1.
Piedmont
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Piedmont is one of the 20 regions of Italy. It has an area of 25,402 square kilometres and a population of about 4.6 million, the capital of Piedmont is Turin. The name Piedmont comes from medieval Latin Pedemontium or Pedemontis, i. e. ad pedem montium, meaning “at the foot of the mountains”. Other towns of Piedmont with more than 20,000 inhabitants sorted by population and it borders with France, Switzerland and the Italian regions of Lombardy, Liguria, Aosta Valley and for a very small fragment with Emilia Romagna. The geography of Piedmont is 43. 3% mountainous, along with areas of hills. Piedmont is the second largest of Italys 20 regions, after Sicily and it is broadly coincident with the upper part of the drainage basin of the river Po, which rises from the slopes of Monviso in the west of the region and is Italy’s largest river. The Po collects all the waters provided within the semicircle of mountains which surround the region on three sides, from the highest peaks the land slopes down to hilly areas, and then to the upper, and then to the lower great Padan Plain. 7. 6% of the territory is considered protected area. There are 56 different national or regional parks, one of the most famous is the Gran Paradiso National Park located between Piedmont and the Aosta Valley, Piedmont was inhabited in early historic times by Celtic-Ligurian tribes such as the Taurini and the Salassi. They were later subdued by the Romans, who founded several colonies there including Augusta Taurinorum, after the fall of the Western Roman Empire, the region was repeatedly invaded by the Burgundians, the Goths, Byzantines, Lombards, Franks. In the 9th–10th centuries there were incursions by the Magyars. At the time Piedmont, as part of the Kingdom of Italy within the Holy Roman Empire, was subdivided into several marks, in 1046, Oddo of Savoy added Piedmont to their main territory of Savoy, with a capital at Chambéry. Other areas remained independent, such as the powerful comuni of Asti and Alessandria, the County of Savoy was elevated to a duchy in 1416, and Duke Emanuele Filiberto moved the seat to Turin in 1563. In 1720, the Duke of Savoy became King of Sardinia, founding what evolved into the Kingdom of Sardinia, the Republic of Alba was created in 1796 as a French client republic in Piedmont. A new client republic, the Piedmontese Republic, existed between 1798 and 1799 before it was reoccupied by Austrian and Russian troops, in June 1800 a third client republic, the Subalpine Republic, was established in Piedmont. It fell under full French control in 1801 and it was annexed by France in September 1802, in the congress of Vienna, the Kingdom of Sardinia was restored, and furthermore received the Republic of Genoa to strengthen it as a barrier against France. Piedmont was a springboard for Italys unification in 1859–1861, following earlier unsuccessful wars against the Austrian Empire in 1820–1821 and this process is sometimes referred to as Piedmontisation. However, the efforts were countered by the efforts of rural farmers
2.
Turin
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Turin is a city and an important business and cultural centre in northern Italy, capital of the Piedmont region and was the first capital city of Italy. The city is located mainly on the bank of the Po River, in front of Susa Valley and surrounded by the western Alpine arch. The population of the city proper is 892,649 while the population of the area is estimated by Eurostat to be 1.7 million inhabitants. The Turin metropolitan area is estimated by the OECD to have a population of 2.2 million, in 1997 a part of the historical center of Torino was inscribed in the World Heritage List under the name Residences of the Royal House of Savoy. Turin is well known for its Renaissance, Baroque, Rococo, Neo-classical, many of Turins public squares, castles, gardens and elegant palazzi such as Palazzo Madama, were built between the 16th and 18th centuries. This was after the capital of the Duchy of Savoy was moved to Turin from Chambery as part of the urban expansion, the city used to be a major European political center. Turin was Italys first capital city in 1861 and home to the House of Savoy, from 1563, it was the capital of the Duchy of Savoy, then of the Kingdom of Sardinia ruled by the Royal House of Savoy and finally the first capital of the unified Italy. Turin is sometimes called the cradle of Italian liberty for having been the birthplace and home of notable politicians and people who contributed to the Risorgimento, such as Cavour. The city currently hosts some of Italys best universities, colleges, academies, lycea and gymnasia, such as the University of Turin, founded in the 15th century, in addition, the city is home to museums such as the Museo Egizio and the Mole Antonelliana. Turins attractions make it one of the worlds top 250 tourist destinations, Turin is ranked third in Italy, after Milan and Rome, for economic strength. With a GDP of $58 billion, Turin is the worlds 78th richest city by purchasing power, as of 2010, the city has been ranked by GaWC as a Gamma World city. Turin is also home to much of the Italian automotive industry, the Taurini were an ancient Celto-Ligurian Alpine people, who occupied the upper valley of the Po River, in the center of modern Piedmont. In 218 BC, they were attacked by Hannibal as he was allied with their long-standing enemies, the Taurini chief town was captured by Hannibals forces after a three-day siege. As a people they are mentioned in history. It is believed that a Roman colony was established in 27 BC under the name of Castra Taurinorum, both Livy and Strabo mention the Taurinis country as including one of the passes of the Alps, which points to a wider use of the name in earlier times. In the 1st century BC, the Romans created a military camp, the typical Roman street grid can still be seen in the modern city, especially in the neighborhood known as the Quadrilatero Romano. Via Garibaldi traces the path of the Roman citys decumanus which began at the Porta Decumani. The Porta Palatina, on the side of the current city centre, is still preserved in a park near the Cathedral
3.
