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Mathematics | Perspective Time | Space DESMA 9: Art, Science and Technology "“Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.” - Benoit Mandelbrot TODAY 1/13/09: • • • • • ZERO Perspective Escher Buckminster Fuller Mandelbrot ZERO "zero" derives ultimately from the Arabic sifr which also gives us the word "cipher" Around 700 BC, uses three hooks to denote an empty place in the positional notation. On a tablet found at Kish, an ancient Mesopotamian city located east of Babylon in what is today south-central Iraq. Not used as a number but a punctuation mark Greek astronomers began to use the symbol O. Some historians favor the explanation that it is omicron, the first letter of the Greek word for nothing -- "ouden". Zero appeared in India and spread to China and Islamic countries. It was not until the 1600’s the zero was used in the West, after encountering a lot of resistance! ZERO during the Middle Ages, zero was disparaged as a mark of infidel sorcery, the sign of the Devil himself, the canceller of all meaning. For the Mayans, Zero was the Death God among their lords of the underworld, and men adopting the persona of Zero were ritualistically sacrificed in hopes of staving off the day of zero, the time when time itself would stop. Only much later was zero reinterpreted as a symbol of God's power to create a lot out of naught. Nothingness or Emptiness “zero = infinity”, as alluded to in the Upanishads and reflected in the Sanskrit word “poornam” which means both “zero” and “full” Y2K bug: zero causes problems even recently! Recently many people throughout the world celebrated the new millennium on 1 January 2000. Of course they celebrated the passing of only 1999 years since when the calendar was set up no year zero was specified. Although one might forgive the original error, it is a little surprising that most people seemed unable to understand why the third millennium and the 21st century begin on 1 January 2001. PERSPECTIVE al-Haytham Al Haytham, around 1000 A.D. gave the first correct explanation of vision, showing that light is reflected from an object into the eye. He studied the complete science of vision, called perspectiva in medieval times, and although he did not apply his ideas to painting, the Renaissance artists later made important use of his study. Duccio: first attempt Annunciation 1316 13th century: GIOTTO The Visitation Not a precise formulation – Intuitive approach Albrecht Dürer (German, 1471-1528), Melancholia 1, 1514 BRUNELLESCHI: Credited with the first correct formulation of linear perspective, about 1413 His perspective paintings were lost but a “Trinity” fresco by Massacio from this same period still exists. He used Brunelleschi’s perspective principles. VANISHING POINT Developed by Brunellesci, Massacio preserved Alberti described Alberti’s treatise “On Painting”, 1435 Written in Latin for scholars and Italian for the general audience Perspective is necessary in order to understand painting. It is completely mathematical, concerning the roots in nature from which arise this graceful and noble art…. A painting is the intersection of a visual pyramid at a given distance, with a fixed centre and a defined position of light, represented by art with lines and colours on a given surface. If you place the intersection one metre from the eye, the first object, being four metres from the eye, will diminish by three-quarters of its height on the intersection; and if it is eight metres from the eye it will diminish by seven-eighths and if it is sixteen metres away it will diminish by fifteen-sixteenths, and so on. As the distance doubles so the diminution will double. Piero de la Francesca – leading artist of his time, also leading Mathematician in 15tn century Rennaisance Piero’s illustration of a dodecahedron Leonardo da Vinci Mathematics & Art Fused into a Single concept GOLDEN RATIO "Vitruvian Man" by Leonardo da Vinci Gallerie dell'Accademia, Venice, Italy ARCHITECTURE The Parthenon, Athens, superimposed with a diagram demonstrating the use of the Golden Mean in the design Architect Charles E. Jeanneret, known as "Le Corbusier" (French, 1887-1965), often used golden rectangles in his designs for buildings. One of these is the United Nations building in New York. The proportions of its vertical and horizontal dimensions correspond to the Golden Mean. Golden ratio and Hollywood Phi: Nature of Growth DIMENSIONALITY: MOVING AWAY FROM FLATLAND Maurits Escher In a lecture in 1953, Escher said: I have often felt closer to people who work scientifically (though I certainly do not do so myself) than to my fellow artists. Circle Limit III In 1958 he met mathematician Coxeter and they became life-long friends. In 1995, Coxeter published a paper which proved that Escher reached mathematical perfection in one of his etchings. [Escher] got it absolutely right to the millimetre, absolutely to the millimetre .... Unfortunately he didn't live long enough to see my mathematical vindication. In mathematical quarters, the regular division of the plane has been considered theoretically. ... [Mathematicians] have opened the gate leading to an extensive domain, but they have not entered this domain themselves. By their very nature they are more interested in the way in which the gate is opened than in the garden lying behind it. Buckminster Fuller Octet Truss Tensegrity Geodesic dome How they work Synergetics Artists influenced by Buckminster Fuller: John Cage Joseph Albers Kenneth Snelson Others impressed by him Bucky class we Elaine de Kooning and Joseph Albers, Black Mountain college, 1948 Piet Mondrian 1942 1926 Mandelbrot: The Psychedelic world of FRACTALS Fibonacci, Fractals and Financial Markets - Socionomics.net Contemporary Artists using Mathematics http://www.sciencenews.org/articles/20061118/mathtrek.asp http://www.sciencenews.org/articles/20060506/mathtrek.asp Sol Lewitt John Sims Casey Reas Mark Pomilio MUSIC & MATHEMATICS MATH ROCK – Gil Kuno presents on Wednesday Yoshida Tatsuya, one of the originators of the RUINS and the cited as the "indisputable master drummer of the Japanese underground" On campus: IPAM In the city: