In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship. Expressed algebraically, for quantities a and b with a > b > 0, a + b a = a b = def φ, where the Greek letter phi represents the golden ratio. It is an irrational number, a solution to the quadratic equation x 2 − x − 1 = 0, with a value of: φ = 1 + 5 2 = 1.6180339887 …. The golden ratio is called the golden mean or golden section. Other names include extreme and mean ratio, medial section, divine proportion, divine section, golden proportion, golden cut, golden number. Mathematicians since Euclid have studied the properties of the golden ratio, including its appearance in the dimensions of a regular pentagon and in a golden rectangle, which may be cut into a square and a smaller rectangle with the same aspect ratio; the golden ratio has been used to analyze the proportions of natural objects as well as man-made systems such as financial markets, in some cases based on dubious fits to data.
The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other plant parts. Some twentieth-century artists and architects, including Le Corbusier and Salvador Dalí, have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing. Two quantities a and b are said to be in the golden ratio φ if a + b a = a b = φ. One method for finding the value of φ is to start with the left fraction. Through simplifying the fraction and substituting in b/a = 1/φ, a + b a = a a + b a = 1 + b a = 1 + 1 φ. Therefore, 1 + 1 φ = φ. Multiplying by φ gives φ + 1 = φ 2 which can be rearranged to φ 2 − φ − 1 = 0. Using the quadratic formula, two solutions are obtained: 1 + 5 2 = 1.618 033 988 7 … and 1 − 5 2 = − 0.618 033 988 7 … Because φ is the ratio between positive quantities, φ is positive: φ = 1 + 5 2 = 1.61803 39887 … The golden ratio has been claimed to have held a special fascination for at least 2,400 years, although without reliable evidence.
According to Mario Livio: Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties. But the fascination with the Golden Ratio is not confined just to mathematicians. Biologists, musicians, architects and mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics. Ancient Greek mathematicians first studied what we now call the golden ratio because of its frequent appearance in geometry. According to one story, 5th-century BC mathematician Hippasus discovered that the golden ratio was neither a whole number nor a fraction, surprising Pythagoreans. Euclid's Elements provides several propositions and their proofs employing the golden ratio and contains the first known definition: A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.
The golden ratio was studied peripherally over the next millennium. Abu Kamil employed it in his geometric calculati
Charles-Édouard Jeanneret, known as Le Corbusier, was a Swiss-French architect, painter, urban planner and one of the pioneers of what is now called modern architecture. He was born in Switzerland and became a French citizen in 1930, his career spanned five decades, he designed buildings in Europe, Japan and North and South America. Dedicated to providing better living conditions for the residents of crowded cities, Le Corbusier was influential in urban planning, was a founding member of the Congrès International d'Architecture Moderne. Le Corbusier prepared the master plan for the city of Chandigarh in India, contributed specific designs for several buildings there. On 17 July 2016, seventeen projects by Le Corbusier in seven countries were inscribed in the list of UNESCO World Heritage Sites as The Architectural Work of Le Corbusier, an Outstanding Contribution to the Modern Movement. Charles-Édouard Jeanneret was born on 6 October 1887 in La Chaux-de-Fonds, a small city in the French-speaking Neuchâtel canton in north-western Switzerland, in the Jura mountains, just 5 kilometres across the border from France.
It was an industrial town, devoted to the manufacture of watches. His father was an artisan who enameled watches, while his mother gave piano lessons, his elder brother Albert was an amateur violinist. He attended a kindergarten. Like his contemporaries Frank Lloyd Wright and Mies van der Rohe, Le Corbusier did not have formal academic training as an architect, he was attracted to the visual arts and at the age of fifteen he entered the municipal art school in La-Chaux-de-Fonds which taught the applied arts connected with watchmaking. Three years he attended the higher course of decoration, founded by the painter Charles L'Eplattenier, who had studied in Budapest and Paris. Le Corbusier wrote that L'Eplattenier had made him "a man of the woods" and taught him painting from nature, his father took him into the mountains around the town. He wrote "we were on mountaintops, his architecture teacher in the Art School was the architect René Chapallaz, who had a large influence on Le Corbusier's earliest house designs.
However, he reported that it was the art teacher L'Eplattenier who made him choose architecture. "I had a horror of architecture and architects," he wrote. "... I was sixteen, I accepted the verdict and I obeyed. I moved into architecture." Le Corbusier began teaching himself by going to the library to read about architecture and philosophy, by visiting museums, by sketching buildings, by constructing them. In 1905, he and two other students, under the supervision of their teacher, René Chapallaz and built his first house, the Villa Fallet, for the engraver Louis Fallet, a friend of his teacher Charles L'Eplattenier. Located on the forested hillside near Chaux-de-fonds, it was a large chalet with a steep roof in the local alpine style and crafted colored geometric patterns on the façade. The success of this house led to his construction of two similar houses, the Villas Jacquemet and Stotzer, in the same area. In September 1907, he made his first trip outside of Switzerland. In Florence, he visited the Florence Charterhouse in Galluzzo, which made a lifelong impression on him.
"I would have liked to live in one of what they called their cells," he wrote later. "It was the solution for a unique kind of worker's housing, or rather for a terrestrial paradise." He traveled to Paris, during fourteen months between 1908 until 1910 he worked as a draftsman in the office of the architect Auguste Perret, the pioneer of the use of reinforced concrete in residential construction and the architect of the Art Deco landmark Théâtre des Champs-Élysées. Two years between October 1910 and March 1911, he traveled to Germany and worked four months in the office Peter Behrens, where Ludwig Mies van der Rohe and Walter Gropius were working and learning. In 1911, he traveled again for five months, he spoke of what he saw during this trip in many of his books, it was the subject of his last book, Le Voyage d'Orient. In 1912, he began his most ambitious project. Located on the forested hillside near La-Chaux-de-Fonds; the Jeanneret-Perret house was larger than the others, in a more innovative style.
The interior spaces were organized around the four pillars of the salon in the center, foretelling the open interiors he would create in his buildings. The project was more expensive to build. However, it led to a commission to build an more imposing villa in the nearby village of Le Locle for a wealthy watch manufacturer. Georges Favre-Jacot. Le Corbusier designed the new house in less than a month; the building was designed to fit its hillside site, interior plan was spacious and designed around a courtyard for maximum light, significant de
Lecturer is an academic rank within many universities, though the meaning of the term varies somewhat from country to country. It denotes an academic expert, hired to teach on a full- or part-time basis, they may conduct research. In the UK, the term lecturer covers several academic ranks; the key distinction is between temporary/fixed-term lectureships. A permanent lecturer in UK universities holds an open-ended position that covers teaching and administrative responsibilities. Permanent lectureships are tenure-track or tenured positions that are equivalent to an assistant or associate professorship in North America. After a number of years, a lecturer may be promoted based on his or her research record to become a senior lecturer; this position is below professor. Research lecturers are the equivalent in rank of lecturers and senior lecturers, but reflect a research-intensive orientation. Research lecturers are common in fields such as medicine and biological and physical sciences. In contrast, fixed-term or temporary lecturers are appointed for specific short-term teaching needs.
These positions are non-renewable and are common post-doctoral appointments. In North American terms, a fixed-term lecturer can hold an equivalent rank to assistant professor without tenure. Longer contracts denote greater seniority or higher rank. Teaching fellows may sometimes be referred to as lecturers—for example, Exeter named some of that group as education and scholarship lecturers to recognise the contribution of teaching, elevate the titles of teaching fellows to lecturers; some universities refer to graduate students or others, who undertake ad-hoc teaching for a department sessional lecturers. Like adjunct professors and sessional lecturers in North America, these non-permanent teaching staff are very poorly paid; these varying uses of the term lecturer cause confusion for non-UK academics. As a proportion of UK academic staff, the proportion of permanent lectureships has fallen considerably; this is one reason why permanent lectureships are secured only after several years of post-doctoral experience.
Data from the Higher Education Statistics Agency show that in 2013-14, 36 per cent of full- and part-time academic staff were on fixed-term contracts, down from 45 per cent a decade earlier. Over the same period, the proportion of academic staff on permanent contracts rose from 55 per cent to 64 per cent. Others were on contracts classed as “atypical”.' In the UK, promotion to a senior lectureship reflected prowess in teaching or administration rather than research, the position was much less to lead direct to promotion to professor. In contrast, promotion to senior lecturer nowadays is based on research achievements, is an integral part of the promotion path to a full chair. Promotion to reader is sometimes still necessary before promotion to a full chair. Senior lecturers and readers are sometimes paid on the same salary scale, although readers are recognized as more senior. Readers in pre-1992 universities are considered at least the equivalent, in terms of status, of professors in post-1992 universities.
Many academics consider it more prestigious to have been a reader in a pre-1992 university than a professor in a post-1992 university. Many open-ended lecturers in the UK have a doctorate and have postdoctoral research experience. In all fields, a doctorate is a prerequisite, although this was not the case; some academic positions could have been held on the basis of research merit alone, without a higher degree. The new universities have a different ranking naming scheme from the older universities. Many pre-1992 universities use the grades: Lecturer, Senior Lecturer, Professor. Meanwhile, post-1992 grades are normally: Lecturer, Senior Lecturer, Principal Lecturer or Reader, Professor. Much confusion surrounds the differing use of the "Senior Lecturer" title. A Senior Lecturer in a post-1992 university is equivalent to a Lecturer in a pre-1992 university, whereas a Senior Lecturer in a pre-1992 university is most equivalent to a Principal Lecturer in a post-1992 university. According to the Times Higher Education, the University of Warwick decided in 2006 "to break away from hundreds of years of academic tradition, renaming lecturers'assistant professors', senior lecturers and readers'associate professors' while still calling professors'professors'.
The radical move will horrify those who believe the "professor" title should be reserved for an academic elite." Nottingham has a mixture of the standard UK system, the system at Warwick, with both lecturers and assistant professors. At Reading, job advertisements and academic staff web pages use the title associate professor, but the ordinances of the university make no reference to these titles, they address only procedures for conferring the traditional UK academic ranks. Since the Conservatives' 1988 Education Reform Act, the ironclad tenure that used to exist in the UK has given way to a less secure form of tenure. Technically, university vice-chancellors can make individual faculty members redundant for poor performance or institute departmental redundancies, but in practice, this is rare; the most noted use of this policy happened in 2012 at Queen Mary University of London where lecturer
De Stijl, Dutch for "The Style" known as Neoplasticism, was a Dutch art movement founded in 1917 in Leiden. De Stijl consisted of architects. In a narrower sense, the term De Stijl is used to refer to a body of work from 1917 to 1931 founded in the Netherlands. Proponents of De Stijl advocated pure abstraction and universality by a reduction to the essentials of form and colour. De Stijl is the name of a journal, published by the Dutch painter, designer and critic Theo van Doesburg that served to propagate the group's theories. Along with van Doesburg, the group's principal members were the painters Piet Mondrian, Vilmos Huszár, Bart van der Leck, the architects Gerrit Rietveld, Robert van't Hoff, J. J. P. Oud; the artistic philosophy that formed a basis for the group's work is known as Neoplasticism—the new plastic art. According to Theo van Doesburg in the introduction of the magazine "De Stijl" 1917 no.1, the "De Stijl"-movement was a reaction to the "Modern Baroque" of the Amsterdam School movement with the magazine "Wendingen".
Mondrian sets forth the delimitations of Neoplasticism in his essay "Neo-Plasticism in Pictorial Art". He writes, "this new plastic idea will ignore the particulars of appearance, to say, natural form and colour. On the contrary, it should find its expression in the abstraction of form and colour, to say, in the straight line and the defined primary colour". With these constraints, his art allows only primary colours and non-colours, only squares and rectangles, only straight and horizontal or vertical lines; the De Stijl movement posited the fundamental principle of the geometry of the straight line, the square, the rectangle, combined with a strong asymmetricality. The name De Stijl is derived from Gottfried Semper's Der Stil in den technischen und tektonischen Künsten oder Praktische Ästhetik, which Curl suggests was mistakenly believed to advocate materialism and functionalism; the "plastic vision" of De Stijl artists called Neo-Plasticism, saw itself as reaching beyond the changing appearance of natural things to bring an audience into intimate contact with an immutable core of reality, a reality, not so much a visible fact as an underlying spiritual vision.
In general, De Stijl proposed ultimate simplicity and abstraction, both in architecture and painting, by using only straight horizontal and vertical lines and rectangular forms. Furthermore, their formal vocabulary was limited to the primary colours, red and blue, the three primary values, black and grey; the works attained aesthetic balance by the use of opposition. This element of the movement embodies the second meaning of stijl: "a post, jamb or support". In many of the group's three-dimensional works and horizontal lines are positioned in layers or planes that do not intersect, thereby allowing each element to exist independently and unobstructed by other elements; this feature can be found in the Red and Blue Chair. De Stijl was influenced by Cubist painting as well as by the mysticism and the ideas about "ideal" geometric forms in the neoplatonic philosophy of mathematician M. H. J. Schoenmaekers; the De Stijl movement was influenced by Neopositivism. The works of De Stijl would influence the Bauhaus style and the international style of architecture as well as clothing and interior design.
However, it did not follow the general guidelines of an "-ism", nor did it adhere to the principles of art schools like the Bauhaus. In music, De Stijl was an influence only on the work of composer Jakob van Domselaer, a close friend of Mondrian. Between 1913 and 1916, he composed his Proeven van Stijlkunst, inspired by Mondrian's paintings; this minimalistic—and, at the time, revolutionary—music defined "horizontal" and "vertical" musical elements and aimed at balancing those two principles. Van Domselaer was unknown in his lifetime, did not play a significant role within De Stijl. From the flurry of new art movements that followed the Impressionist revolutionary new perception of painting, Cubism arose in the early 20th century as an important and influential new direction. In the Netherlands, there was interest in this "new art". However, because the Netherlands remained neutral in World War I, Dutch artists were not able to leave the country after 1914 and were thus isolated from the international art world—and in particular, from Paris, its centre then.