Kingdom of Italy
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The state was founded as a result of the unification of Italy under the influence of the Kingdom of Sardinia, which can be considered its legal predecessor state. Italy declared war on Austria in alliance with Prussia in 1866, Italian troops entered Rome in 1870, ending more than one thousand years of Papal temporal power. Italy entered into a Triple Alliance with Germany and Austria-Hungary in 1882, victory in the war gave Italy a permanent seat in the Council of the League of Nations. Fascist Italy is the era of National Fascist Party rule from 1922 to 1943 with Benito Mussolini as head of government, according to Payne, Fascist regime passed through several relatively distinct phases. The first phase was nominally a continuation of the parliamentary system, then came the second phase, the construction of the Fascist dictatorship proper from 1925 to 1929. The third phase, with activism, was 1929–34. The war itself was the phase with its disasters and defeats. Italy was allied with Nazi Germany in World War II until 1943 and it switched sides to the Allies after ousting Mussolini and shutting down the Fascist party in areas controlled by the Allied invaders. Shortly after the war, civil discontent led to the referendum of 1946 on whether Italy would remain a monarchy or become a republic. Italians decided to abandon the monarchy and form the Italian Republic, the Kingdom of Italy claimed all of the territory which is modern-day Italy. The development of the Kingdoms territory progressed under Italian re-unification until 1870, the state for a long period of time did not include Trieste or Trentino-Alto Adige/Südtirol, which are in Italy today, and only annexed them in 1919. After the Second World War, the borders of present-day Italy were founded, the Kingdom of Italy was theoretically a constitutional monarchy. Executive power belonged to the monarch, as executed through appointed ministers, two chambers of parliament restricted the monarchs power—an appointive Senate and an elective Chamber of Deputies. The kingdoms constitution was the Statuto Albertino, the governing document of the Kingdom of Sardinia. In theory, ministers were responsible to the king. However, in practice, it was impossible for an Italian government to stay in office without the support of Parliament, members of the Chamber of Deputies were elected by plurality voting system elections in uninominal districts. A candidate needed the support of 50% of those voting, and of 25% of all enrolled voters, if not all seats were filled on the first ballot, a runoff was held shortly afterwards for the remaining vacancies. After a brief multinominal experimentation in 1882, proportional representation into large, regional, Socialists became the major party, but they were unable to form a government in a parliament split into three different factions, with Christian Populists and classical liberals
4.
