelie Vol. 2. Former collaborator Tanja joined them at a number of shows during their tour, and as of 2020 is a permanent member of the band. The planned album, Rebelie rebelů, was finally released towards the end of 2016. A year later, Citron finally released their 1981 album Obratník Raka, which had until then only been published in its English version.

Radim Pařízek, the last remaining member of the band's original lineup, died on 2 March 2021 after complications from a recent surgery; he was 67 years old.

Band members
Current members
 Ladislav Křížek - vocals
 Djordje Erič - guitar
 Pavel Hanus - guitar
 Jiří "George" Rain - bass
 Tanja - vocals

Past members

 Olda Říha - guitar
 Tolja Kohout - drums
 Miloš Nop - keyboards
 Petr Michalík - bass
 Vladimír Kubala - vocals
 Jiří Schmutz - guitar
 Milan Tutsch - drums
 Miloslav Benýšek - guitar
 Stanislav Hranický - vocals
 Jindřich Kvita - guitar
 Jiří Krejčí - bass
 Václav Vlasák - bass
 Jaroslav Bartoň - guitar
 Tibor Šándor - vocals
 Pavol Chodelka - guitar
 František Michalík - vocals
 Henry Tóth - guitar
 Libor Kozelský - guitar
 Aleš Hubáček - vocals
 Pavel Silber - bass
 Pavel Novotný - guitar
 Ladislav Krečmer - guitar
 Stanislav Balko - guitar
 Jiří Šperl - guitar
 Radim Pařízek - drums

Discography

Studio albums
 Obratník Raka (1981 - unreleased)
 Tropic of Cancer (1983 - English version of Obratník Raka)
 Plni energie (1986)
 Full of Energy (1987 - English version of Plni energie)
 Radegast (1987)
 Radegast (1989 - English version)
 Vypusťte psy! (1990)
 Sex bomby (1992)
 Síla návratů (2001)
 Bigbítový pánbů (2010)
 Jen si od plic zanadávej (2013)
 Rebelie rebelů (2016)
 Obratník Raka (2017 - previously unreleased 1981 album)

EPs
 Rebelie Vol. 1 (2015)
 Rebelie Vol. 2 (2016)

Compilations
 Best of Citron (1999)
 Singly (2017)
 1979–2017 (2017 - box set)

References

External links

 Studio Citron official page

Czechoslovak rock music groups
Czech heavy metal musical groups
Czech rock music groups
Musical groups from Ostrava
Musical groups established in 1976
1976 establishments in Czechoslovakia

Citron (band)