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Interview With Math Legend Benoit Mandelbrot
Vertigo01 writes "New Scientist is currently featuring an interview with Benoit Mandelbrot the father of the Mandelbrot set, and the man who discovered fractals. 'What motivates me now are ideas I developed 10, 20 or 30 years ago, and the feeling that these ideas may be lost if I don't push them a little bit further.'"
then you should loak at this and this and this and ...
The first version of the Mandlebrot set was printed on a flat bed plotter in the 60's, if I remember my history correctly.
The printouts are reproduced in a book, but I don't recall which one. Might be in Mandelbrot's own book.
I *think* this might be one: http://coco.ccu.uniovi.es/geofractal/capitulos/01/ imagenes/MandelbrotOriginal.gif
--- Ban humanity.
If anyone is interested, a great book on the subject is Peitgen and Richter's The Beauty of Fractals. It presents a good mathematical background, but it also has tons of pictures demonstrating the math.
(S(SKK)(SKK))(S(SKK)(SKK))
Printed out on a teletype terminal at 132x66 if I remember correctly from the SciAm article.
SMQ 90AE4B2BC4F6BEAF7340F0B40BA2DEF7340F6BC2D0392
It is called wavelets and people are beginning to apply them to video and audio compression. Its tricky stuff though. The neat thing is that unlike FFT, these things operate on equations that tend to zero at plus/minus infinity. That may not seem like a big deal, but it tells you a lot about how good your approximation is and how many more calculations you should do before it is good. It is a very interesting concept - I wish I had learned more about them in my DSP class.
I submitted this story last night, and it didn't get posted.
The most interesting part of Mandelbrot's work revolved around the Hausdorff Dimension, which was a way to describe geometry using a real number as opposed to the integers of Euclidian geometry.
I admit I never understood all of the (somewhat convoluted) description Mandelbrot gave in "Fractal Geometry of Nature", but it seemed to boil down to the idea that you could get rid of infinities and zeros if you allowed fractions of a dimension.
ie: A coastline has an infinite length, if you measure it in just one dimension, and zero area if you measure it in two, but a finite value that you can usefully compare to other objects if you use a dimension between 1 and 2.
IIRC, the Hausdorff Dimension is calculated by measuring the object at different scales. You then took the ratio of the change in scale and the change in measured length. As you went to finer and finer scales, this ratio tends to a limit, which is always equal to or greater than the Euclidian dimension and always strictly less than the Euclidian dimension plus 1.
Where the Hausdorff Dimension is a value strictly greater than the Euclidian dimension, the object is considered a fractal. Fractals are never "random", they are always self-similar. That appears to be a universal law, though I've yet to see a clear explanation as to why.
Another interesting characteristic is that self-similarity does not occur at random intervals. The ratio between the intervals is always an integer multiple of the Feigenbaum Number.
The Feigenbaum Number is itself interesting. It was first observed by Michael Feigenbaum, when he examined chaotic systems that were in an oscillating state. (Chaotic systems, when given insufficient initial conditions to become chaotic will oscillate.) As you increase the inputs, the oscillations exactly double. They don't change smoothly.
The ratio of the change in inputs necessary to double the oscillations is the same between all doublings and between all chaotic systems. This ratio is the Feigenbaum Number. Many properties of chaos and fractals are tightly bound to this value.
The Feigenbaum Number is considered evidence that chaos is not so much a property of the system, but rather that chaos and fractals are the more universal/abstract and the systems are merely products.
It's a small world and it smells funny; I'd buy another if it wasn't for the money; Take back what I paid (SoM)
The original name should have been David Benoit.
Hope that clears things up for you.
And if you RTFA you'd see: "The Mandelbrot set is the modern development of a theory developed independently in 1918 by Gaston Julia and Pierre Fatou. Julia wrote an enormous book - several hundred pages long - and was very hostile to his rival Fatou. That killed the subject for 60 years because nobody had a clue how to go beyond them. My uncle didn't know either, but he said it was the most beautiful problem imaginable and that it was a shame to neglect it. He insisted that it was important to learn Julia's work and he pushed me hard to understand how equations behave when you iterate them rather than solve them. At first, I couldn't find anything to say. But later, I decided a computer could take over where Julia had stopped 60 years previously."
Q:How many libertarians does it take to stop a Panzer division? A:None. Obviously market forces will take care of it.
OK... if you remember way back when to vector spaces, for a given space, there are lots of "bases" (plural of basis), minimal sets of vectors that collectively "span" the space, i.e. pick any vector in the space and I can hand you a weighted sum of vectors in the basis that adds up to the vector you picked.
OK... now, let's go on to vector spaces (or is this that further generalization thereof, namely Hilbert spaces?) where the "vectors" are functions! Those have bases, too. For functions with a particular period (i.e. there's some number p such that for any x and any integer k, f(x + kp) = f(x)), you can finagle {sin kx, cos kx | k in N} to maneuver the period from 2 * pi to p and position it appropriately so that they form a basis for that space of functions. ("My photo of Aunt Sarah isn't periodic!" you say? Then we pretend it's periodic, i.e. it infinitely repeats like a Warhol Marilyn Monroe, and just never show the repetitions.)
Here's the trick: if you can arrange your basis so that those weights (remember the weighted sum?) get smaller and smaller as you go on, you can do lossy compression by throwing away all the terms past a certain point.
People did it with Chebyshev polynomials to get decent results for power series approximations (at a cost of spreading around the error) with fewer terms, and you can do it with {sin kx, cos kx | k in N}, because as k gets bigger, sin kx and cos kx wiggle faster and faster, and most pictures don't look like Moire patterns or op art. (The reason that you don't want JPEG for line art is that sharp edges are guaranteed to require lots of terms, so they're guaranteed to look bad when you leave them out.)
fractal image compression is a separate and distinct technique from wavelet transforms. I do recall that there was a company called Iterated Systems that had a browser plugin for viewing their proprietary image filetype. It looks like they've dropped off the face of the planet. Anyway, here's a nice bibliography on the subject.
The "cue the foo posts in 3, 2, 1..." posts will commence with no subsequent foo posts in 3, 2, 1...
I guess no one ever learned how to make a fractal equation that looked like a given image on the fly.
I may be mistaken, but I think somebody did, and called it JPEG.
JPEG and fractal compression are completely different, I'm afraid.
JPEG transforms blocks of the image from the spatial domain to the frequency domain, and keeps only the strongest spatial frequencies. To look at it another way, it tries to express each block as the sum of various functions that look like bands or ripple patterns.
Fractal compression tries to find similarities between different parts of the image, and to express the image as a bunch of these similarity relations (affine transforms, or different types of mapping).
There's more detail for each type of algorithm, but that's the basic approach for each. Some versions of fractal compression to a frequency transform of blocks during the compression stage, but that's just to make it easier to compare blocks to each other when sifting possibilities, as opposed to part of the mechanism of compression itself.
It is called wavelets
Actually, no.
Wavelet transforms involve expressing the input data as the sum of wavelet basis functions (much as a Fourier transform uses sine/cosine waves).
Fractal compression involves looking for self-similar features in the image itself, removing this redundancy by expressing it as a series of affine transformations, or something similar.
Frequency- and wavelet-transforms can make the search for self-similar structures easier, but they represent fundamentally different approaches (the best you can do to draw an analogy is to consider fractals to be a different type of parameterized basis function that you're doing a transform with).
Comeon mods, it's the Cantor set, it's self similar, get it?
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Mandelbrot is not the inventor of fractals!
Three people whose work on fractals predated Mandelbrot's by some time, and IMNSHO was infinitely more impressive because it was done without the help of computers, are Felix Hausdorff, inventor of the Hausdorff dimension, Georg Cantor, inventor of the fractal Cantor "middle thirds" Set, and Gaston Julia, who discovered/invented the Julia Set, to which the Mandelbrot Set is closely related.
Think about how amazing the work of these three mathematicians was, given that they, unlike Mandelbrot, didn't have computers to iterate maps or visualize sets, and yet they were able to characterize these sets, including their fractal nature. I find Julia's accomplishment especially impressive.
Mandelbrot is better than these three at self-promotion. When he fiddled a bit with the Julia Set and produced a new set from it, he called it the "M Set" in his work, and waited for somebody else to fill in the remaining 9 letters after "M."
There was a joke among physicists messing around with fractal stuff in the late 1980s that while the most common letter in the English language is "e," the most common letter in Mandelbrot's work was either "I" or "M" (the probable winner, given that "me," "my," "mine," and "Mandelbrot" all begin with "M").
That said, Mandelbrot's work was interesting, and he did acknowledge Julia's work in his own. After all, the Mandelbrot Set is a map where each point on the complex plane represents a Julia Set, where the points inside the Mandelbrot Set represent connected Julia Sets and the points outside represent disconnected Julia Sets. And Mandelbrot took advantage of the computer technology available to him to plot some of these sets, giving us visual representations of these things. But to give him credit for inventing fractals is unfair to the great mathematicians who worked on fractals long before Mandelbrot.
--Mark
"It is nice to know that the computer understands the problem. But I would like to understand it too." --Eugene Wigner
Z_ doesn't have to approach zero to be in the set... it can also settle down to a finite value, or cycle between 2 or more values, or even jump around randomly within a range of values until you hit your iteration maximum.
All you can safely say is that if the absolute value of Z_ gets above a certain value (4) then it will approach infinity, and that value is NOT in the set.
-CausticPuppy "Of all the people I know, you're certainly one of them." -Somebody I don't know
Mandelbrot gives Gaston Julia proper attribution in TFA. But it took this extraordinary man to bring new life to this field.
I think the first fractal discovered should be... the Golden Ratio. It may not be derived from the same mathmatics, but the end result is the same
Although fractals are self-similar, a self-similar pattern isn't necessarily fractal. Golden spirals/rectangles/triangles aren't fractal because they can be described using classical geometry.
For a detailed breakdown of such distinctions, see Manfred Schroeder's Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise.