1.
United States
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Forty-eight of the fifty states and the federal district are contiguous and located in North America between Canada and Mexico. The state of Alaska is in the northwest corner of North America, bordered by Canada to the east, the state of Hawaii is an archipelago in the mid-Pacific Ocean. The U. S. territories are scattered about the Pacific Ocean, the geography, climate and wildlife of the country are extremely diverse. At 3.8 million square miles and with over 324 million people, the United States is the worlds third- or fourth-largest country by area, third-largest by land area. It is one of the worlds most ethnically diverse and multicultural nations, paleo-Indians migrated from Asia to the North American mainland at least 15,000 years ago. European colonization began in the 16th century, the United States emerged from 13 British colonies along the East Coast. Numerous disputes between Great Britain and the following the Seven Years War led to the American Revolution. On July 4,1776, during the course of the American Revolutionary War, the war ended in 1783 with recognition of the independence of the United States by Great Britain, representing the first successful war of independence against a European power. The current constitution was adopted in 1788, after the Articles of Confederation, the first ten amendments, collectively named the Bill of Rights, were ratified in 1791 and designed to guarantee many fundamental civil liberties. During the second half of the 19th century, the American Civil War led to the end of slavery in the country. By the end of century, the United States extended into the Pacific Ocean. The Spanish–American War and World War I confirmed the status as a global military power. The end of the Cold War and the dissolution of the Soviet Union in 1991 left the United States as the sole superpower. The U. S. is a member of the United Nations, World Bank, International Monetary Fund, Organization of American States. The United States is a developed country, with the worlds largest economy by nominal GDP. It ranks highly in several measures of performance, including average wage, human development, per capita GDP. While the U. S. economy is considered post-industrial, characterized by the dominance of services and knowledge economy, the United States is a prominent political and cultural force internationally, and a leader in scientific research and technological innovations. In 1507, the German cartographer Martin Waldseemüller produced a map on which he named the lands of the Western Hemisphere America after the Italian explorer and cartographer Amerigo Vespucci

2.
VJing
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VJing is a broad designation for realtime visual performance. Characteristics of VJing are the creation or manipulation of imagery in realtime through technological mediation and for an audience, VJing often takes place at events such as concerts, nightclubs, music festivals and sometimes in combination with other performative arts. This results in a multimedia performance that can include music. The term VJing became popular in its association with MTVs Video Jockey, in both situations VJing is the manipulation or selection of visuals, the same way DJing is a selection and manipulation of audio. In addition to the selection of media, VJing mostly implies realtime processing of the visual material, the term is also used to describe the performative use of generative software, although the word becomes dubious since no video is being mixed. Historically, VJing gets its references from art forms that deal with the experience of vision. The color organ is a mechanism to make colors correspond to sound through mechanical, bainbridge Bishop, who contributed to the development of the color organ, was dominated with the idea of painting music. Between 1919 and 1927, Mary Hallock-Greenewalt, a piano soloist, created a new art form called Nourathar. Her light music consisted of environmental color fields that produced a scale of light intensities, in place of a keyboard, the Sarabet had a console with graduated sliders and other controls, more like a modern mixing board. Lights could be adjusted directly via the sliders, through the use of a pedal, in clubs and private events in the 1960s people used liquid-slides, disco balls and light projections on smoke to give the audience new sensations. Some of these experiments were linked to the music, but most of the time they functioned as decorations and these came to be known as liquid light shows. The Exploding Plastic Inevitable, between 1966 and 1967, organized by Andy Warhol contributed to the fusion of music and visuals in a party context, at concerts, a few bands started to have regular film/video along with their music. Experimental film maker Tony Potts was considered a member of The Monochrome Set for his work on lighting design. Test Department initially worked with Bert Turnball as their resident visual artist, creating slideshows, the organization, Ministry of Power included collaborations with performance groups, traditional choirs and various political activists. Industrial bands would perform in art contexts, as well as in concert halls, groups like Cabaret Voltaire started to use low cost video editing equipment to create their own time-based collages for their sound works. In their words, before, you had to do collages on paper, the film collages made by and for groups such as the Test Dept, Throbbing Gristle and San Franciscos Tuxedomoon became part of their live shows. An example of mixing film with live performance is that of Public Image Ltd. at the Ritz Riot in 1981 and this club, located on the East 9th St in New York, had a state of the art video projection system. It was used to show a combination of prerecorded and live video on the clubs screen, piL played behind this screen with lights rear projecting their shadows on to the screen

3.
Stephen Hawking
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Hawking was the first to set forth a theory of cosmology explained by a union of the general theory of relativity and quantum mechanics. He is a supporter of the many-worlds interpretation of quantum mechanics. In 2002, Hawking was ranked number 25 in the BBCs poll of the 100 Greatest Britons, Hawking has a rare early-onset, slow-progressing form of amyotrophic lateral sclerosis that has gradually paralysed him over the decades. He now communicates using a single cheek muscle attached to a speech-generating device, Hawking was born on 8 January 1942 in Oxford, England to Frank and Isobel Hawking. Despite their families financial constraints, both attended the University of Oxford, where Frank read medicine and Isobel read Philosophy. The two met shortly after the beginning of the Second World War at a research institute where Isobel was working as a secretary. They lived in Highgate, but, as London was being bombed in those years, Hawking has two younger sisters, Philippa and Mary, and an adopted brother, Edward. In 1950, when Hawkings father became head of the division of parasitology at the National Institute for Medical Research, Hawking and his moved to St Albans. In St Albans, the family were considered intelligent and somewhat eccentric. They lived an existence in a large, cluttered, and poorly maintained house. During one of Hawkings fathers frequent absences working in Africa, the rest of the family spent four months in Majorca visiting his mothers friend Beryl and her husband, Hawking began his schooling at the Byron House School in Highgate, London. He later blamed its progressive methods for his failure to learn to read while at the school, in St Albans, the eight-year-old Hawking attended St Albans High School for Girls for a few months. At that time, younger boys could attend one of the houses, the family placed a high value on education. Hawkings father wanted his son to attend the well-regarded Westminster School and his family could not afford the school fees without the financial aid of a scholarship, so Hawking remained at St Albans. From 1958 on, with the help of the mathematics teacher Dikran Tahta, they built a computer from clock parts, although at school Hawking was known as Einstein, Hawking was not initially successful academically. With time, he began to show aptitude for scientific subjects and, inspired by Tahta. Hawkings father advised him to medicine, concerned that there were few jobs for mathematics graduates. He wanted Hawking to attend University College, Oxford, his own alma mater, as it was not possible to read mathematics there at the time, Hawking decided to study physics and chemistry

4.
New York City
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The City of New York, often called New York City or simply New York, is the most populous city in the United States. With an estimated 2015 population of 8,550,405 distributed over an area of about 302.6 square miles. Located at the tip of the state of New York. Home to the headquarters of the United Nations, New York is an important center for international diplomacy and has described as the cultural and financial capital of the world. Situated on one of the worlds largest natural harbors, New York City consists of five boroughs, the five boroughs – Brooklyn, Queens, Manhattan, The Bronx, and Staten Island – were consolidated into a single city in 1898. In 2013, the MSA produced a gross metropolitan product of nearly US$1.39 trillion, in 2012, the CSA generated a GMP of over US$1.55 trillion. NYCs MSA and CSA GDP are higher than all but 11 and 12 countries, New York City traces its origin to its 1624 founding in Lower Manhattan as a trading post by colonists of the Dutch Republic and was named New Amsterdam in 1626. The city and its surroundings came under English control in 1664 and were renamed New York after King Charles II of England granted the lands to his brother, New York served as the capital of the United States from 1785 until 1790. It has been the countrys largest city since 1790, the Statue of Liberty greeted millions of immigrants as they came to the Americas by ship in the late 19th and early 20th centuries and is a symbol of the United States and its democracy. In the 21st century, New York has emerged as a node of creativity and entrepreneurship, social tolerance. Several sources have ranked New York the most photographed city in the world, the names of many of the citys bridges, tapered skyscrapers, and parks are known around the world. Manhattans real estate market is among the most expensive in the world, Manhattans Chinatown incorporates the highest concentration of Chinese people in the Western Hemisphere, with multiple signature Chinatowns developing across the city. Providing continuous 24/7 service, the New York City Subway is one of the most extensive metro systems worldwide, with 472 stations in operation. Over 120 colleges and universities are located in New York City, including Columbia University, New York University, and Rockefeller University, during the Wisconsinan glaciation, the New York City region was situated at the edge of a large ice sheet over 1,000 feet in depth. The ice sheet scraped away large amounts of soil, leaving the bedrock that serves as the foundation for much of New York City today. Later on, movement of the ice sheet would contribute to the separation of what are now Long Island and Staten Island. The first documented visit by a European was in 1524 by Giovanni da Verrazzano, a Florentine explorer in the service of the French crown and he claimed the area for France and named it Nouvelle Angoulême. Heavy ice kept him from further exploration, and he returned to Spain in August and he proceeded to sail up what the Dutch would name the North River, named first by Hudson as the Mauritius after Maurice, Prince of Orange

5.
Brown University
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Brown is the seventh-oldest institution of higher education in the United States and one of the nine Colonial Colleges established before the American Revolution. At its foundation, Brown was the first college in the United States to accept students regardless of their religious affiliation and its engineering program was established in 1847 and was the first in the Ivy League. It was one of the early doctoral-granting U. S. institutions in the late 19th century, adding master, Browns New Curriculum is sometimes referred to in education theory as the Brown Curriculum and was adopted by faculty vote in 1969 after a period of student lobbying. In 1971, Browns coordinate womens institution Pembroke College was fully merged into the university, Pembroke Campus now operates as a place for dorms and classrooms. Undergraduate admissions is very selective, with a rate of 8.3 percent for the class of 2021. The University comprises The College, the Graduate School, Alpert Medical School, the School of Engineering, the School of Public Health, and the School of Professional Studies. The Brown/RISD Dual Degree Program, offered in conjunction with the Rhode Island School of Design, is a course that awards degrees from both institutions. Browns main campus is located in the College Hill Historic District in the city of Providence, the Universitys neighborhood is a federally listed architectural district with a dense concentration of Colonial-era buildings. On the western edge of the campus, Benefit Street contains one of the finest cohesive collections of restored seventeenth-, Browns faculty and alumni include eight Nobel Prize laureates, five National Humanities Medalists, and ten National Medal of Science laureates. Other notable alumni include eight billionaire graduates, a U. S. Supreme Court Chief Justice, to erect a public Building or Buildings for the boarding of the youth & the Residence of the Professors. Stiles and Ellery were co-authors of the Charter of the College two years later, there is further documentary evidence that Stiles was making plans for a college in 1762. On January 20, Chauncey Whittelsey, pastor of the First Church of New Haven, answered a letter from Stiles, should you make any Progress in the Affair of a Colledge, I should be glad to hear of it, I heartily wish you Success therein. Isaac Backus was the historian of the New England Baptists and an inaugural Trustee of Brown, Mr. James Manning, who took his first degree in New-Jersey college in September,1762, was esteemed a suitable leader in this important work. Manning arrived at Newport in July 1763 and was introduced to Stiles, stiless first draft was read to the General Assembly in August 1763 and rejected by Baptist members who worried that the College Board of Fellows would under-represent the Baptists. A revised Charter written by Stiles and Ellery was adopted by the Assembly on March 3,1764, in September 1764, the inaugural meeting of the College Corporation was held at Newport. Governor Stephen Hopkins was chosen chancellor, former and future governor Samuel Ward was vice chancellor, John Tillinghast treasurer, the Charter stipulated that the Board of Trustees be composed of 22 Baptists, five Quakers, five Episcopalians, and four Congregationalists. Of the 12 Fellows, eight should be Baptists—including the College president—and the rest indifferently of any or all Denominations, the Charter was not the grant of King George III, as is sometimes supposed, but rather an Act of the colonial General Assembly. In two particulars, the Charter may be said to be a uniquely progressive document, the oft-repeated statement is inaccurate that Browns Charter alone prohibited a religious test for College membership, other college charters were also liberal in that particular

6.
Carnegie Mellon University
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Carnegie Mellon University is a private research university in Pittsburgh, Pennsylvania. Founded in 1900 by Andrew Carnegie as the Carnegie Technical Schools, in 1967, the Carnegie Institute of Technology merged with the Mellon Institute of Industrial Research to form Carnegie Mellon University. The universitys 140-acre main campus is 3 miles from Downtown Pittsburgh, john Heinz III College of Information Systems and Public Policy, and the School of Computer Science. The university also has campuses in Qatar and Silicon Valley, with degree-granting programs in six continents, Carnegie Mellon consistently ranks in the top 25 in the national U. S. News & World Report rankings. It is home to the world’s first degree-granting Robotics and Drama programs, the university spent $242 million on research in 2015. Carnegie Mellon counts 13,650 students from 114 countries, over 100,000 living alumni, Carnegies vision was to open a vocational training school for the sons and daughters of working-class Pittsburghers. Carnegie was inspired for the design of his school by the Pratt Institute in Brooklyn, in 1912 the institution changed its name to Carnegie Institute of Technology and began offering four-year degrees. The Mellon Institute of Industrial Research was founded in 1913 by banker and industrialist brothers Andrew, Mellon in honor of their father, Thomas Mellon, the patriarch of the Mellon family. The Institute began as an organization which performed work for government. In 1927, the Mellon Institute incorporated as an independent nonprofit, in 1938, the Mellon Institutes iconic building was completed and it moved to its new, and current, location on Fifth Avenue. In 1967, with support from Paul Mellon, Carnegie Tech merged with the Mellon Institute of Industrial Research to become Carnegie Mellon University, Carnegie Mellons coordinate womens college, the Margaret Morrison Carnegie College closed in 1973 and merged its academic programs with the rest of the university. The industrial research mission of the Mellon Institute survived the merger as the Carnegie Mellon Research Institute and continued doing work on contract to industry, CMRI closed in 2001 and its programs were subsumed by other parts of the university or spun off into independent entities. Carnegie Mellons 140-acre main campus is three miles from downtown Pittsburgh, between Schenley Park and the Squirrel Hill, Shadyside, and Oakland neighborhoods, Carnegie Mellon is bordered to the west by the campus of the University of Pittsburgh. Carnegie Mellon owns 81 buildings in the Oakland and Squirrel Hill neighborhoods of Pittsburgh, for decades the center of student life on campus was Skibo Hall, the Universitys student union. Built in the 1950s, Skibo Halls design was typical of Mid-Century Modern architecture, the original Skibo was razed in the summer of 1994 and replaced by a new student union that is fully wi-fi enabled. Known as University Center, the building was dedicated in 1996, in 2014, Carnegie Mellon re-dedicated the University Center as the Cohon University Center in recognition of the eighth president of the university, Jared Cohon. A large grassy area known as the Cut forms the backbone of the campus, the Cut was formed by filling in a ravine with soil from a nearby hill that was leveled to build the College of Fine Arts building. The northwestern part of the campus was acquired from the United States Bureau of Mines in the 1980s, the sculpture was controversial for its placement, the general lack of input that the campus community had, and its aesthetic appeal

7.
Mathematics and art
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Mathematics and art are related in a variety of ways. Mathematics has itself been described as an art motivated by beauty, Mathematics can be discerned in arts such as music, dance, painting, architecture, sculpture, and textiles. This article focuses, however, on mathematics in the visual arts, Mathematics and art have a long historical relationship. Artists have used mathematics since the 5th century BC when the Greek sculptor Polykleitos wrote his Canon, prescribing proportions based on the ratio 1, persistent popular claims have been made for the use of the golden ratio in ancient art and architecture, without reliable evidence. In the Italian Renaissance, Luca Pacioli wrote the influential treatise De Divina Proportione, illustrated with woodcuts by Leonardo da Vinci, another Italian painter, Piero della Francesca, developed Euclids ideas on perspective in treatises such as De Prospectiva Pingendi, and in his paintings. The engraver Albrecht Dürer made many references to mathematics in his work Melencolia I, in modern times, the graphic artist M. C. Mathematics has inspired textile arts such as quilting, knitting, cross-stitch, crochet, embroidery, weaving, Turkish and other carpet-making, as well as kilim. In Islamic art, symmetries are evident in forms as varied as Persian girih and Moroccan zellige tilework, Mughal jaali pierced stone screens, and widespread muqarnas vaulting. Mathematics has directly influenced art with conceptual tools such as perspective, the analysis of symmetry, and mathematical objects such as polyhedra. Magnus Wenninger creates colourful stellated polyhedra, originally as models for teaching, mathematical concepts such as recursion and logical paradox can be seen in paintings by Rene Magritte and in engravings by M. C. Computer art often makes use of including the Mandelbrot set. Polykleitos the elder was a Greek sculptor from the school of Argos, and his works and statues consisted mainly of bronze and were of athletes. While his sculptures may not be as famous as those of Phidias, Polykleitos uses the distal phalanx of the little finger as the basic module for determining the proportions of the human body. Next, he takes the length and multiplies that by √2 to get the length of the palm from the base of the finger to the ulna. This geometric series of measurements progresses until Polykleitos has formed the arm, chest, body, the influence of the Canon of Polykleitos is immense in Classical Greek, Roman, and Renaissance sculpture, many sculptors following Polykleitoss prescription. While none of Polykleitoss original works survive, Roman copies demonstrate his ideal of physical perfection, some scholars argue that Pythagorean thought influenced the Canon of Polykleitos. In classical times, rather than making distant figures smaller with linear perspective, painters sized objects, in the Middle Ages, some artists used reverse perspective for special emphasis. The Muslim mathematician Alhazen described a theory of optics in his Book of Optics in 1021, the Renaissance saw a rebirth of Classical Greek and Roman culture and ideas, among them the study of mathematics to understand nature and the arts

