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.
Fractal flame
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Fractal flames are a member of the iterated function system class of fractals created by Scott Draves in 1992. Draves open-source code was ported into Adobe After Effects graphics software. Fractal flames differ from ordinary iterated function systems in three ways, Nonlinear functions are iterated in addition to affine transforms, log-density display instead of linear or binary Color by structure instead of monochrome or by density. The tone mapping and coloring are designed to display as much of the detail of the fractal as possible, the algorithm consists of two steps, creating a histogram and then rendering the histogram. First one iterates a set of functions, starting from a chosen point P =. Set of flame functions, { F1, p 1 F2, p 2 … F n, p n In each iteration, then one computes the next iteration of P by applying Fj on. Each individual function has the form, F j = ∑ V k ∈ V a r i a t i o n s w k ⋅ V k where the parameter wk is called the weight of the variation Vk. Draves suggests that all w k, s are non-negative and sum to one, the functions Vk are a set of predefined functions. This is done as follows, The colors in the image will therefore reflect what functions were used to get to that part of the image, to increase the quality of the image, one can use supersampling to decrease the noise. This involves creating a larger than the image so each pixel has multiple data points to pull from. For example, creating a histogram with 300×300 cells in order to draw an 100×100 px image, each pixel would use a 3×3 group of histogram buckets to calculate its value. For each pixel in the image, do the following computations. This is implemented in for example the Apophysis software, to increase the quality even more, one can use gamma correction on each individual color channel, but this is a very heavy computation, since the log function is slow. A simplified algorithm would be to let the brightness be linearly dependent on the frequency, but this would make some parts of the fractal lose detail, which is undesirable. The flame algorithm is like a Monte Carlo simulation, with the flame quality directly proportional to the number of iterations of the simulation, the noise that results from this stochastic sampling can be reduced by blurring the image, to get a smoother result in less time. One does not however want to lose resolution in the parts of the image that receive many samples and this problem can be solved with adaptive density estimation to increase image quality while keeping render times to a minimum. FLAM3 uses a simplification of the methods presented in *Adaptive Filtering for Progressive Monte Carlo Image Rendering*, the idea is to vary the width of the filter inversely proportional to the number of samples available. As a result, areas with few samples and lots of noise get blurred and smoothed, not all Flame implementations use density estimation

3.
Distributed computing
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Distributed computing is a field of computer science that studies distributed systems. A distributed system is a model in which components located on networked computers communicate and coordinate their actions by passing messages, the components interact with each other in order to achieve a common goal. Three significant characteristics of distributed systems are, concurrency of components, lack of a global clock, examples of distributed systems vary from SOA-based systems to massively multiplayer online games to peer-to-peer applications. A computer program that runs in a system is called a distributed program. There are many alternatives for the message passing mechanism, including pure HTTP, RPC-like connectors, Distributed computing also refers to the use of distributed systems to solve computational problems. In distributed computing, a problem is divided into many tasks, each of which is solved by one or more computers, which communicate with each other by message passing. The terms are used in a much wider sense, even referring to autonomous processes that run on the same physical computer. The entities communicate with each other by message passing, a distributed system may have a common goal, such as solving a large computational problem, the user then perceives the collection of autonomous processors as a unit. Other typical properties of distributed systems include the following, The system has to tolerate failures in individual computers. The structure of the system is not known in advance, the system may consist of different kinds of computers and network links, each computer has only a limited, incomplete view of the system. Each computer may know one part of the input. Distributed systems are groups of networked computers, which have the goal for their work. The terms concurrent computing, parallel computing, and distributed computing have a lot of overlap, the same system may be characterized both as parallel and distributed, the processors in a typical distributed system run concurrently in parallel. Parallel computing may be seen as a tightly coupled form of distributed computing. In distributed computing, each processor has its own private memory, Information is exchanged by passing messages between the processors. The figure on the right illustrates the difference between distributed and parallel systems, figure shows a parallel system in which each processor has a direct access to a shared memory. The situation is complicated by the traditional uses of the terms parallel and distributed algorithm that do not quite match the above definitions of parallel. The use of concurrent processes that communicate by message-passing has its roots in operating system architectures studied in the 1960s, the first widespread distributed systems were local-area networks such as Ethernet, which was invented in the 1970s

4.
Electric Sheep
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Electric Sheep is a distributed computing project for animating and evolving fractal flames, which are in turn distributed to the networked computers, which display them as a screensaver. The process is transparent to the user, who can simply install the software as a screensaver. As the screensaver entertains the user, their computer is used for rendering commercial projects. There are about 500,000 active users, according to Mitchell Whitelaw in his Metacreation, Art and Artificial Life, On the screen they are luminous, twisting, elastic shapes, abstract tangles and loops of glowing filaments. The name Electric Sheep is taken from the title of Philip K. Dicks novel Do Androids Dream of Electric Sheep, the title mirrors the nature of the project, computers who have started running the screensaver begin rendering the fractal movies. Users may vote on sheep that they like or dislike, each movie is a fractal flame with several of its parameters animated. Both are automatically downloaded by the screen saver, the underlying copyright issues raised by generative, distributed digital art projects involve novel legal issues that the current copyright system can not understand or handle. The screensaver was created and released as free software by Scott Draves in 1999 and continues to be developed by him, the 2.7. x series differs from the old versions. It has a new logo, higher quality sheep and other features and it has switched to a freemium model in which the server software is not available and much of the computed data is not available under a free license, which led to its removal from Debian

5.
Texture synthesis
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Texture synthesis is the process of algorithmically constructing a large digital image from a small digital sample image by taking advantage of its structural content. It is an object of research in graphics and is used in many fields, amongst others digital image editing, 3D computer graphics. Texture synthesis can be used to fill in holes in images, create large non-repetitive background images, procedural textures are a related technique which may synthesise textures from scratch with no source material. By contrast, texture synthesis refers to techniques where some source image is being matched or extended. Texture is a word and in the context of texture synthesis may have one of the following meanings, In common speech. Texture has been described by five different properties in the psychology of perception, coarseness, contrast, directionality, in 3D computer graphics, a texture is a digital image applied to the surface of a three-dimensional model by texture mapping to give the model a more realistic appearance. Often, the image is a photograph of a real texture, in image processing, every digital image composed of repeated elements is called a texture. Texture can be arranged along a spectrum going from regular to stochastic, connected by a smooth transition and these textures look like somewhat regular patterns. An example of a texture is a stonewall or a floor tiled with paving stones. Texture images of stochastic textures look like noise, colour dots that are scattered over the image, barely specified by the attributes minimum and maximum brightness. Many textures look like stochastic textures when viewed from a distance, an example of a stochastic texture is roughcast. Texture synthesis algorithms are intended to create an image that meets the following requirements. The output should be as similar as possible to the sample, the output should not have visible artifacts such as seams, blocks and misfitting edges. The output should not repeat, i. e. the same structures in the image should not appear multiple places. Like most algorithms, texture synthesis should be efficient in computation time, the following methods and algorithms have been researched or developed for texture synthesis, The simplest way to generate a large image from a sample image is to tile it. This means multiple copies of the sample are copied and pasted side by side. Except in rare cases, there will be the seams in between the tiles and the image will be highly repetitive and these algorithms perform well with stochastic textures only, otherwise they produce completely unsatisfactory results as they ignore any kind of structure within the sample image. Algorithms of that use a fixed procedure to create an output image

6.
Video art
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Video art is an art form which relies on moving pictures in a visual and audio medium. Video art came into existence during the late 1960s and early 1970s as new video technology became available outside corporate broadcasting. Video art is named after the analog video tape, which was most commonly used recording technology in the forms early years. With the advent of recording equipment, many artists began to explore digital technology as a new way of expression. One of the key differences between art and theatrical cinema is that video art does not necessarily rely on many of the conventions that define theatrical cinema. This distinction also distinguishes video art from cinemas subcategories, Nam June Paik, a Korean-American artist who studied in Germany, is widely regarded as a pioneer in video art. In March 1963 Nam June Paik showed at the Galerie Parnass in Wuppertal the Exposition of Music – Electronic Television, in May 1963 Wolf Vostell showed the installation 6 TV Dé-coll/age at the Smolin Gallery in New York and created the video Sun in your head in Cologne. Originally Sun in your head was made on 16mm film and transferred 1967 to videotape, prior to the introduction of consumer video equipment, moving image production was only available non-commercially via 8mm film and 16mm film. After the Portapaks introduction and its subsequent update every few years, many of the early prominent video artists were those involved with concurrent movements in conceptual art, performance, and experimental film. There were also such as Steina and Woody Vasulka who were interested in the formal qualities of video. Kate Craig, Vera Frenkel and Michael Snow were important to the development of art in Canada. Much video art in the mediums heyday experimented formally with the limitations of the video format, for example, American artist Peter Campus Double Vision combined the video signals from two Sony Portapaks through an electronic mixer, resulting in a distorted and radically dissonant image. Much video art in America was produced out of New York City, with The Kitchen, founded in 1972 by Steina and Woody Vasulka, an early multi-channel video art work was Wipe Cycle by Ira Schneider and Frank Gillette. Wipe Cycle was first exhibited at the Howard Wise Gallery in New York in 1969 as part of an exhibition titled TV as a Creative Medium. An installation of nine screens, Wipe Cycle combined live images of gallery visitors, found footage from commercial television. The material was alternated from one monitor to the next in an elaborate choreography, on the West coast, the San Jose State television studios in 1970, Willoughby Sharp began the Videoviews series of videotaped dialogues with artists. The Videoviews series consists of Sharps’ dialogues with Bruce Nauman, Joseph Beuys, Vito Acconci, Chris Burden, Lowell Darling, in Europe, Valie Exports groundbreaking video piece, Facing a Family was one of the first instances of television intervention and broadcasting video art. Export believed the television could complicate the relationship between subject, spectator, and television, in the United Kingdom David Halls TV Interruptions were transmitted intentionally unannounced and uncredited on Scottish TV, the first artist interventions on British television

