Benoît Mandelbrot

RImaginary numbers are used to represent the 2nd dimension. The x-axis is the real part, and the y-axis is the imaginary part.The iteration is Z = Z^2 + C. Z is a set of numbers, but not the "mandelbrot set".The mandelbrot set is actually the set of C's, where "iteration is bounded". For example, if C=1, your set of Z = {0,1,2,5,26...}; if C=i, your set of Z = {0,i,-1+i,-i,-1+i,-i...}. Notice for C=i, the series bounces back and forth between -i and -1+i. This means that C=i leads to a bounded set, so i is in the mandelbrot set. On the other hand, C=1 does not lead to a bounded set, as the set goes to infinity, so 1 is not in the mandelbrot set.If you plot all the values of C that are bounded in the complex coordinate system (y=imaginary, x=real), you'll get the "core" of the mandelbrot fractal; i.e. the black part. For example, C=i (real part = 0, imaginary part = 1) would be plotted as x=0, y=1; and C=1 would be plotted as x=1, y=0.When they add the pretty colors, they are taking points that are not part of the mandelbrot set, and coloring them based on how many interations before Z > 2. (For the example of C=1, Z={0,1,2...}, then the color would be based on three iterations, a color would have to be assigned for each positive integer 1, 2, 3, etc).

RNot true. The idea of fractals has been around since Archimedes (at the latest). How else do you think he managed to get a good approximation of pi? Similarly with defining the golden ratio (Phi).In the meantime, try these:http://www.skytopia.com/project/imath/imath.htmlNote: dividing by zero is NOT a problem - there's just no proper answerwriting out a full version of pi is not impossible - it's just not having enough time to do itAs for primes.... we do know that 2^n - 1 (merseme prime approach) is often a prime...

RMan·del·brot set : (mändl-brǒt) http://education.yahoo.com/reference/dictionary/entry/Mandelbrot+setEdit: After Benoit B. Mandelbrot (born 1924), Polish-born American mathematician.

RHe just took his first dip in the Beer Volcano, and afterward he's going to team up with Einstein to dunk Euclid.

RIn 1958 Mandelbrot moved to the U.S. to take up a research position at IBM's Thomas J. Watson Reasearch Center. It was here that he was asked to tackle the problem of line noise. The reigning joke was that the noise was generated by a "guy with a screwdriver" fiddling with some piece of connected equipment. Meanwhile engineers sought to solve the problem by increasing signal strength to drown out the noise. But Mandelbrot's eventually showed that the noise was both consistent and erratic, some kind of inescapable natural feature of the system that did not disappear with increased signal strength. But more remarkably he also showed that every burst of noise also contained within it bursts of clear signal (a situation he conceived of in terms of the Cantor set). Stranger still, he found that the ratio of periods of noise to periods of clean transmission remained constant, regardless of the scale of time used to plot the phenomenon (i.e. months, days, seconds).In 1961, while working in economics, Mandelbrot traveled to Harvard to give a talk on income distribution. As he entered the lecture hall he found that Hendrik Houthakker, who had invited him, had apparently already written the chart Mandelbrot would use to illustrate his lecture on the chalkboard. But Houthakker explained that the chart was actually a description of the results of his own research into the fluctuations in the cotton market. How was this possible?Mandelbrot returned home and turned his attention to analyzing cotton prices. Using records dating back to 1900, he began to perceive an astonishing pattern -- one that hearkened back to his work on line noise a decade earlier. He discovered that cotton prices followed a pattern that was both erratic and regular. That is, although price changes were erratic in terms of normal distribution and no one could predict the exact amount of any particular price change, the changes themselves followed a symmetrical pattern with regards to scaling. Regardless of whether the scale of time was hourly, daily, or monthly, the curve was the same. And it had remained so for at least 60 years (the length of time covered by his records).