Italy
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Italy, officially the Italian Republic, is a unitary parliamentary republic in Europe. Located in the heart of the Mediterranean Sea, Italy shares open land borders with France, Switzerland, Austria, Slovenia, San Marino, Italy covers an area of 301,338 km2 and has a largely temperate seasonal climate and Mediterranean climate. Due to its shape, it is referred to in Italy as lo Stivale. With 61 million inhabitants, it is the fourth most populous EU member state, the Italic tribe known as the Latins formed the Roman Kingdom, which eventually became a republic that conquered and assimilated other nearby civilisations. The legacy of the Roman Empire is widespread and can be observed in the distribution of civilian law, republican governments, Christianity. The Renaissance began in Italy and spread to the rest of Europe, bringing a renewed interest in humanism, science, exploration, Italian culture flourished at this time, producing famous scholars, artists and polymaths such as Leonardo da Vinci, Galileo, Michelangelo and Machiavelli. The weakened sovereigns soon fell victim to conquest by European powers such as France, Spain and Austria. Despite being one of the victors in World War I, Italy entered a period of economic crisis and social turmoil. The subsequent participation in World War II on the Axis side ended in defeat, economic destruction. Today, Italy has the third largest economy in the Eurozone and it has a very high level of human development and is ranked sixth in the world for life expectancy. The country plays a prominent role in regional and global economic, military, cultural and diplomatic affairs, as a reflection of its cultural wealth, Italy is home to 51 World Heritage Sites, the most in the world, and is the fifth most visited country. The assumptions on the etymology of the name Italia are very numerous, according to one of the more common explanations, the term Italia, from Latin, Italia, was borrowed through Greek from the Oscan Víteliú, meaning land of young cattle. The bull was a symbol of the southern Italic tribes and was often depicted goring the Roman wolf as a defiant symbol of free Italy during the Social War. Greek historian Dionysius of Halicarnassus states this account together with the legend that Italy was named after Italus, mentioned also by Aristotle and Thucydides. The name Italia originally applied only to a part of what is now Southern Italy – according to Antiochus of Syracuse, but by his time Oenotria and Italy had become synonymous, and the name also applied to most of Lucania as well. The Greeks gradually came to apply the name Italia to a larger region, excavations throughout Italy revealed a Neanderthal presence dating back to the Palaeolithic period, some 200,000 years ago, modern Humans arrived about 40,000 years ago. Other ancient Italian peoples of undetermined language families but of possible origins include the Rhaetian people and Cammuni. Also the Phoenicians established colonies on the coasts of Sardinia and Sicily, the Roman legacy has deeply influenced the Western civilisation, shaping most of the modern world
5.
University of Turin
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The University of Turin is a university in the city of Turin in the Piedmont region of north-western Italy. It is one of the oldest universities in Europe, and continues to play an important role in research, the University of Turin was founded as a studium in 1404, under the initiative of Prince Ludovico di Savoia. From 1427 to 1436 the seat of the university was transferred to Chieri and it was closed in 1536, and reestablished by Duke Emmanuel Philibert thirty years later. With the reforms carried out by Victor Amadeus II, the University of Turin became a new model for many other universities. During the 18th century, the University faced an enormous growth in faculty and endowment size, notable scholars of this period include Cesare Lombroso, Carlo Forlanini and Arturo Graf. In the 20th century, the University of Turin was one of the centers of the Italian anti-fascism, the new impulse was performed in collaboration with other national and international research centers, as well as with local organizations and the Italian Minister of Public Instruction. By the end of the 1990s, the campuses of Alessandria, Novara. The new institution, which only held courses in civil. The Bishop, as Rector of Studies, proclaimed and conferred the title on the new doctors, after a series of interruptions in its activities, the university was moved to Chieri and later, in 1434, to Savigliano. In 1436, when the returned to Turin, Ludovico di Savoia. The ducal licenses of 6 October 1436 set up the three faculties of Theology, Arts and Medicine, and Law, and twenty-five lectureships or chairs. The growth and development of the role of Turin as the capital led to the consolidation of the University. The Study, closed at the beginning of 1536 with the French occupation, reopened in 1558 with lecturers at Mondovì, it was re-established in Turin in 1566. With Emmanuel Philibert and Charles Emmanuel I, the University enjoyed a season of great prosperity due to the presence of illustrious teachers, the opening of the new premises marked a major turning point in the history of the greatest Piedmontese educational institution. This had an effect on the cultural linguistic models of the Duchy. At the time, the Piedmontese Studium became a point of reference for university reforms at Parma and Modena, Charles Emmanuel III continued the policy of innovation and consolidation commenced by Victor Amadeus II and created a University Museum in 1739. However, in the last decades of the 18th century, the course of events at the University, closely connected to international developments, led to urban unrest. The revolt of university students in 1791 joined by artisans who stormed the Collegio delle Province in 1792 causing numerous victims, was an instance of this conflict
6.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
7.