7.
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

8.
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

9.
The Grand Design (book)
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The Grand Design is a popular-science book written by physicists Stephen Hawking and Leonard Mlodinow and published by Bantam Books in 2010. The book examines the history of knowledge about the universe. The authors of the point out that a Unified Field Theory may not exist. It argues that invoking God is not necessary to explain the origins of the universe, in response to criticism, Hawking has said, One cant prove that God doesnt exist, but science makes God unnecessary. When pressed on his own religious views by the Channel 4 documentary Genius of Britain, published in the United States on September 7,2010, the book became the number one bestseller on Amazon. com just a few days after publication. It was published in the United Kingdom on September 9,2010 and it topped the list of adult non-fiction books of The New York Times Non-fiction Best Seller list in Sept-Oct 2010. The book examines the history of knowledge about the universe. It starts with the Ionian Greeks, who claimed that nature works by laws and it later presents the work of Nicolaus Copernicus, who advocated the concept that the Earth is not located in the center of the universe. The authors then describe the theory of quantum mechanics using, as an example, the presentation has been described as easy to understand by some reviewers, but also as sometimes impenetrable, by others. The central claim of the book is that the theory of quantum mechanics, the book concludes with the statement that only some universes of the multiple universes support life forms. We, of course, are located in one of those universes, the laws of nature that are required for life forms to exist appear in some universes by pure chance, Hawking and Mlodinow explain. Evolutionary biologist and advocate for atheism Richard Dawkins welcomed Hawkings position and said that Darwinism kicked God out of biology, Hawking is now administering the coup de grace. Theoretical physicist Sean M. questions that are part of human curiosity. If our universe arose spontaneously from nothing at all, one might predict that its total energy should be zero, and when we measure the total energy of the universe, which could have been anything, the answer turns out to be the only one consistent with this possibility. But data like this coming in from our revolutionary new tools promise to turn much of what is now metaphysics into physics, whether God survives is anyones guess. James Trefil, a professor of physics at George Mason University, said in his Washington Post review and it gets into the deepest questions of modern cosmology without a single equation. The reader will be able to get through it without bogging down in a lot of detail and will. Maybe in the end the whole multiverse idea will actually turn out to be right, canada Press journalist Carl Hartman said, Cosmologists, the people who study the entire cosmos, will want to read British physicist and mathematician Stephen Hawkings new book

10.
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

11.
Center for Art and Media Karlsruhe
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The ZKM houses under one roof two museums, three research institutes as well as a media center, in this way it groups research and production, exhibitions and events, archives and collections. It works on the interface of art and science, and takes up cutting-edge insights in media technologies with the objective of developing them further, since the death of founding director Heinrich Klotz, the ZKM has been directed by Prof. Peter Weibel. In addition to the ZKM, the associated Karlsruhe University of Arts and Design, the founding of the Center for Art and Media goes back to the early 1980s. With the founding of a board of trustees in 1989, and the appointment of Heinrich Klotz as founding director, when initially founded, the ZKM was located in various buildings around the city. Prior to the move to its present location, the art festival MultiMediale took place at various sites. For some considerable time, an area to the south of the Karlsruhe Central Station had been designated, to this end, an international architect’s competition for the new building was announced on March 1989, from which the visionary design by Dutch architect Rem Koolhaas was to result. However, the construction of the so-called Koolhaas-Cube was abandoned in 1992 for reasons of costs, Karlsruhe opted for the conversion of the so-called “Hallenbau A”, an industrial ruin erected between 1914 and 1918 by architect Philipp Jakob Manz as a weapons and munitions factory. The conversion, based on plans drafted by the Hamburg office of Schweger, as well as the extension of the Media Cube which takes account of the Koolhaas design, started in 1993. In a second stage of construction, the spaces for the Museum of Contemporary Art, from 2004 to 2005 the Museum of Contemporary Art was integrated into the ZKM. “The task envisaged for the ZKM is the out of the creative possibilities between the traditional arts and media technologies for the purpose of achieving innovative results. The objective is the enrichment of the arts, not their technical amputation, for this reason both traditional and media arts must compete with one another. At the ZKM either aspect – each for itself and with one another – are given a voice. The Bauhaus, founded in Weimar in 1919, may serve as a model. ”The basic idea as formulated by founding director Heinrich Klotz in 1992, was implemented and developed further in the years that followed, today, the ZKM is chiefly characterized by four guiding ideas. The ZKM is a location for all forms of contemporary art and it is a platform for cross-border experiments between the fine arts and the performative arts. Research, production and presentation comprise all medial forms and methods, from oil painting through to App, exhibitions, publications and symposia open up new perspectives to new questions, the objectives of which are the setting of innovative and trend-setting themes. At the ZKM, people all over the world and of all ages are invited to discover art. It is a house that encourages its visitors to active participation, exchange. Actors from all social spheres – from the arts, sciences, politics, as a center of research and development both in theory and practice, the ZKM gathers together artists and scientists from various disciplines

12.
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

13.
Andries van Dam
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Andries Andy van Dam is a Dutch-born American professor of computer science and former vice-president for research at Brown University in Providence, Rhode Island. Together with Ted Nelson he contributed to the first hypertext system and he co-authored Computer Graphics, Principles and Practice along with J. D. Foley, S. K. Feiner, and John Hughes. He also co-founded the precursor of todays ACM SIGGRAPH conference, van Dam serves on several technical boards and committees. He teaches a course in computer science and courses in computer graphics at Brown University. Van Dam received his B. S. degree with Honors in Engineering Sciences from Swarthmore College in 1960 and his M. S. and Ph. D. from the University of Pennsylvania in 1963 and 1966, respectively. Van Dam has mentored undergraduates, other scholars, and practitioners in hypertext, one of his students was Randy Pausch, who gained national renown during his ultimately unsuccessful two-year fight with pancreatic cancer. Pauschs Last Lecture in September 2007 was the basis for the bestseller Last Lecture, van Dam was the final speaker after the hour-plus talk. He praised Pausch for his courage and leadership, calling him a role model, Pausch died on July 25,2008. Danah Boyd, Scott Draves, Dick Bulterman and Andy Hertzfeld also were students of Andy van Dam, when the program was promoted to a full department, van Dam served as its first chair, from 1979 to 1985. In 1995 van Dam was appointed Thomas J. Watson, Jr, University Professor of Technology and Education as well as professor of computer science. At the University of Pennsylvania in 1966, he became the person to receive a Ph. D. in Computer Science. Van Dam is perhaps most known as the co-designer, along with Ted Nelson, of the first hypertext system, HES, with it and its immediate successor, FRESS, he was an early proponent of the use of hypertext in the humanities and in pedagogy. The term hypertext was coined by Ted Nelson, who was working with him at the time and he is also known for co-authoring Computer Graphics, Principles and Practice with J. D. Foley, S. K. Feiner, and J. F. Hughes. This popular textbook in computer graphics and is called the Bible of computer graphics. In 1967, van Dam co-founded ACM SICGRAPH, the precursor of todays ACM SIGGRAPH, in 1983 he was one of the founders of IRIS, which developed a hypertext scholars workstation. Van Dam teaches an Introduction to Computer Graphics course, as well as one course every fall. He is also serving on the board of Microsoft Research, as chairman of the Rhode Island Governors Science and Technology Advisory Council. In 1994 he was inducted as a Fellow of the Association for Computing Machinery, when the Brown Center for Informational Technology was built, van Dam demanded it include showers and a Chinese restaurant

14.
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

15.
Andrew Witkin
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Andrew Paul Witkin was an American computer scientist who made major contributions in computer vision and computer graphics. Witkin studied psychology at Columbia College, Columbia University for his bachelors degree, after MIT, Witkin worked briefly at SRI International on computer vision. From 1988 to 1998 he was a professor of science, robotics. At CMU and Pixar, with his colleagues he developed the methods and simulators used to model and render natural-looking cloth, hair, water, the paper Snakes, Active Contour Models achieved an honorable mention for the Marr Prize in 1987. According to CiteSeer, this paper is the 11th most cited paper ever in computer science, the 1987 paper Constraints on deformable models, Recovering 3D shape and nonrigid motion was also a prize winner. In 1992, Witkin and Kass where awarded the Prix Ars Electronica computer graphics award for Reaction–Diffusion Texture Buttons, Witkin received the ACM SIGGRAPH Computer Graphics Achievement Award in 2001 for his pioneering work in bringing a physics based approach to computer graphics. Andrew Witkin was the son of psychologist Herman A. Witkin and he was married to psychologist Sharon Witkin, their children are Emily Witkin and Anna Witkin. He died in a diving accident off the coast of Monterey. Witkin, A. Scale-space filtering, A new approach to multi-scale description, IEEE International Conference on Acoustics, Speech, and Signal Processing. Andy Witkin, From Computer Vision to Computer Graphics, Witkin, A. Kass, M. Spacetime constraints. Proceedings of the 15th annual conference on Computer graphics and interactive techniques - SIGGRAPH88, Andrew Witkin author profile page at the ACM Digital Library Andrew Witkins publications indexed by the Scopus bibliographic database, a service provided by Elsevier. Andrew P. Witkin at DBLP Bibliography Server Andy Witkin at the Internet Movie Database