RHere's the Wikipedia page on vexel vs. vector. http://en.wikipedia.org/wiki/Vexel Vector art is usually created in programs like Adobe Illustrator. Vectors are mathematically generated by the program and are RESOLUTION INDEPENDENT. This means you can create them at one size and resize them without loss of image quality.Pixels are the dots on your screen, and are the dots that make up the image in a photograph or any image generated by programs like Adobe Photoshop. Pixels very much depend on resolution: if you try to enlarge a small pixel-based image, all you are doing is making the pixels bigger, so the image loses quality. Fractals are another mathematical based art. There are various programs, both free and for pay that generate fractals. Fractals appear in nature, and were first described by Benoit Mandelbrot. One of the most famous fractal sets is his Mandelbrot Set, and once you see it, you will probably recognize it because it has been used in all sorts of art. Here's the Wikipedia page on fractals:http://en.wikipedia.org/wiki/FractalsMake sure you scroll through the page because there are some very interesting images, including one of my favorite examples of fractals that occur in nature, the romanesco broccoli.

RI would have a look at ........... the Mandelbrot set, named after Benoît Mandelbrot, is a set of points in the complex plane, the boundary of which forms a fractal. Mathematically the Mandelbrot set can be defined as the set of complex values of c for which the orbit of 0 under iteration of the complex quadratic polynomial zn+1 = zn2 + c remains bounded. That is, a complex number, c, is in the Mandelbrot set if, when starting with z0=0 and applying the iteration repeatedly, the absolute value of zn never exceeds a certain number (that number depends on c) however large n gets.For example, letting c = 1 gives the sequence 0, 1, 2, 5, 26,…, which tends to infinity. As this sequence is unbounded, 1 is not an element of the Mandelbrot set.On the other hand, c = i (where i is the square root of -1) gives the sequence 0, i, (−1 + i), −i, (−1 + i), −i…, which is bounded and so i belongs to the Mandelbrot set.When computed and graphed on the complex plane the Mandelbrot Set is seen to have an elaborate boundary which does not simplify at any given magnification. This qualifies the boundary as a fractal.The Mandelbrot set has become popular outside mathematics both for its aesthetic appeal and for being a complicated structure arising from a simple definition. Benoît Mandelbrot and others worked hard to communicate this area of mathematics to the public.This is an amazing topic with plenty of illustrations.Mandelbrot himself is quite a character, suggest that you also write about him as part of your work.

RFractals in cinema and graphic designAfter Loren Carpenter, co-founder of Pixar Animation Studios, read Benoit Mandelbrot’s Fractals: Form, Chance, and Dimension, he began experimenting with fractals to make his computer graphics look more realistic. This technique gave rise to software programs now widely used across the computer graphics industry to create special effects, including fictitious landscapes and imaginary worlds—such as the Genesis planet sequence in Star Trek II: The Wrath of Khan and the damaged Death Star in Return of the Jedi.More at this interesting pagehttp://www-03.ibm.com/ibm/history/ibm100/us/en/icons/fractal/transform/Complex numbers and some related theorems are used in the amazing JPEG formatKey word is DCT, Discrete Cosine Transformhttp://www.image-engineering.de/index.php?option=com_content&view=article&id=522

ROff the top of my head, these names come to mind:Paul Erdös is widely written about, but I don't know of any specific works of his.Kurt Godel pioneered in some fundamental work about which problems can and cannot be solved.Wiles solving Fermat's Last Theorem certainly guarantees him everlasting fame, but that was just one problem, large and famous as it was.Einstein was a physicist, not a mathematician.From this page: http://web.centre.edu/mat/century.htmlwe get the following list, about half of which I am familiar with:Bourbaki Henri L. LebesgueRichard Courant Benoit MandelbrotKarmarkar Robert L. MoorePaul Erdös Emmy NoetherSir Ronald Fisher Karl PearsonKurt F. Gödel J. Henri PoincaréRichard W. Hamming John W. TukeyDavid Hilbert Alan M. TuringFritz John John von NeumannJohn Kemeny Hermann WeylDonald Knuth Norbert WienerAndrei N. Kolomogorov Andrew J. WilesIt is interesting to see some prominent computer science names there.Knuth is associated with computer science almost exclusively.Turing, Wiener, and von Neumann all came out of mathematics backgrounds and laid the foundations for computing.It's hard to pick just one name.

RA fractal is generally "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole," a property called self-similarity. The term was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured." A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion.A fractal often has the following features:It has a fine structure at arbitrarily small scales. It is too irregular to be easily described in traditional Euclidean geometric language. It is self-similar (at least approximately or stochastically). It has a Hausdorff dimension which is greater than its topological dimension (although this requirement is not met by space-filling curves such as the Hilbert curve). It has a simple and recursive definition.