Euclid
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Euclid, sometimes called Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the father of geometry. He was active in Alexandria during the reign of Ptolemy I, in the Elements, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory, Euclid is the anglicized version of the Greek name Εὐκλείδης, which means renowned, glorious. Very few original references to Euclid survive, so little is known about his life, the date, place and circumstances of both his birth and death are unknown and may only be estimated roughly relative to other people mentioned with him. He is rarely mentioned by name by other Greek mathematicians from Archimedes onward, the few historical references to Euclid were written centuries after he lived by Proclus c.450 AD and Pappus of Alexandria c.320 AD. Proclus introduces Euclid only briefly in his Commentary on the Elements, Proclus later retells a story that, when Ptolemy I asked if there was a shorter path to learning geometry than Euclids Elements, Euclid replied there is no royal road to geometry. This anecdote is questionable since it is similar to a story told about Menaechmus, a detailed biography of Euclid is given by Arabian authors, mentioning, for example, a birth town of Tyre. This biography is generally believed to be completely fictitious, however, this hypothesis is not well accepted by scholars and there is little evidence in its favor. The only reference that historians rely on of Euclid having written the Elements was from Proclus, although best known for its geometric results, the Elements also includes number theory. The geometrical system described in the Elements was long known simply as geometry, today, however, that system is often referred to as Euclidean geometry to distinguish it from other so-called non-Euclidean geometries that mathematicians discovered in the 19th century. In addition to the Elements, at least five works of Euclid have survived to the present day and they follow the same logical structure as Elements, with definitions and proved propositions. Data deals with the nature and implications of information in geometrical problems. On Divisions of Figures, which only partially in Arabic translation. It is similar to a first-century AD work by Heron of Alexandria, catoptrics, which concerns the mathematical theory of mirrors, particularly the images formed in plane and spherical concave mirrors. The attribution is held to be anachronistic however by J J OConnor, phaenomena, a treatise on spherical astronomy, survives in Greek, it is quite similar to On the Moving Sphere by Autolycus of Pitane, who flourished around 310 BC. Optics is the earliest surviving Greek treatise on perspective, in its definitions Euclid follows the Platonic tradition that vision is caused by discrete rays which emanate from the eye. One important definition is the fourth, Things seen under a greater angle appear greater, proposition 45 is interesting, proving that for any two unequal magnitudes, there is a point from which the two appear equal. Other works are attributed to Euclid, but have been lost
8.
Bertrand Russell
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Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, writer, social critic, political activist and Nobel laureate. At various points in his life he considered himself a liberal, a socialist, and a pacifist and he was born in Monmouthshire into one of the most prominent aristocratic families in the United Kingdom. In the early 20th century, Russell led the British revolt against idealism and he is considered one of the founders of analytic philosophy along with his predecessor Gottlob Frege, colleague G. E. Moore, and protégé Ludwig Wittgenstein. He is widely held to be one of the 20th centurys premier logicians, with A. N. Whitehead he wrote Principia Mathematica, an attempt to create a logical basis for mathematics. His philosophical essay On Denoting has been considered a paradigm of philosophy, Russell mostly was a prominent anti-war activist, he championed anti-imperialism. Occasionally, he advocated preventive nuclear war, before the opportunity provided by the monopoly is gone. He went to prison for his pacifism during World War I, in 1950 Russell was awarded the Nobel Prize in Literature in recognition of his varied and significant writings in which he champions humanitarian ideals and freedom of thought. Bertrand Russell was born on 18 May 1872 at Ravenscroft, Trellech, Monmouthshire and his parents, Viscount and Viscountess Amberley, were radical for their times. Lord Amberley consented to his wifes affair with their childrens tutor, both were early advocates of birth control at a time when this was considered scandalous. Lord Amberley was an atheist and his atheism was evident when he asked the philosopher John Stuart Mill to act as Russells secular godfather, Mill died the year after Russells birth, but his writings had a great effect on Russells life. His paternal grandfather, the Earl Russell, had asked twice by Queen Victoria to form a government. The Russells had been prominent in England for several centuries before this, coming to power, Lady Amberley was the daughter of Lord and Lady Stanley of Alderley. Russell often feared the ridicule of his grandmother, one of the campaigners for education of women. Russell had two siblings, brother Frank, and sister Rachel, in June 1874 Russells mother died of diphtheria, followed shortly by Rachels death. In January 1876, his father died of bronchitis following a period of depression. Frank and Bertrand were placed in the care of their staunchly Victorian paternal grandparents and his grandfather, former Prime Minister Earl Russell, died in 1878, and was remembered by Russell as a kindly old man in a wheelchair. His grandmother, the Countess Russell, was the dominant family figure for the rest of Russells childhood, the countess was from a Scottish Presbyterian family, and successfully petitioned the Court of Chancery to set aside a provision in Amberleys will requiring the children to be raised as agnostics. Her favourite Bible verse, Thou shalt not follow a multitude to do evil, the atmosphere at Pembroke Lodge was one of frequent prayer, emotional repression, and formality, Frank reacted to this with open rebellion, but the young Bertrand learned to hide his feelings
9.