During that period, Theo van Doesburg started looking for other artists to set up a journal and start an art movement. Van Doesburg was a writer and critic, more successful writing about art than working as an independent artist. Quite adept at making new contacts due to his flamboyant personality and outgoing nature, he had many useful connections in the art world. Around 1915, Van Doesburg started meeting the artists who would become the founders of the journal, he first met Piet Mondrian at an exhibition in Stedelijk Museum Amsterdam. Mondrian, who had moved to Paris in 1912, had be
In mathematics, the Fibonacci numbers denoted Fn form a sequence, called the Fibonacci sequence, such that each number is the sum of the two preceding ones, starting from 0 and 1. That is, F 0 = 0, F 1 = 1, F n = F n − 1 + F n − 2, for n > 1. One has F2 = 1. In some books, in old ones, F0, the "0" is omitted, the Fibonacci sequence starts with F1 = F2 = 1; the beginning of the sequence is thus: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, … Fibonacci numbers are related to the golden ratio: Binet's formula expresses the nth Fibonacci number in terms of n and the golden ratio, implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as n increases. Fibonacci numbers are named after Italian mathematician Leonardo of Pisa known as Fibonacci, they appear to have first arisen as early as 200 BC in work by Pingala on enumerating possible patterns of poetry formed from syllables of two lengths. In his 1202 book Liber Abaci, Fibonacci introduced the sequence to Western European mathematics, although the sequence had been described earlier in Indian mathematics.
Fibonacci numbers appear unexpectedly in mathematics, so much so that there is an entire journal dedicated to their study, the Fibonacci Quarterly. Applications of Fibonacci numbers include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure, graphs called Fibonacci cubes used for interconnecting parallel and distributed systems, they appear in biological settings, such as branching in trees, the arrangement of leaves on a stem, the fruit sprouts of a pineapple, the flowering of an artichoke, an uncurling fern and the arrangement of a pine cone's bracts. Fibonacci numbers are closely related to Lucas numbers L n in that they form a complementary pair of Lucas sequences U n = F n and V n = L n. Lucas numbers are intimately connected with the golden ratio; the Fibonacci sequence appears in Indian mathematics in connection with Sanskrit prosody, as pointed out by Parmanand Singh in 1985. In the Sanskrit poetic tradition, there was interest in enumerating all patterns of long syllables of 2 units duration, juxtaposed with short syllables of 1 unit duration.
Counting the different patterns of successive L and S with a given total duration results in the Fibonacci numbers: the number of patterns of duration m units is Fm + 1. Knowledge of the Fibonacci sequence was expressed as early as Pingala. Singh cites Pingala's cryptic formula misrau cha and scholars who interpret it in context as saying that the number of patterns for m beats is obtained by adding one to the Fm cases and one to the Fm−1 cases. Bharata Muni expresses knowledge of the sequence in the Natya Shastra. However, the clearest exposition of the sequence arises in the work of Virahanka, whose own work is lost, but is available in a quotation by Gopala: Variations of two earlier meters... For example, for four, variations of meters of two three being mixed, five happens.... In this way, the process should be followed in all mātrā-vṛttas. Hemachandra is credited with knowledge of the sequence as well, writing that "the sum of the last and the one before the last is the number... of the next mātrā-vṛtta."
Outside India, the Fibonacci sequence first appears in the book Liber Abaci by Fibonacci. Using it to calculate the growth of rabbit populations. Fibonacci considers the growth of a hypothetical, idealized rabbit population, assuming that: a newly born pair of rabbits, one male, one female, are put in a field. Fibonacci posed the puzzle: how many pairs will there be in one year? At the end of the first month, they mate. At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field. At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field. At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair, making 5 pairs. At the end of the nth month, the number of pairs of rabbits is equal to the number of new pairs plus the number of pairs alive last month; this is the nth Fibonacci number. The name "Fibonacci sequence" was first used by the 19th
Système universitaire de documentation
The système universitaire de documentation or SUDOC is a system used by the libraries of French universities and higher education establishments to identify and manage the documents in their possession. The catalog, which contains more than 10 million references, allows students and researcher to search for bibliographical and location information in over 3,400 documentation centers, it is maintained by the Bibliographic Agency for Higher Education. Official website