8.
Catenary
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In physics and geometry, a catenary is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends. The curve has a U-like shape, superficially similar in appearance to a parabola, the curve appears in the design of certain types of arches and as a cross section of the catenoid—the shape assumed by a soap film bounded by two parallel circular rings. The catenary is also called the alysoid, chainette, or, particularly in the materials sciences, mathematically, the catenary curve is the graph of the hyperbolic cosine function. The surface of revolution of the curve, the catenoid, is a minimal surface. The mathematical properties of the curve were first studied by Robert Hooke in the 1670s. Catenaries and related curves are used in architecture and engineering, in the design of bridges and arches, in the offshore oil and gas industry, catenary refers to a steel catenary riser, a pipeline suspended between a production platform and the seabed that adopts an approximate catenary shape. The word catenary is derived from the Latin word catēna, which means chain and it appears to be a very scientifical work. I have not yet had time to engage in it, but I find that the conclusions of his demonstrations are and it is often said that Galileo thought the curve of a hanging chain was parabolic. That the curve followed by a chain is not a parabola was proven by Joachim Jungius, some much older arches approximate catenaries, an example of which is the Arch of Taq-i Kisra in Ctesiphon. David Gregory wrote a treatise on the catenary in 1697 in which he provided an incorrect derivation of the differential equation. Euler proved in 1744 that the catenary is the curve which, nicolas Fuss gave equations describing the equilibrium of a chain under any force in 1796. Catenary arches are used in the construction of kilns. To create the desired curve, the shape of a chain of the desired dimensions is transferred to a form which is then used as a guide for the placement of bricks or other building material. The Gateway Arch in St. Louis, Missouri, United States is sometimes said to be an catenary and it is close to a more general curve called a flattened catenary, with equation y = A cosh, which is a catenary if AB =1. While a catenary is the shape for a freestanding arch of constant thickness. According to the U. S. National Historic Landmark nomination for the arch and its shape corresponds to the shape that a weighted chain, having lighter links in the middle, would form. In free-hanging chains, the force exerted is uniform with respect to length of the chain, the same is true of a simple suspension bridge or catenary bridge, where the roadway follows the cable. A stressed ribbon bridge is a sophisticated structure with the same catenary shape

9.
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

10.
Golden ratio
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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. Its value is, φ =1 +52 =1.6180339887 …, A001622 The golden ratio is also called the golden mean or golden section. Other names include extreme and mean ratio, medial section, divine proportion, divine section, golden proportion, golden cut, the golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other plant parts. The golden ratio has also used to analyze the proportions of natural objects as well as man-made systems such as financial markets. 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 =1 + b a =1 +1 φ, multiplying by φ gives φ +1 = φ2 which can be rearranged to φ2 − φ −1 =0. First, the line segment A B ¯ is about doubled and then the semicircle with the radius A S ¯ around the point S is drawn, now the semicircle is drawn with the radius A B ¯ around the point B. The arising intersection point E corresponds 2 φ, next up, the perpendicular on the line segment A E ¯ from the point D will be establish. The subsequent parallel F S ¯ to the line segment C M ¯, produces, as it were and it is well recognizable, this triangle and the triangle M S C are similar to each other. The hypotenuse F S ¯ has due to the cathetuses S D ¯ =1 and D F ¯ =2 according the Pythagorean theorem, finally, the circle arc is drawn with the radius 5 around the point F. The golden ratio has been claimed to have held a fascination for at least 2,400 years. But the fascination with the Golden Ratio is not confined just to mathematicians, biologists, artists, musicians, historians, architects, psychologists, and even 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, the division of a line into extreme and mean ratio is important in the geometry of regular pentagrams and pentagons. Euclid explains a construction for cutting a line in extreme and mean ratio, throughout the Elements, several propositions and their proofs employ the golden ratio. The golden ratio is explored in Luca Paciolis book De divina proportione, since the 20th century, the golden ratio has been represented by the Greek letter φ or less commonly by τ. Timeline according to Priya Hemenway, Phidias made the Parthenon statues that seem to embody the golden ratio, plato, in his Timaeus, describes five possible regular solids, some of which are related to the golden ratio