16.
Dana Scott
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His research career involved computer science, mathematics, and philosophy. He has worked also on modal logic, topology, and category theory and he received his BA in Mathematics from the University of California, Berkeley, in 1954. He wrote his Ph. D. thesis on Convergent Sequences of Complete Theories under the supervision of Alonzo Church while at Princeton, solomon Feferman writes of this period, Scott began his studies in logic at Berkeley in the early 50s while still an undergraduate. Scott was clearly in line to do a Ph. D. with Tarski, upset by that, Scott left for Princeton where he finished with a Ph. D. under Alonzo Church. But it was not long before the relationship between them was mended to the point that Tarski could say to him, I hope I can call you my student. After completing his Ph. D. studies, he moved to the University of Chicago and this work led to the joint bestowal of the Turing Award on the two, for the introduction of this fundamental concept of computational complexity theory. During this period he started supervising Ph. D. students, such as James Halpern, Scott also began working on modal logic in this period, beginning a collaboration with John Lemmon, who moved to Claremont, California, in 1963. Later, Scott and Montague independently discovered an important generalisation of Kripke semantics for modal and tense logic, John Lemmon and Scott began work on a modal-logic textbook that was interrupted by Lemmons death in 1966. Scott eventually published the work as An Introduction to Modal Logic, following an initial observation of Robert Solovay, Scott formulated the concept of Boolean-valued model, as Solovay and Petr Vopěnka did likewise at around the same time. This work led to the award of the Leroy P. Steele Prize in 1972, Scott took up a post as Professor of Mathematical Logic on the Philosophy faculty of Oxford University in 1972. He was member of Merton College while at Oxford and is now an Honorary Fellow of the college, one of Scotts contributions is his formulation of domain theory, allowing programs involving recursive functions and looping-control constructs to be given denotational semantics. Additionally, he provided a foundation for the understanding of infinitary and continuous information through domain theory, the 2007 EATCS Award for his contribution to theoretical computer science. In 1994, he was inducted as a Fellow of the Association for Computing Machinery, in 2012 he became a fellow of the American Mathematical Society. Finite Automata and Their Decision Problem, a proof of the independence of the continuum hypothesis. In Philosophical Problems in Logic, ed. K. Lambert, gierz, G. Hofmann, K. H. Keimel, K. Lawson, J. D. Mislove, M. W. Scott, D. S. Encyclopedia of Mathematics and its Applications, Scotts trick Scott–Potter set theory Blackburn, de Rijke and Venema. In the Stanford Encyclopedia of Philosophy, solomon Feferman and Anita Burdman Feferman. Cambridge University Press, ISBN 0-521-80240-7, ISBN 978-0-521-80240-6, denotational Semantics, The Scott-Strachey Approach to Programming Language Theory

17.
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

18.
Algorithm
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In mathematics and computer science, an algorithm is a self-contained sequence of actions to be performed. Algorithms can perform calculation, data processing and automated reasoning tasks, an algorithm is an effective method that can be expressed within a finite amount of space and time and in a well-defined formal language for calculating a function. The transition from one state to the next is not necessarily deterministic, some algorithms, known as randomized algorithms, giving a formal definition of algorithms, corresponding to the intuitive notion, remains a challenging problem. In English, it was first used in about 1230 and then by Chaucer in 1391, English adopted the French term, but it wasnt until the late 19th century that algorithm took on the meaning that it has in modern English. Another early use of the word is from 1240, in a manual titled Carmen de Algorismo composed by Alexandre de Villedieu and it begins thus, Haec algorismus ars praesens dicitur, in qua / Talibus Indorum fruimur bis quinque figuris. Which translates as, Algorism is the art by which at present we use those Indian figures, the poem is a few hundred lines long and summarizes the art of calculating with the new style of Indian dice, or Talibus Indorum, or Hindu numerals. An informal definition could be a set of rules that precisely defines a sequence of operations, which would include all computer programs, including programs that do not perform numeric calculations. Generally, a program is only an algorithm if it stops eventually, but humans can do something equally useful, in the case of certain enumerably infinite sets, They can give explicit instructions for determining the nth member of the set, for arbitrary finite n. An enumerably infinite set is one whose elements can be put into one-to-one correspondence with the integers, the concept of algorithm is also used to define the notion of decidability. That notion is central for explaining how formal systems come into being starting from a set of axioms. In logic, the time that an algorithm requires to complete cannot be measured, from such uncertainties, that characterize ongoing work, stems the unavailability of a definition of algorithm that suits both concrete and abstract usage of the term. Algorithms are essential to the way computers process data, thus, an algorithm can be considered to be any sequence of operations that can be simulated by a Turing-complete system. Although this may seem extreme, the arguments, in its favor are hard to refute. Gurevich. Turings informal argument in favor of his thesis justifies a stronger thesis, according to Savage, an algorithm is a computational process defined by a Turing machine. Typically, when an algorithm is associated with processing information, data can be read from a source, written to an output device. Stored data are regarded as part of the state of the entity performing the algorithm. In practice, the state is stored in one or more data structures, for some such computational process, the algorithm must be rigorously defined, specified in the way it applies in all possible circumstances that could arise. That is, any conditional steps must be dealt with, case-by-case

19.
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

20.
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

21.
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

22.
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

23.
Minimal surface
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In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature, the term minimal surface is used because these surfaces originally arose as surfaces that minimized total surface area subject to some constraint. Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a solution, forming a soap film. However the term is used for general surfaces that may self-intersect or do not have constraints. For a given constraint there may also exist several minimal surfaces with different areas, Minimal surfaces can be defined in several equivalent ways in R3. Local least area definition, A surface M ⊂ R3 is minimal if, note that this property is local, there might exist other surfaces that minimize area better with the same global boundary. Variational definition, A surface M ⊂ R3 is minimal if and this definition makes minimal surfaces a 2-dimensional analogue to geodesics. Note that spherical bubbles are not minimal surfaces as per this definition, while they minimize total area subject to a constraint on internal volume, mean curvature definition, A surface M ⊂ R3 is minimal if and only if its mean curvature vanishes identically. A direct implication of this definition is that point on the surface is a saddle point with equal. This definition ties minimal surfaces to harmonic functions and potential theory, harmonic definition, If X =, M → R3 is an isometric immersion of a Riemann surface into 3-space, then X is said to be minimal whenever xi is a harmonic function on M for each i. A direct implication of this definition and the principle for harmonic functions is that there are no compact complete minimal surfaces in R3. This definition uses that the curvature is half of the trace of the shape operator. If the projected Gauss map obeys the Cauchy–Riemann equations then either the trace vanishes or every point of M is umbilic, mean curvature flow definition, Minimal surfaces are the critical points for the mean curvature flow. The local least area and variational definitions allow extending minimal surfaces to other Riemannian manifolds than R3, Minimal surface theory originates with Lagrange who in 1762 considered the variational problem of finding the surface z = z of least area stretched across a given closed contour. He derived the Euler–Lagrange equation for the solution d d x + d d y =0 He did not succeed in finding any solution beyond the plane. By expanding Lagranges equation to z y y −2 z x z y z x y + z x x =0 Gaspard Monge, while these were successfully used by Heinrich Scherk in 1830 to derive his surfaces, they were generally regarded as practically unusable. Catalan proved in 1842/43 that the helicoid is the only ruled minimal surface, progress had been fairly slow until the middle of the century, when the Björling problem was solved using complex methods. The first golden age of minimal surfaces began, schwarz found the solution of the Plateau problem for a regular quadrilateral in 1865 and for a general quadrilateral in 1867 using complex methods

24.
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

25.
Perspective (graphical)
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Perspective in the graphic arts is an approximate representation, on a flat surface, of an image as it is seen by the eye. If viewed from the spot as the windowpane was painted. Each painted object in the scene is thus a flat, scaled down version of the object on the side of the window. All perspective drawings assume the viewer is a distance away from the drawing. Objects are scaled relative to that viewer, an object is often not scaled evenly, a circle often appears as an ellipse and a square can appear as a trapezoid. This distortion is referred to as foreshortening, Perspective drawings have a horizon line, which is often implied. This line, directly opposite the viewers eye, represents objects infinitely far away and they have shrunk, in the distance, to the infinitesimal thickness of a line. It is analogous to the Earths horizon, any perspective representation of a scene that includes parallel lines has one or more vanishing points in a perspective drawing. A one-point perspective drawing means that the drawing has a vanishing point, usually directly opposite the viewers eye. All lines parallel with the line of sight recede to the horizon towards this vanishing point. This is the standard receding railroad tracks phenomenon, a two-point drawing would have lines parallel to two different angles. Any number of vanishing points are possible in a drawing, one for each set of lines that are at an angle relative to the plane of the drawing. Perspectives consisting of parallel lines are observed most often when drawing architecture. In contrast, natural scenes often do not have any sets of parallel lines, the only method to indicate the relative position of elements in the composition was by overlapping, of which much use is made in works like the Parthenon Marbles. Chinese artists made use of perspective from the first or second century until the 18th century. It is not certain how they came to use the technique, some authorities suggest that the Chinese acquired the technique from India, oblique projection is also seen in Japanese art, such as in the Ukiyo-e paintings of Torii Kiyonaga. This was detailed within Aristotles Poetics as skenographia, using flat panels on a stage to give the illusion of depth, the philosophers Anaxagoras and Democritus worked out geometric theories of perspective for use with skenographia. Alcibiades had paintings in his house designed using skenographia, so this art was not confined merely to the stage, Euclids Optics introduced a mathematical theory of perspective, but there is some debate over the extent to which Euclids perspective coincides with the modern mathematical definition