Calculus
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Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. It has two branches, differential calculus, and integral calculus, these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the notions of convergence of infinite sequences. Generally, modern calculus is considered to have developed in the 17th century by Isaac Newton. Today, calculus has widespread uses in science, engineering and economics, Calculus is a part of modern mathematics education. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, Calculus has historically been called the calculus of infinitesimals, or infinitesimal calculus. Calculus is also used for naming some methods of calculation or theories of computation, such as calculus, calculus of variations, lambda calculus. The ancient period introduced some of the ideas that led to integral calculus, the method of exhaustion was later discovered independently in China by Liu Hui in the 3rd century AD in order to find the area of a circle. In the 5th century AD, Zu Gengzhi, son of Zu Chongzhi, indian mathematicians gave a non-rigorous method of a sort of differentiation of some trigonometric functions. In the Middle East, Alhazen derived a formula for the sum of fourth powers. He used the results to carry out what would now be called an integration, Cavalieris work was not well respected since his methods could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first. The formal study of calculus brought together Cavalieris infinitesimals with the calculus of finite differences developed in Europe at around the same time, pierre de Fermat, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal error term. The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, in other work, he developed series expansions for functions, including fractional and irrational powers, and it was clear that he understood the principles of the Taylor series. He did not publish all these discoveries, and at this time infinitesimal methods were considered disreputable. These ideas were arranged into a calculus of infinitesimals by Gottfried Wilhelm Leibniz. He is now regarded as an independent inventor of and contributor to calculus, unlike Newton, Leibniz paid a lot of attention to the formalism, often spending days determining appropriate symbols for concepts. Leibniz and Newton are usually credited with the invention of calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the used in calculus today
10.
Refugium Peccatorum
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Refugium Peccatorum meaning Refuge of Sinners is a Roman Catholic title for the Blessed Virgin Mary. Its use goes back to Saint Germanus of Constantinople in the 8th century and she is the new Eve, who cannot eliminate the damage created by Eve, but limit it. Her fullness of grace, her position among the disciples of Christ, Refugium Peccatorum is one part of four Marian advocations in the Litany of Loreto, the others being Salus Infirmorum Consolatrix Afflictorum and Auxilium Christianorum. The Archconfraternity of the Immaculate Heart of Mary, Refuge of Sinners was established at Notre-Dame-des-Victoires in the 19th century, and spread throughout the world. In some 19th-century images, biblical scenes are included below the image of the Madonna, the term Refugium peccatorum is also used other works of Roman Catholic Marian art. For instance, there is a statue representing the Virgin Mary, on the grand staircase of the old municipal palace in Venice. The name came from the fact that the convicts were allowed to stop in front of the Virgin Mary’s statue to pray for their soul on the way to the scaffold, the traditional feast day of Our Lady, Refuge of Sinners is August 16. In Mexico, the feast day is observed on July 4 and she is the patroness of California, where the dioceses there celebrate the feast on July 5. Our Lady of Laus, which the Vatican approved in 2008, is named Refuge of Sinners because she asked for the conversion of sinners
11.