11.
Hyperboloid structure
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Hyperboloid structures are architectural structures designed using a hyperboloid in one sheet. The first hyperboloid structures were built by Russian engineer Vladimir Shukhov, the worlds first hyperboloid tower is located in Polibino, Dankovsky District, Lipetsk Oblast, Russia. Hyperbolic structures have a negative Gaussian curvature, meaning they curve inward rather than outward or being straight. As doubly ruled surfaces, they can be made with a lattice of beams, hence are easier to build than curved surfaces that do not have a ruling. With cooling towers, a structure is preferred. At the bottom, the widening of the tower provides an area for installation of fill to promote thin film evaporative cooling of the circulated water. In the 1880s, Shukhov began to work on the problem of the design of systems to use a minimum of materials, time. His calculations were most likely derived from mathematician Pafnuty Chebyshevs work on the theory of best approximations of functions, Shukhovs mathematical explorations of efficient roof structures led to his invention of a new system that was innovative both structurally and spatially. The steel gridshells of the pavilions of the 1896 All-Russian Industrial. Two pavilions of this type were built for the Nizhni Novgorod exposition, one oval in plan, the roofs of these pavilions were doubly curved gridshells formed entirely of a lattice of straight angle-iron and flat iron bars. Shukhov himself called them azhurnaia bashnia, the patent of this system, for which Shukhov applied in 1895, was awarded in 1899. Shukhov also turned his attention to the development of an efficient and his solution was inspired by observing the action of a woven basket holding up a heavy weight. Again, it took the form of a curved surface constructed of a light network of straight iron bars. Over the next twenty years, he designed and built close to two hundred of these towers, no two alike, most with heights in the range of 12m to 68m. At least as early as 1911, Shukhov began experimenting with the concept of forming a tower out of stacked sections of hyperboloids. Stacking the sections permitted the form of the tower to taper more at the top, increasing the number of sections would increase the tapering of the overall form, to the point that it began to resemble a cone. By 1918 Shukhov had developed this concept into the design of a nine-section stacked hyperboloid radio transmission tower for Moscow, Shukhov designed a 350m tower, which would have surpassed the Eiffel Tower in height by 50m, while using less than a quarter of the amount of material. In July 1919, Lenin decreed that the tower should be built to a height of 150m, construction of the smaller tower with six stacked hyperboloids began within a few months, and Shukhov Tower was completed by March 1922

12.
Paraboloid
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In geometry, a paraboloid is a quadric surface that has one axis of symmetry and no center of symmetry. The term paraboloid is derived from parabola, which refers to a section that has the same property of symmetry. In a suitable coordinate system with three axes x, y, and z, it can be represented by the equation z = x 2 a 2 + y 2 b 2. Where a and b are constants that dictate the level of curvature in the xz, in this position, the elliptic paraboloid opens upward. A hyperbolic paraboloid is a ruled surface shaped like a saddle. In a suitable system, a hyperbolic paraboloid can be represented by the equation z = y 2 b 2 − x 2 a 2. In this position, the hyperbolic paraboloid opens down along the x-axis, obviously both the paraboloids contain a lot of parabolas. But there are differences, too, an elliptic paraboloid contains ellipses. With a = b an elliptic paraboloid is a paraboloid of revolution and this shape is also called a circular paraboloid. This also works the way around, a parallel beam of light incident on the paraboloid parallel to its axis is concentrated at the focal point. This applies also for other waves, hence parabolic antennas, for a geometrical proof, click here. The hyperbolic paraboloid is a ruled surface, it contains two families of mutually skew lines. The lines in each family are parallel to a common plane, hence the hyperbolic paraboloid is a conoid. This property makes easy to realize a hyperbolic paraboloid with concrete, the widely sold fried snack food Pringles potato crisps resemble a truncated hyperbolic paraboloid. The distinctive shape of these allows them to be stacked in sturdy tubular containers. Examples in architecture St. a point, if the plane is a tangent plane, remark, an elliptic paraboloid is projectively equivalent to a sphere. Remarks, A hyperbolic paraboloid is a surface, but not developable. The Gauss curvature at any point is negative, hence it is a saddle surface