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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 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)
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.)