26.
Camera lucida
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A camera lucida is an optical device used as a drawing aid by artists. The camera lucida performs an optical superimposition of the subject being viewed upon the surface upon which the artist is drawing, the artist sees both scene and drawing surface simultaneously, as in a photographic double exposure. This allows the artist to duplicate key points of the scene on the drawing surface, the camera lucida was patented in 1807 by William Hyde Wollaston. The basic optics were described 200 years earlier by Johannes Kepler in his Dioptrice, by the 19th century, Kepler’s description had fallen into oblivion, so Wollaston’s claim was never challenged. The term camera lucida is Wollastons, while on honeymoon in Italy in 1833, the photographic pioneer William Fox Talbot used a camera lucida as a sketching aid. He later wrote that it was a disappointment with his efforts which encouraged him to seek a means to cause these natural images to imprint themselves durably. In 2001, artist David Hockneys book Secret Knowledge, Rediscovering the Lost Techniques of the Old Masters was met with controversy and their evidence is based largely on the characteristics of the paintings by great artists of later centuries, such as Ingres, Van Eyck, and Caravaggio. The camera lucida is still available today through art-supply channels but is not well known or widely used and it has enjoyed a resurgence recently through a number of Kickstarter campaigns. The name camera lucida is obviously intended to recall the much older drawing aid, there is no optical similarity between the devices. The camera lucida is a light, portable device that not require special lighting conditions. No image is projected by the camera lucida, in the simplest form of camera lucida, the artist looks down at the drawing surface through a half-silvered mirror tilted at 45 degrees. This superimposes a direct view of the surface beneath. This design produces an image which is right-left reversed when turned the right way up. Also, light is lost in the imperfect reflection, Wollastons design used a prism with four optical faces to produce two successive reflections, thus producing an image that is not inverted or reversed. Angles ABC and ADC are 67. 5° and BCD is 135°, hence, the reflections occur through total internal reflection, so very little light is lost. It is not possible to see straight through the prism, so it is necessary to look at the edge to see the paper. The instrument often includes a weak negative lens, creating an image of the scene at about the same distance as the drawing surface. If white paper is used with the camera lucida, the superimposition of the paper with the scene tends to wash out the scene, when working with a camera lucida, it is often beneficial to use black paper and to draw with a white pencil

27.
Camera obscura
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The surroundings of the projected image have to be relatively dark for the image to be clear, so many historical camera obscura experiments were performed in dark rooms. The term camera obscura also refers to constructions or devices that use of the principle within a box. Camerae obscurae with a lens in the opening have been used since the second half of the 16th century, before the term camera obscura was first used in 1604, many other expressions were used including cubiculum obscurum, cubiculum tenebricosum, conclave obscurum and locus obscurus. Rays of light travel in straight lines and change when they are reflected and partly absorbed by an object, retaining information about the color, lit objects reflect rays of light in all directions. The human eye itself works much like a camera obscura with an opening, a biconvex lens, a camera obscura device consists of a box, tent or room with a small hole in one side. Light from a scene passes through the hole and strikes a surface inside, where the scene is reproduced, inverted and reversed. The image can be projected onto paper, and can then be traced to produce an accurate representation. In order to produce a reasonably clear projected image, the aperture has to be about 1/100th the distance to the screen, many camerae obscurae use a lens rather than a pinhole because it allows a larger aperture, giving a usable brightness while maintaining focus. As the pinhole is made smaller, the image gets sharper, with too small a pinhole, however, the sharpness worsens, due to diffraction. Using mirrors, as in an 18th-century overhead version, it is possible to project a right-side-up image, another more portable type is a box with an angled mirror projecting onto tracing paper placed on the glass top, the image being upright as viewed from the back. There are theories that occurrences of camera obscura effects inspired paleolithic cave paintings and it is also suggested that camera obscura projections could have played a role in Neolithic structures. Perforated gnomons projecting an image of the sun were described in the Chinese Zhoubi Suanjing writings. The location of the circle can be measured to tell the time of day. In Arab and European cultures its invention was later attributed to Egyptian astronomer. Some ancient sightings of gods and spirits, especially in worship, are thought to possibly have been conjured up by means of camera obscura projections. In these writings it is explained how the image in a collecting-point or treasure house is inverted by an intersecting point that collected the light. Light coming from the foot of a person would partly be hidden below. Rays from the head would partly be hidden above and partly form the part of the image

28.
Projective geometry
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Projective geometry is a topic of mathematics. It is the study of properties that are invariant with respect to projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than can be expressed by a transformation matrix and translations. The first issue for geometers is what kind of geometry is adequate for a novel situation, one source for projective geometry was indeed the theory of perspective. Another difference from elementary geometry is the way in which parallel lines can be said to meet in a point at infinity, again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a perspective drawing. See projective plane for the basics of geometry in two dimensions. While the ideas were available earlier, projective geometry was mainly a development of the 19th century and this included the theory of complex projective space, the coordinates used being complex numbers. Several major types of more abstract mathematics were based on projective geometry and it was also a subject with a large number of practitioners for its own sake, as synthetic geometry. Another topic that developed from axiomatic studies of projective geometry is finite geometry, the topic of projective geometry is itself now divided into many research subtopics, two examples of which are projective algebraic geometry and projective differential geometry. Projective geometry is an elementary form of geometry, meaning that it is not based on a concept of distance. In two dimensions it begins with the study of configurations of points and lines and that there is indeed some geometric interest in this sparse setting was first established by Desargues and others in their exploration of the principles of perspective art. In higher dimensional spaces there are considered hyperplanes, and other linear subspaces, Projective geometry can also be seen as a geometry of constructions with a straight-edge alone. Since projective geometry excludes compass constructions, there are no circles, no angles, no measurements, no parallels and it was realised that the theorems that do apply to projective geometry are simpler statements. For example, the different conic sections are all equivalent in projective geometry, during the early 19th century the work of Jean-Victor Poncelet, Lazare Carnot and others established projective geometry as an independent field of mathematics. Its rigorous foundations were addressed by Karl von Staudt and perfected by Italians Giuseppe Peano, Mario Pieri, Alessandro Padoa, after much work on the very large number of theorems in the subject, therefore, the basics of projective geometry became understood. The incidence structure and the cross-ratio are fundamental invariants under projective transformations, Projective geometry can be modeled by the affine plane plus a line at infinity and then treating that line as ordinary. An algebraic model for doing projective geometry in the style of geometry is given by homogeneous coordinates. In a foundational sense, projective geometry and ordered geometry are elementary since they involve a minimum of axioms and either can be used as the foundation for affine, Projective geometry is not ordered and so it is a distinct foundation for geometry

29.
Proportion (architecture)
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Proportion is a central principle of architectural theory and an important connection between mathematics and art. It is the effect of the relationships of the various objects and spaces that make up a structure to one another. These relationships are governed by multiples of a standard unit of length known as a module. Proportion in architecture was discussed by Vitruvius, Alberti, Andrea Palladio, Architecture in Roman antiquity was rarely documented except in the writings of Vitruvius treatise De Architectura. Vitruvius served as an engineer under Julius Caesar during the first Gallic Wars, the treatise was dedicated to Emperor Augustus. Moreover, Vitruvius identified the Six Principles of Design as order, arrangement, proportion, symmetry, propriety, among the six principles, proportion interrelates and supports all the other factors in geometrical forms and arithmetical ratios. The word symmetria, usually translated to symmetry in modern renderings, in ancient times meant something more closely related to mathematical harmony, Vitruvius tried to describe his theory in the makeup of the human body, which he referred to as the perfect or golden ratio. The principles of measurement units digit, foot, and cubit also came from the dimensions of a Vitruvian Man, more specifically, Vitruvius used the total height of 6 feet of a person, and each part of the body takes up a different ratio. For example, the face is about 1/10 of the height. Vitruvius used these ratios to prove that the composition of classical orders mimicked the human body, in classical architecture, the module was established as the radius of the lower shaft of a classical column, with proportions expressed as a fraction or multiple of that module. In his Le Modulor, Le Corbusier presented a system of proportions which took the golden ratio, history of architecture Mathematics and architecture Mathematics and art P. H. Scholfield. The Theory of Proportion in Architecture

30.
Body proportions
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While there is significant variation in anatomical proportions between people, there are many references to body proportions that are intended to be canonical, either in art, measurement, or medicine. In measurement, body proportions are used to relate two or more measurements based on the body. A cubit, for instance, is supposed to be six palms, a span is taken to be 9 inches and was previously considered as half a cubit. While convenient, these ratios may not reflect the variation of the individuals using them. Similarly, in art, body proportions are the study of relation of human or animal parts to each other. These ratios are used in depictions of the figure. It is important in drawing to draw the human figure in proportion. In modern figure drawing, the unit of measurement is the head. This unit of measurement is reasonably standard, and has long used by artists to establish the proportions of the human figure. Ancient Egyptian art used a canon of proportion based on the fist, measured across the knuckles, with 18 fists from the ground to the hairline on the forehead. This was already established by the Narmer Palette from about the 31st century BC, the proportions used in figure drawing are, An average person, is generally 7-and-a-half heads tall. An ideal figure, used when aiming for an impression of nobility or grace, is drawn at 8 heads tall, a heroic figure, used in the heroic for the depiction of gods and superheroes, is eight-and-a-half heads tall. Most of the length comes from a bigger chest and longer legs. A study using Polish participants by Sorokowski found 5% longer legs than a used as a reference was considered most attractive. The study concluded this preference might stem from the influence of leggy runway models, the Sorokowski study was criticized for using a picture of the same person with digitally altered leg lengths which Marco Bertamini felt were unrealistic. Another study using British and American participants, found mid-ranging leg-to-body ratios to be most ideal, the Swami et al. study was criticized for using a picture of the same person with digitally altered leg lengths which Marco Bertamini felt were unrealistic. Bertamini also criticized the Swami study for only changing the leg length while keeping the arm length constant, bertaminis own study which used stick figures mirrored Swamis study, however, by finding a preference for leggier women. Another common measurement of related to leg-to-body ratio is sitting-height ratio, sitting height ratio is the ratio of the head plus spine length to total height which is highly correlated to leg-to-body ratio

31.
Symmetry
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Symmetry in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, symmetry has a precise definition, that an object is invariant to any of various transformations. Although these two meanings of symmetry can sometimes be told apart, they are related, so they are discussed together. The opposite of symmetry is asymmetry, a geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion. This means that an object is symmetric if there is a transformation that moves individual pieces of the object, an object has rotational symmetry if the object can be rotated about a fixed point without changing the overall shape. An object has symmetry if it can be translated without changing its overall shape. An object has symmetry if it can be simultaneously translated and rotated in three-dimensional space along a line known as a screw axis. An object has symmetry if it does not change shape when it is expanded or contracted. Fractals also exhibit a form of symmetry, where small portions of the fractal are similar in shape to large portions. Other symmetries include glide reflection symmetry and rotoreflection symmetry, a dyadic relation R is symmetric if and only if, whenever its true that Rab, its true that Rba. Thus, is the age as is symmetrical, for if Paul is the same age as Mary. Symmetric binary logical connectives are and, or, biconditional, nand, xor, the set of operations that preserve a given property of the object form a group. In general, every kind of structure in mathematics will have its own kind of symmetry, examples include even and odd functions in calculus, the symmetric group in abstract algebra, symmetric matrices in linear algebra, and the Galois group in Galois theory. In statistics, it appears as symmetric probability distributions, and as skewness, symmetry in physics has been generalized to mean invariance—that is, lack of change—under any kind of transformation, for example arbitrary coordinate transformations. This concept has one of the most powerful tools of theoretical physics. See Noethers theorem, and also, Wigners classification, which says that the symmetries of the laws of physics determine the properties of the found in nature. Important symmetries in physics include continuous symmetries and discrete symmetries of spacetime, internal symmetries of particles, in biology, the notion of symmetry is mostly used explicitly to describe body shapes. Bilateral animals, including humans, are more or less symmetric with respect to the plane which divides the body into left

32.
Tessellation
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A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions and a variety of geometries, a periodic tiling has a repeating pattern. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups, a tiling that lacks a repeating pattern is called non-periodic. An aperiodic tiling uses a set of tile shapes that cannot form a repeating pattern. In the geometry of higher dimensions, a space-filling or honeycomb is called a tessellation of space. A real physical tessellation is a made of materials such as cemented ceramic squares or hexagons. Such tilings may be decorative patterns, or may have such as providing durable and water-resistant pavement. Historically, tessellations were used in Ancient Rome and in Islamic art such as in the decorative geometric tiling of the Alhambra palace, in the twentieth century, the work of M. C. Escher often made use of tessellations, both in ordinary Euclidean geometry and in geometry, for artistic effect. Tessellations are sometimes employed for decorative effect in quilting, Tessellations form a class of patterns in nature, for example in the arrays of hexagonal cells found in honeycombs. Tessellations were used by the Sumerians in building wall decorations formed by patterns of clay tiles, decorative mosaic tilings made of small squared blocks called tesserae were widely employed in classical antiquity, sometimes displaying geometric patterns. In 1619 Johannes Kepler made a documented study of tessellations. He wrote about regular and semiregular tessellations in his Harmonices Mundi, he was possibly the first to explore and to explain the structures of honeycomb. Some two hundred years later in 1891, the Russian crystallographer Yevgraf Fyodorov proved that every periodic tiling of the features one of seventeen different groups of isometries. Fyodorovs work marked the beginning of the mathematical study of tessellations. Other prominent contributors include Shubnikov and Belov, and Heinrich Heesch, in Latin, tessella is a small cubical piece of clay, stone or glass used to make mosaics. The word tessella means small square and it corresponds to the everyday term tiling, which refers to applications of tessellations, often made of glazed clay. Tessellation or tiling in two dimensions is a topic in geometry that studies how shapes, known as tiles, can be arranged to fill a plane without any gaps, according to a given set of rules

33.
Wallpaper group
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A wallpaper group is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art, There are 17 possible distinct groups. Wallpaper groups are two-dimensional symmetry groups, intermediate in complexity between the simpler frieze groups and the space groups. Wallpaper groups categorize patterns by their symmetries, subtle differences may place similar patterns in different groups, while patterns that are very different in style, color, scale or orientation may belong to the same group. Consider the following examples, Examples A and B have the same group, it is called p4m in the IUC notation. Example C has a different wallpaper group, called p4g or 4*2, a complete list of all seventeen possible wallpaper groups can be found below. A symmetry of a pattern is, loosely speaking, a way of transforming the pattern so that it exactly the same after the transformation. For example, translational symmetry is present when the pattern can be translated some finite distance, think of shifting a set of vertical stripes horizontally by one stripe. Strictly speaking, a true symmetry only exists in patterns that repeat exactly, a set of only, say, five stripes does not have translational symmetry—when shifted, the stripe on one end disappears and a new stripe is added at the other end. In practice, however, classification is applied to finite patterns, sometimes two categorizations are meaningful, one based on shapes alone and one also including colors. When colors are ignored there may be more symmetry, the types of transformations that are relevant here are called Euclidean plane isometries. This type of symmetry is called a translation, Examples A and C are similar, except that the smallest possible shifts are in diagonal directions. If we turn example B clockwise by 90°, around the centre of one of the squares, Examples A and C also have 90° rotations, although it requires a little more ingenuity to find the correct centre of rotation for C. We can also flip example B across a horizontal axis that runs across the middle of the image, example B also has reflections across a vertical axis, and across two diagonal axes. The same can be said for A. However, example C is different and it only has reflections in horizontal and vertical directions, not across diagonal axes. If we flip across a line, we do not get the same pattern back. This is part of the reason that the group of A and B is different from the wallpaper group of C. A proof that there were only 17 possible patterns was first carried out by Evgraf Fedorov in 1891, the proof that the list of wallpaper groups was complete only came after the much harder case of space groups had been done

34.
Algorithmic art
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Algorithmic art, also known as algorithm art, is art, mostly visual art, of which the design is generated by an algorithm. Algorithmic artists are sometimes called algorists, algorithmic art, also known as computer-generated art, is a subset of generative art and is related to systems art. Fractal art is an example of algorithmic art, the final output is typically displayed on a computer monitor, printed with a raster-type printer, or drawn using a plotter. Variability can be introduced by using pseudo-random numbers, there is no consensus as to whether the product of an algorithm that operates on an existing image can still be considered computer-generated art, as opposed to computer-assisted art. Some of the earliest known examples of computer-generated algorithmic art were created by Georg Nees, Frieder Nake, A. Michael Noll, Manfred Mohr and these artworks were executed by a plotter controlled by a computer, and were therefore computer-generated art but not digital art. The act of creation lay in writing the program, which specified the sequence of actions to be performed by the plotter and her early work with copier and telematic art focused on the differences between the human hand and the algorithm. Aside from the work of Roman Verostko and his fellow algorists. These are important here because they use a different means of execution, whereas the earliest algorithmic art was drawn by a plotter, fractal art simply creates an image in computer memory, it is therefore digital art. The native form of an artwork is an image stored on a computer –this is also true of very nearly all equation art. However, in a stricter sense fractal art is not considered algorithmic art, from one point of view, for a work of art to be considered algorithmic art, its creation must include a process based on an algorithm devised by the artist. This input may be mathematical, computational, or generative in nature, inasmuch as algorithms tend to be deterministic, meaning that their repeated execution would always result in the production of identical artworks, some external factor is usually introduced. This can either be a number generator of some sort. By this definition, fractals made by a program are not art. However, defined differently, algorithmic art can be seen to include fractal art, the artist Kerry Mitchell stated in his 1999 Fractal Art Manifesto, Fractal Art is not. Computer Art, in the sense that the computer does all the work. The work is executed on a computer, but only at the direction of the artist, turn a computer on and leave it alone for an hour. When you come back, no art will have been generated, algorist is a term used for digital artists who create algorithmic art. Algorists formally began correspondence and establishing their identity as artists following a panel titled Art, the co-founders were Roman Verostko and Jean-Pierre Hébert. Fractal art consists of varieties of computer-generated fractals with colouring chosen to give an attractive effect, especially in the western world, it is not drawn or painted by hand

35.
Anamorphosis
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Anamorphosis is a distorted projection or perspective requiring the viewer to use special devices or occupy a specific vantage point to reconstitute the image. The word anamorphosis is derived from the Greek prefix ana‑, meaning back or again, an optical anamorphism is the visualization of a mathematical operation called an affine transformation. There are two types of anamorphosis, perspective and mirror. More-complex anamorphoses can be devised using distorted lenses, mirrors, or other optical transformations, examples of perspectival anamorphosis date to the early Renaissance. Examples of mirror anamorphosis were first seen in the late Renaissance, the deformed image is painted on a plane surface surrounding the mirror. By looking into the mirror, a viewer can see the image undeformed, leonardos Eye is the earliest known definitive example of perspective anamorphosis in modern times. The prehistoric cave paintings at Lascaux may also use this technique, Hans Holbein the Younger is well known for incorporating an oblique anamorphic transformation into his painting The Ambassadors. In this artwork, a distorted shape lies diagonally across the bottom of the frame, viewing this from an acute angle transforms it into the plastic image of a human skull, a symbolic memento mori. During the seventeenth century, Baroque trompe loeil murals often used anamorphism to combine actual architectural elements with illusory painted elements, when a visitor views the art work from a specific location, the architecture blends with the decorative painting. The dome and vault of the Church of St. Ignazio in Rome, painted by Andrea Pozzo, due to neighboring monks complaining about blocked light, Pozzo was commissioned to paint the ceiling to look like the inside of a dome, instead of building a real dome. As the ceiling is flat, there is one spot where the illusion is perfect. Mirror anamorphosis emerged early in the 17th century in Italy and China and it remains uncertain whether Jesuit missionaries imported or exported the technique. Anamorphosis could be used to conceal images for privacy or personal safety, a secret portrait of Bonnie Prince Charlie is painted in a distorted manner on a tray and can only be recognized when a polished cylinder is placed in the correct position. To possess such an image would have seen as treason in the aftermath of the 1746 Battle of Culloden. In the eighteenth and nineteenth centuries, anamorphic images had come to be used more as childrens games than fine art, in the twentieth century, some artists wanted to renew the technique of anamorphosis. Marcel Duchamp was interested in anamorphosis, and some of his installations are visual paraphrases of anamorphoses, Jan Dibbets conceptual works, the so-called perspective corrections are examples of linear anamorphoses. In the late century, mirror anamorphosis was revived as childrens toys. Beginning in 1967, Dutch artist Jan Dibbets based a series of photographic work titled Perspective Corrections on the distortion of reality through perspective anamorphosis

36.
Computer art
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Computer art is any art in which computers play a role in production or display of the artwork. Such art can be an image, sound, animation, video, CD-ROM, DVD-ROM, video game, website, algorithm, performance or gallery installation. Many traditional disciplines are now integrating digital technologies and, as a result, for instance, an artist may combine traditional painting with algorithm art and other digital techniques. As a result, defining computer art by its end product can thus be difficult, Computer art is by its nature evolutionary since changes in technology and software directly affect what is possible. Notable artists in this vein include Manfred Mohr, Ronald Davis, Harold Cohen, Joseph Nechvatal, George Grie, Olga Kisseleva, John Lansdown, and Jean-Pierre Hébert. On the title page of the magazine Computers and Automation, January 1963, Edmund Berkeley published a picture by Efraim Arazi from 1962 and this picture inspired him to initiate the first Computer Art Contest in 1963. The annual contest was a key point in the development of art up to the year 1973. The precursor of computer art dates back to 1956-1958, with the generation of what is probably the first image of a human being on a computer screen, a pin-up girl at a SAGE air defense installation. Desmond Paul Henry invented the Henry Drawing Machine in 1960, his work was shown at the Reid Gallery in London in 1962, many artists tentatively began to explore the emerging computing technology for use as a creative tool. In the summer of 1962, A. Michael Noll programmed a computer at Bell Telephone Laboratories in Murray Hill. His later computer-generated patterns simulated paintings by Piet Mondrian and Bridget Riley, Noll also used the patterns to investigate aesthetic preferences in the mid-1960s. The Stuttgart exhibit featured work by Georg Nees, the New York exhibit featured works by Bela Julesz, a third exhibition was put up in November 1965 at Galerie Wendelin Niedlich in Stuttgart, Germany, showing works by Frieder Nake and Georg Nees. Analogue computer art by Maughan Mason along with computer art by Noll were exhibited at the AFIPS Fall Joint Computer Conference in Las Vegas toward the end of 1965. In 1968, the Institute of Contemporary Arts in London hosted one of the most influential exhibitions of computer art called Cybernetic Serendipity. The exhibition included many of whom often regarded as the first digital artists, Nam June Paik, Frieder Nake, Leslie Mezei, Georg Nees, A. Michael Noll, John Whitney, one year later, the Computer Arts Society was founded, also in London. At the time of the opening of Cybernetic Serendipity, in August 1968 and it took up the European artists movement of New Tendencies that had led to three exhibitions in Zagreb of concrete, kinetic, and constructive art as well as op art and conceptual art. New Tendencies changed its name to Tendencies and continued with more symposia, exhibitions, a competition, katherine Nash and Richard Williams published Computer Program for Artists, ART1 in 1970. Xerox Corporation’s Palo Alto Research Center designed the first Graphical User Interface in the 1970s, the first Macintosh computer is released in 1984, since then the GUI became popular

37.
Fourth dimension in art
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New possibilities opened up by the concept of four-dimensional space helped inspire many modern artists in the first half of the twentieth century. Early Cubists, Surrealists, Futurists, and abstract artists took ideas from higher-dimensional mathematics, french mathematician Maurice Princet was known as le mathématicien du cubisme. Picassos Portrait of Daniel-Henry Kahnweiler in 1910 was an important work for the artist, early cubist Max Weber wrote an article entitled In The Fourth Dimension from a Plastic Point of View, for Alfred Stieglitzs July 1910 issue of Camera Work. Another influence on the School of Paris was that of Jean Metzinger and Albert Gleizes, in 1936 in Paris, Charles Tamkó Sirató published his Manifeste Dimensioniste, which described how the Dimensionist tendency has led to, Literature leaving the line and entering the plane. Painting leaving the plane and entering space, sculpture stepping out of closed, immobile forms. …The artistic conquest of space, which to date has been completely art-free. The manifesto was signed by prominent modern artists worldwide. In 1953, the surrealist Salvador Dalí proclaimed his intention to paint an explosive, nuclear and he said that, This picture will be the great metaphysical work of my summer. Completed the next year, Crucifixion depicts Jesus Christ upon the net of a hypercube, the unfolding of a tesseract into eight cubes is analogous to unfolding the sides of a cube into six squares. The Metropolitan Museum of Art describes the painting as a new interpretation of an oft-depicted subject, christs spiritual triumph over corporeal harm. Some of Piet Mondrians abstractions and his practice of Neoplasticism are said to be rooted in his view of a utopian universe, the fourth dimension has been the subject of numerous fictional stories. De Stijl Five-dimensional space Four-dimensional space Duration Philosophy of space and time Octacube Clair, spirits, Art, and the Fourth Dimension. Dalí, Salvador, Gómez de la Serna, Ramón, Maurice Princet, Le Mathématicien du Cubisme. Overview of The Fourth Dimension And Non-Euclidean Geometry In Modern Art, traité élémentaire de géométrie à quatre dimensions et introduction à la géométrie à n dimensions. Art in the Fourth Dimension, Giving Form to Form – The Abstract Paintings of Piet Mondrian, spaces of Utopia, An Electronic Journal, 23–35. Einstein, Picasso, space, time, and beauty that causes havoc, making Music Modern, New York in the 1920s. Shadows of Reality, The Fourth Dimension in Relativity, Cubism, in The Fourth Dimension from a Plastic Point of View. The Fourth Dimension And Non-Euclidean Geometry In Modern Art, duchamp in Context, Science and Technology in the Large Glass and Related Works

38.
Fractal art
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Fractal art is a form of algorithmic art created by calculating fractal objects and representing the calculation results as still images, animations, and media. Fractal art developed from the mid-1980s onwards and it is a genre of computer art and digital art which are part of new media art. The Julia set and Mandelbrot sets can be considered as icons of fractal art, Fractal art is rarely drawn or painted by hand. In some cases, other programs are used to further modify the images produced. Non-fractal imagery may also be integrated into the artwork and it was assumed that fractal art could not have developed without computers because of the calculative capabilities they provide. Fractals are generated by applying iterative methods to solving non-linear equations or polynomial equations, Fractals are any of various extremely irregular curves or shapes for which any suitably chosen part is similar in shape to a given larger or smaller part when magnified or reduced to the same size. There are many different kinds of images and can be subdivided into several groups. Fractals derived from standard geometry by using iterative transformations on a common figure like a straight line. The first fractal figures invented near the end of the 19th, IFS Strange attractors Fractal flame L-system fractals Fractals created by the iteration of complex polynomials, perhaps the most famous fractals. Newton fractals, including Nova fractals Quaternionic and hypernionic fractals Fractal terrains generated by random fractal processes Mandelbulbs are a kind of three dimensional fractal, Fractal Expressionism is a term used to differentiate traditional visual art that incorporates fractal elements such as self-similarity for example. Perhaps the best example of fractal expressionism is found in Jackson Pollocks dripped patterns and they have been analysed and found to contain a fractal dimension which has been attributed to his technique. Fractals of all kinds have used as the basis for digital art. High resolution color graphics became available at scientific research labs in the mid-1980s. Scientific forms of art, including art, have developed separately from mainstream culture. Many fractal images are admired because of their perceived harmony and this is typically achieved by the patterns which emerge from the balance of order and chaos. Similar qualities have been described in Chinese painting and miniature trees and rockeries, Fractal rendering programs used to make fractal art include Ultra Fractal, Apophysis, Bryce and Sterling. Fractint was the first widely used fractal generating program, the first fractal image that was intended to be a work of art was probably the famous one on the cover of Scientific American, August 1985. This image showed a landscape formed from the function on the domain outside the Mandelbrot set

39.
Islamic geometric patterns
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Islamic decoration, which tends to avoid using figurative images, makes frequent use of geometric patterns which have developed over the centuries. These may constitute the entire decoration, may form a framework for floral or calligraphic embellishments, interest in Islamic geometric patterns is increasing in the West, both among craftsmen and artists including M. C. Islamic art mostly avoids figurative images to avoid becoming objects of worship, Islamic geometric patterns derived from simpler designs used in earlier cultures, Greek, Roman, and Sasanian. They are one of three forms of Islamic decoration, the others being the based on curving and branching plant forms. Geometric designs and arabesques are forms of Islamic interlace patterns, david Wade states that Much of the art of Islam, whether in architecture, ceramics, textiles or books, is the art of decoration – which is to say, of transformation. Wade argues that the aim is to transfigure, turning mosques into lightness and pattern and she argues that beauty, whether in poetry or in the visual arts, was enjoyed for its own sake, without commitment to religious or moral criteria. Many Islamic designs are built on squares and circles, typically repeated, overlapped and interlaced to form intricate, a recurring motif is the 8-pointed star, often seen in Islamic tilework, it is made of two squares, one rotated 45 degrees with respect to the other. The fourth basic shape is the polygon, including pentagons and octagons, all of these can be combined and reworked to form complicated patterns with a variety of symmetries including reflections and rotations. Such patterns can be seen as mathematical tessellations, which can extend indefinitely and they are constructed on grids that require only ruler and compass to draw. Artist and educator Roman Verostko argues that such constructions are in effect algorithms, the circle symbolizes unity and diversity in nature, and many Islamic patterns are drawn starting with a circle. On this basis is constructed a star surrounded by six smaller irregular hexagons to form a tessellating star pattern. This forms the basic design which is outlined in white on the wall of the mosque and that design, however, is overlaid with an intersecting tracery in blue around tiles of other colours, forming an elaborate pattern that partially conceals the original and underlying design. A similar design forms the logo of the Mohammed Ali Research Center and he observed that many different combinations of polygons can be used as long as the residual spaces between the polygons are reasonably symmetrical. For example, a grid of octagons in contact has squares as the residual spaces, every octagon is the basis for an 8-point star, as seen at Akbars tomb, Sikandra. Hankin considered the skill of the Arabian artists in discovering suitable combinations of polygons and he further records that if a star occurs in a corner, exactly one quarter of it should be shown, if along an edge, exactly one half of it. The mathematical properties of the tile and stucco patterns of the Alhambra palace in Granada. Some authors have claimed on dubious grounds to have found most or all of the 17 wallpaper groups there, the earliest geometrical forms in Islamic art were occasional isolated geometric shapes such as 8-pointed stars and lozenges containing squares. These date from 836 in the Great Mosque of Kairouan, Tunisia, the next development, marking the middle stage of Islamic geometric pattern usage, was of 6- and 8-point stars, which appear in 879 at the Ibn Tulun Mosque, Cairo, and then became widespread

40.
Girih
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Girih, also girih sāzī or girih chīnī, is a Persianate Islamic decorative art form used in architecture and handicrafts, consisting of geometric lines that form an interlaced strapwork. In Iranian architecture, gereh sazi patterns were seen in bannai brickwork, stucco, girih has been defined as geometric designs composed upon or generated from arrays of points from which construction lines radiate and at which they intersect. Straight-edged symmetric shapes are used in girih, girih typically consists of a strapwork that form 6-, 8-, 10-, or 12-pointed stars separated by polygons and straps, and often they were drawn in an interlacing manner. Such patterns usually consist of a unit cell with 2-, 3-. The three-dimensional equivalent of girih is called muqarnas and it is used to decorate the underside of domes or squinches. The girih style of ornamentation is thought to have inspired by the Syrian Roman knotwork patterns dating back to the 2nd century AD. The predecessors of the form were curvilinear interlaced strapwork with three-fold rotational symmetry. The Umayyad Mosque, in Damascus, Syria has window screens made of interlacing undulating strapwork in the form of six-pointed stars, early examples of Islamic geometric patterns made of straight strap lines can be seen in the architecture of the surviving gateway of the Ribat-i Malik caravanserai, Uzbekistan. The earliest form of girih on a book is seen in the fronticepiece of a Koran manuscript from the year 1000 and this Koran has an illuminated page with interlacing octagons and thuluth calligraphy. In woodwork, one of the earliest surviving examples of Islamic geometric art is the 13th-century minbar of the Ibn Tulun Mosque in Cairo, in woodwork, girih patterns can be created by two different methods. In one, a grill with geometric shapes would be created first. In the other method, called gereh-chini wooden panels of geometric shapes would be created individually and this woodwork technique was popular during the Safavid period, examples of it can be seen in various historic structures in Esfahan. The term girih was used in Turkish as a polygonal strap pattern used in architecture as early as the late 15th century, also in the late 15th century, girih patterns were compiled by artisans in pattern catalogs such as the Topkapı Scroll. While curvilinear precedants of girih were seen in the 10th century and it became a dominant design element in the 11th and 12th centuries, as for example, the carved stucco panels with interlaced girih seen in Kharraqan towers near Qazvin, Iran. Stylized plant decoration were sometimes co-ordinated with girih, after the Safavid period, the use of girih continued in the Seljuk and later the Ilkhanid period. In the 14th century girih became an element in the decorative arts and was replaced by vegetal patterns during the Timurid era. However, geometrical strap work patterns continued to be an important element of decorative arts in Central Asian monuments after the Timurid period, the first girih patterns were made by copying a pattern template along a regular grid, the pattern was drawn with compass and straightedge. Today, artisans using traditional techniques use a pair of dividers to leave a mark on a paper sheet that has been left in the sun to become brittle

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Jali
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A jali or jaali, is the term for a perforated stone or latticed screen, usually with an ornamental pattern constructed through the use of calligraphy and geometry. This form of decoration is found in Indian architecture, Indo-Islamic Architecture. Early jali work was built by carving stone, generally in geometric patterns, while later the Mughals used very finely carved plant-based designs. They also often added pietra dura inlay to the surrounds, using marble, the jali helps in lowering the temperature by compressing the air through the holes. Also when the air passes through openings, its velocity increases giving profound diffusion. It has been observed that humid areas like Kerala and Konkan have larger holes with overall lower opacity than compared with the dry regions of Gujarat. With compactness of the areas in the modern India, jalis became less frequent for privacy

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Muqarnas
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It is used for domes, and especially half-domes in entrances, iwans and apses, mostly in traditional Islamic and Persian architecture. Where some elements project downwards, the style may be called mocárabe, these are reminiscent of stalactites, Muqarnas developed around the middle of the 10th century in northeastern Iran and almost simultaneously — but apparently independently — in North Africa. Examples can be found across Morocco and by extension, the Alhambra in Granada, Spain, the Abbasid Palace in Baghdad, Iraq, and the mausoleum of Sultan Qaitbay, Cairo, Egypt. Large rectangular roofs in wood with muqarnas-style decoration adorn the 12th century Cappella Palatina in Palermo, Sicily, Muqarnas is also found in Armenian architecture. Muqarnas is typically applied to the undersides of domes, pendentives, cornices, squinches, arches, Muqarnas is a downward-facing shape, that is, a vertical line can be traced from the floor to any point on a muqarnas surface. It is also arranged in courses, as in a corbelled vault. The edges of surfaces can all be traced on a single plan view, architects can thus plan out muqarnas geometrically. Muqarnas does not have a significant structural role, Muqarnas need not be carved into the structural blocks of a corbelled vault, it can be hung from a structural roof as a purely decorative surface. Muqarnas may be made of brick, stone, stucco, or wood, the individual cells may be called alveoles. Some modern muqarnas elements have been designed, if not built, with upwards-facing cells, Architecture of Iran Islamic architecture Islamic geometric patterns Mathematics and art Ottoman architecture Muqarnas, A Three-dimensional Decoration of Islam Architecture. Contains a database of over a thousand plans of extant muqarnas, indexed by location and geometry

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Zellige
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Zellige, zelige or zellij (Arabic, or Mosaic is tilework made from individually chiseled geometric tiles set into a plaster base. This form of Islamic art is one of the characteristics of Berber. It consists of geometrically patterned mosaics, used to ornament walls, ceilings, fountains, floors, pools, the art of zellige flourished during the Hispano-Moresque period of the Maghreb and the area known as Al-Andalus between 711-1492. The technique was developed during the Nasrid Dynasty and Merinid dynasty who gave it more importance around the 14th century and introduced blue, green. Red was added in the 17th century, the old enamels with the natural colours were used until the beginning of the 20th century and the colours had probably not evolved much since the period of Merinids. The cities of Fes and Meknes in Morocco, remain the centers of this art, patrons of the art used zellige historically to decorate their homes as a statement of luxury and the sophistication of the inhabitants. Zellige is typically a series of patterns utilizing colourful geometric patterns and this framework of expression arose from the need of Islamic artists to create spatial decorations that avoided depictions of living things, consistent with the teachings of Islamic law. Fez and Meknes in Morocco are still the centers for zellij tiles due to the Miocene grey clay of Fez. The clay from this region is composed of Kaolinite. For Fez and Meknes, the composition is 2-56% clay minerals. Meriam El Ouahabi states that, From the other sites, the mineral composition shows besides kaolinite the presence of illite, chlorite, smectite. Meknes clays belong to illitic clays, characterized by illite, kaolinite, smectite and chlorite, Fes clays have a homogeneous composition with illite. and kaolinite as the most abundant clay minerals. Chlorite and smectite are generally present as small quantities, mixed layer illite/chlorite is present in trace amounts in all the examined Fes clay materials. The colour palette of the increased, making it possible to multiply the compositions ad infinitum. The most current form of the zellige is a square, other forms are possible, the octagon combined with a cabochon, a star, a cross, etc. It is then moulded with a thickness of approximately 2 centimetres, there are simple squares of 10 by 10 centimeters or with the corners cut to be combined with a coloured cabochon. To pave the grounds, bejmat, a stone of 15 by 5 centimetres approximately and 2 centimetres thick. An encyclopedia could not contain the full array of complex, often individually varied patterns, star-based patterns are identified by their number of points—itnashari for 12, ishrini for 20, arba wa ishrini for 24 and so on, but they are not necessarily named with exactitude