Madonna (art)
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A Madonna is a representation of Mary, either alone or with her child Jesus. These images are central icons for both the Catholic and Orthodox churches, the word is from Italian ma donna, meaning my lady. Or descriptive of the posture, as in Hodegetria, Eleusa. The term Madonna in the sense of picture or statue of the Virgin Mary enters English usage in the 17th century, in an Eastern Orthodox context, such images are typically known as Theotokos. Madonna may be used of representations of Mary, with or without the infant Jesus, is the focus and central figure of the image. Other types of Marian imagery have a context, depicting scenes from the Life of the Virgin. The earliest depictions of Mary date still to Early Christianity, found in the Catacombs of Rome and these are in a narrative context. The Theotokos iconography as it developed in the 6th to 8th century rose to importance in the high medieval period both in the Eastern Orthodox and in the Latin spheres. Liturgy depicting Mary as powerful intercessor was brought from Greek into Latin tradition in the 8th century, the Greek title of Δεσποινα was adopted as Latin Domina Lady. The medieval Italian Ma Donna pronounced reflects Mea Domina, while Nostra Domina was adopted in French and these names signal both the increased importance of the cult of the virgin and the prominence of art in service to Marian devotion during the late medieval period. Madonna was meant more to people of the theological concept which is placing such a high value on purity or virginity. This is also represented by the color of her clothing, the color blue symbolized purity, virginity, and royalty. In this sense, a Madonna, or a Madonna with Child is used of works of art. There are several types of representation of the Madonna. One type of Madonna shows Mary alone, and standing, generally glorified and with a gesture of prayer and this type of image occurs in a number of ancient apsidal mosaics. Full-length standing images of the Madonna more frequently include the infant Jesus, the most famous Byzantine image, the Hodegetria was originally of this type, though most copies are at half-length. There are a number of paintings that depict the Madonna in this manner. The Madonna enthroned is a type of image that dates from the Byzantine period and was used widely in Medieval and these representations of the Madonna and Child often take the form of large altarpieces
12.
Fractal
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A fractal is a mathematical set that exhibits a repeating pattern displayed at every scale. It is also known as expanding symmetry or evolving symmetry, if the replication is exactly the same at every scale, it is called a self-similar pattern. An example of this is the Menger Sponge, Fractals can also be nearly the same at different levels. This latter pattern is illustrated in small magnifications of the Mandelbrot set, Fractals also include the idea of a detailed pattern that repeats itself. Fractals are different from other geometric figures because of the way in which they scale, doubling the edge lengths of a polygon multiplies its area by four, which is two raised to the power of two. Likewise, if the radius of a sphere is doubled, its volume scales by eight, but if a fractals one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an integer. This power is called the dimension of the fractal. As mathematical equations, fractals are usually nowhere differentiable, the term fractal was first used by mathematician Benoît Mandelbrot in 1975. Mandelbrot based it on the Latin frāctus meaning broken or fractured, there is some disagreement amongst authorities about how the concept of a fractal should be formally defined. Mandelbrot himself summarized it as beautiful, damn hard, increasingly useful, Fractals are not limited to geometric patterns, but can also describe processes in time. Fractal patterns with various degrees of self-similarity have been rendered or studied in images, structures and sounds and found in nature, technology, art, Fractals are of particular relevance in the field of chaos theory, since the graphs of most chaotic processes are fractal. The word fractal often has different connotations for laypeople than for mathematicians, the mathematical concept is difficult to define formally even for mathematicians, but key features can be understood with little mathematical background. If this is done on fractals, however, no new detail appears, nothing changes, self-similarity itself is not necessarily counter-intuitive. The difference for fractals is that the pattern reproduced must be detailed, a regular line, for instance, is conventionally understood to be 1-dimensional, if such a curve is divided into pieces each 1/3 the length of the original, there are always 3 equal pieces. In contrast, consider the Koch snowflake and it is also 1-dimensional for the same reason as the ordinary line, but it has, in addition, a fractal dimension greater than 1 because of how its detail can be measured. This also leads to understanding a third feature, that fractals as mathematical equations are nowhere differentiable, in a concrete sense, this means fractals cannot be measured in traditional ways. The history of fractals traces a path from chiefly theoretical studies to modern applications in computer graphics, according to Pickover, the mathematics behind fractals began to take shape in the 17th century when the mathematician and philosopher Gottfried Leibniz pondered recursive self-similarity. In his writings, Leibniz used the term fractional exponents, also in the last part of that century, Felix Klein and Henri Poincaré introduced a category of fractal that has come to be called self-inverse fractals
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