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Knot
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A knot is a method of fastening or securing linear material such as rope by tying or interweaving. It may consist of a length of one or several segments of rope, string, webbing, twine, strap, Knots have been the subject of interest for their ancient origins, their common uses, and the area of mathematics known as knot theory. There is a variety of knots, each with properties that make it suitable for a range of tasks. Some knots are used to attach the rope to other such as another rope, cleat, ring. Some knots are used to bind or constrict objects, decorative knots usually bind to themselves to produce attractive patterns. While some people can look at diagrams or photos and tie the illustrated knots, Knot tying skills are often transmitted by sailors, scouts, climbers, canyoners, cavers, arborists, rescue professionals, stagehands, fishermen, linemen and surgeons. The International Guild of Knot Tyers is a dedicated to the Promotion of Knot tying. Truckers in need of securing a load may use a truckers hitch, Knots can save spelunkers from being buried under rock. Many knots can also be used as tools, for example, the bowline can be used as a rescue loop. The diamond hitch was used to tie packages on to donkeys. In hazardous environments such as mountains, knots are very important, note the systems mentioned typically require carabineers and the use of multiple appropriate knots. These knots include the bowline, double figure eight, munter hitch, munter mule, prusik, autoblock, thus any individual who goes into a mountainous environment should have basic knowledge of knots and knot systems to increase safety and the ability to undertake activities such as rappelling. Knots can be applied in combination to produce objects such as lanyards. In ropework, the end of a rope is held together by a type of knot called a whipping knot. Many types of textiles use knots to repair damage, macrame, one kind of textile, is generated exclusively through the use of knotting, instead of knits, crochets, weaves or felting. Macramé can produce self-supporting three-dimensional textile structures, as well as flat work, Knots weaken the rope in which they are made. When knotted rope is strained to its point, it almost always fails at the knot or close to it. The bending, crushing, and chafing forces that hold a knot in place also unevenly stress rope fibers, the exact mechanisms that cause the weakening and failure are complex and are the subject of continued study

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Mathematics and architecture
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Mathematics and architecture are related, since, as with other arts, architects use mathematics for several reasons. In Ancient Egypt, Ancient Greece, India, and the Islamic world, buildings including pyramids, temples, mosques, palaces, in Islamic architecture, geometric shapes and geometric tiling patterns are used to decorate buildings, both inside and outside. Some Hindu temples have a structure where parts resemble the whole. In Chinese architecture, the tulou of Fujian province are circular, in the twenty-first century, mathematical ornamentation is again being used to cover public buildings. In the twentieth century, styles such as architecture and Deconstructivism explored different geometries to achieve desired effects. But, they argue, the two are connected, and have been since antiquity. A master builder at the top of his profession was given the title of architect or engineer, in the Renaissance, the quadrivium of arithmetic, geometry, music and astronomy became an extra syllabus expected of the Renaissance man such as Leon Battista Alberti. Similarly in England, Sir Christopher Wren, known today as an architect, was firstly a noted astronomer and they argue that architects have avoided looking to mathematics for inspiration in revivalist times. This would explain why in revivalist periods, such as the Gothic Revival in 19th century England, equally, they note that in reactionary times such as the Italian Mannerism of about 1520 to 1580, or the 17th century Baroque and Palladian movements, mathematics was barely consulted. In contrast, the revolutionary early 20th century movements such as Futurism and Constructivism actively rejected old ideas, embracing mathematics and leading to Modernist architecture. Towards the end of the 20th century, too, fractal geometry was quickly seized upon by architects, as was aperiodic tiling, to provide interesting, Architects use mathematics for several reasons, leaving aside the necessary use of mathematics in the engineering of buildings. Firstly, they use geometry because it defines the form of a building. Secondly, they use mathematics to design forms that are considered beautiful or harmonious, thirdly, they may use mathematical objects such as tessellations to decorate buildings. Fourthly, they may use mathematics in the form of computer modelling to meet environmental goals, the influential Ancient Roman architect Vitruvius argued that the design of a building such as a temple depends on two qualities, proportion and symmetria. Proportion ensures that part of a building relates harmoniously to every other part. Symmetria in Vitruviuss usage means something closer to the English term modularity than mirror symmetry, in his Basilica at Fano, he uses ratios of small integers, especially the triangular numbers to proportion the structure into modules. Thus the Basilicas width to length is 1,2, the aisle around it is as high as it is wide,1,1, the columns are five feet thick and fifty feet high,1,10. Vitruvius named three qualities required of architecture in his De architectura, c.15 B. C. firmness, usefulness and these can be used as categories for classifying the ways in which mathematics is used in architecture

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Geodesic dome
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A geodesic dome is a hemispherical thin-shell structure based on a geodesic polyhedron. The triangular elements of the dome are structurally rigid and distribute the stress throughout the structure. A first, small dome was patented, constructed by the firm of Dykerhoff and Wydmann on the roof of the Zeiss plant in Jena, a larger dome, called The Wonder of Jena opened to the public in July 1926. Some 20 years later, R. Buckminster Fuller named the dome geodesic from field experiments with artist Kenneth Snelson at Black Mountain College in 1948 and 1949. Although Fuller was not the inventor, he is credited with the U. S. popularization of the idea for which he received U. S. patent 2,682,23529 June 1954. Howard of Synergetics, Inc. and specialty buildings like the Kaiser Aluminum domes, auditoriums, weather observatories, the dome was soon breaking records for covered surface, enclosed volume, and construction speed. Marines experimented with helicopter-deliverable geodesic domes, a 30-foot wood and plastic geodesic dome was lifted and carried by helicopter at 50 knots without damage, leading to the manufacture of a standard magnesium dome by Magnesium Products of Milwaukee. The dome was introduced to an audience as a pavilion for the 1964 Worlds Fair in New York City designed by Thomas C. Howard of Synergetics. This dome is now used as an aviary by the Queens Zoo in Flushing Meadows Corona Park after it was redesigned by TC Howard of Synergetics, another dome is from Expo 67 at the Montreal Worlds Fair, where it was part of the American Pavilion. The structures covering later burned, but the structure still stands and, under the name Biosphère. In the 1970s, Zomeworks licensed plans for structures based on other geometric solids, such as the Johnson solids, Archimedean solids and these structures may have some faces that are not triangular, being squares or other polygons. In 1975, a dome was constructed at the South Pole, on October 1,1982, one of the most famous geodesic domes, Spaceship Earth at the EPCOT Center in Walt Disney World, opened. The building is Epcots icon, and is included in the parks logo. In the year 2000 the worlds first fully sustainable geodesic dome hotel, the hotels dome design is key to resisting the regions strong winds and is based on the dwellings of the indigenous Kaweskar people. Wooden domes have a hole drilled in the width of a strut, a stainless steel band locks the struts hole to a steel pipe. With this method, the struts may be cut to the length needed. Triangles of exterior plywood are then nailed to the struts, the dome is wrapped from the bottom to the top with several stapled layers of tar paper, in order to shed water, and finished with shingles. This type of dome is called a hub-and-strut dome because of the use of steel hubs to tie the struts together

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Islamic architecture
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Islamic architecture encompasses a wide range of both secular and religious styles from the foundation of Islam to the present day. What today is known as Islamic architecture was influenced by Persian, Roman, Byzantine, further east, it was also influenced by Chinese and Indian architecture as Islam spread to Southeast Asia. The principal Islamic architectural types are, the Mosque, the Tomb, the Palace, from these four types, the vocabulary of Islamic architecture is derived and used for other buildings such as public baths, fountains and domestic architecture. Symbolic views of scholars on Islamic architecture have consistently been criticized by historians for lacking historical evidence. The Dome of the Rock in Jerusalem is one of the most important buildings in all of Islamic architecture and it is patterned after the nearby Church of the Holy Sepulchre and Byzantine Christian artists were employed to create its elaborate mosaics against a golden background. The great epigraphic vine frieze was adapted from the pre-Islamic Syrian style, the Dome of the Rock featured interior vaulted spaces, a circular dome, and the use of stylized repeating decorative arabesque patterns. Desert palaces in Jordan and Syria served the caliphs as living quarters, reception halls, and baths, the horseshoe arch became a popular feature in Islamic structures. After the Moorish invasion of Spain in 711 AD the form was taken by the Umayyads who accentuated the curvature of the horseshoe. The Great Mosque of Damascus, built on the site of the basilica of John the Baptist after the Islamic invasion of Damascus, certain modifications were implemented, including expanding the structure along the transversal axis which better fit with the Islamic style of prayer. The Abbasid dynasty witnessed the movement of the capital from Damascus to Baghdad, the shift to Baghdad influenced politics, culture, and art. The Great Mosque of Samarra, once the largest in the world, was built for the new capital, other major mosques built in the Abbasid Dynasty include the Mosque of Ibn Tulun in Cairo, Abu Dalaf in Iraq, the great mosque in Tunis. Abbasid architecture in Iraq as exemplified in the Fortress of Al-Ukhaidir demonstrated the despotic, the Great Mosque of Kairouan is considered the ancestor of all the mosques in the western Islamic world. Its original marble columns and sculptures were of Roman workmanship brought in from Carthage and it is one of the best preserved and most significant examples of early great mosques, founded in 670 AD and dating in its present form largely from the Aghlabid period. The Great Mosque of Kairouan is constituted of a square minaret, a large courtyard surrounded by porticos. The Great Mosque of Samarra in Iraq, completed in 847 AD, the Hagia Sophia in Istanbul also influenced Islamic architecture. When the Ottomans captured the city from the Byzantines, they converted the basilica to a mosque, the Hagia Sophia also served as a model for many Ottoman mosques such as the Shehzade Mosque, the Suleiman Mosque, and the Rüstem Pasha Mosque. Domes are a structural feature of Islamic architecture. Domes remain in use, being a significant feature of many mosques, the distinctive pointed domes of Islamic architecture, also originating with the Byzantines and Persians, have remained a distinguishing feature of mosques into the 21st century