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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.'"
I hope to be like him when I get to be that old. In case any of you haven't heard of Mandelbrot, you should take a look here.
Mandelbrot fractal sets are cool, but I think the first fractal discovered should be considered phi, aka the Golden Ratio. It may not be derived from the same mathmatics, but the end result is the same...
I guess no one ever learned how to make a fractal equation that looked like a given image on the fly.
sigs, as if you care.
I wrote my first Mandelbrot set explorer on an Atari 800. :-) Yeah... fractal exploration in interpreted BASIC at 1.79 Megahertz. Good times.
SLOW times, but good times.
Fuck, I feel old. :-(
--- Ban humanity.
note to mods (and people scratching their heads): this is funny (or trying to be) because the mandelbrot set is generated by a function over the complex plane, which has one axis of real numbers, and one axis of the "imaginary" numbers, multiples of i=sqrt(-1).
"A witty saying proves nothing." ~Voltaire
"d'Oh!" ~Homer
Gaston Julia, from circa 1920, investigated fractals before Mandelbrot. His work is the basis of Mandelbrot sets as the points in the Mandelbrot set are exactly those parameters for the corresponding Julia sets that are connected. If anyone should attribute fractals to any one man, Julia is more pronounced than Mandelbrot. Granted, Mandelbrot popularized fractals but the analysis stems from Julia's work.
Wow, I fondly remember the days when I, as a wide-eyed six year old, typed in a Mandlebrot-graphics generation program from Compute! magazine into my Commodore 64.
My friends didn't get it. But I loved it. It made a great backdrop to leave on the screen while I did other, more "normal" kid things. (Legos, drawing, etc.)
Now that I appreciate the mathematics behind it, I must give my respect to the man. Thanks for the childhood brain food, Mandlebrot, even if I didn't get it at the time.
I hope the land around you yields, a crop like all the other fields, and then your waiting might make sense...
New Scientist: How did you feel when you discovered it?
Mandelbrot: Its astounding complication was completely out of proportion with what I was expecting. Here is the curious thing: the first night I saw the set, it was just wild. The second night, I became used to it. After a few nights, I became familiar with it.
I wonder what he means by "saw" it.
What graphics computers were popular in the 1940's?
Q:Fractals seem to appear all over nature and in economics. Even the internet is fractal. What does that say about the underlying nature of these phenomena?
A:Well, it depends on the field. Circles and straight lines also appear everywhere. Does this mean that all those phenomena have something in common? Of course not. The roughly circular trajectory of a planet around the sun is due to gravitational interactions. Berries are round because a sphere has a smaller skin. The beauty of geometry is that it is a language of extraordinary subtlety that serves many purposes.
Q:So fractals don't point to a single rule underlying reality?
A:There is no single rule that governs the use of geometry. I don't think that one exists.
----
If I believed in a God, I'd say God bless Mr Mandelbrot. As it is, I'll just say, "Damn skippy."
I suppose it's not right that i'm more irritated about the new-age whackos who think fractals really *MEAN* something than the guy who invented the Mandelbrot set is.
(Invented? Discovered? Well, whatever, you know what I mean.)
Now I've got a nice little quote of The Man Himself telling them all they're f-ing idiots.
I LOVE THIS MAN!
I like the work the guy has done in the past, but I sometimes I'm dismayed by a little too much self-promotion by academics these days. Recall in his open letter in Wired:
Wired article
Here he mentions the need to conduct fundamental research, which I applaud, but he fails to mention that many, many people are already doing this, and has come across as championing an idea which has already been pursued for decades. If there's one thing I know about life, it's that people with money will almost always do their best to make more of it, and that includes learning how to use the market via financial research. Most mathematically inclined graduate students in Mandelbrot's own university, Yale, go on to financial research.
It reminds me a little of another widely regarded expert, David Gelernter, who has published lots of grandoise nonsense which are devoured readily by people who don't stop to think about what is actually said. For example, in his article about the future ("The Second Coming: A Manifesto"), he says at one point:
"Everything is up for grabs. Everything will change. There is a magnificent sweep of intellectual landscape right in front of us."
Well, that's nice. What's it mean? Perhaps I shouldn't fault the researchers, since getting your name out there seems to be the only way to attract lots of research funds, but every once in a while, it'd be nice to see someone slightly in touch with reality talk about what they want to do and why.
In the article, Mandelbrot says it's simple to understand how some spaces can be more empty than others, once it is explained. Can someone explain it?
The interview reminds me of an old joke that a "mandelbrot" would become a standard unit for measuring ego. Like Farad, one Mandelbrot would be a very large amount of ego, in common usage you would typically see pico- and micro-mandelbrots.
jeff
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))
I spent a significant amount of time in highschool playing with a mandelbrot program and color cycling. In this time, I fell into a trance, and lost a good 4 hours of my life.
When do you plan on giving me these hours of my life back?
*hypnotised by color cycling mandelbrot sets*
*drooool*
no
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)
I had a friend at the University that made a postscript program that would print a mandelbrot set.
He sent the file to be printed to the laser printer in the mac lab (the original apple laser writer).
And then nothing.
And then nothing.
13 hours later it printed a mandelbrot picture at the very highest resolution.
Pretty cool.
The Internet is full. Go Away!!!
i won 1st prize in the connecticut science fair computer division based on my work doing that and john conway's game of life in assembly language on the trs-80 color computer!
based on that success, i was accepted into yale university
where i met benoit mandelbrot in person... he was on the faculty and still is i believe... 17 year old awe...
this is all for real!
dude, memories of plugging in the assembler cartridge... i had one of those 4 cartridge switchers, so i could also run lode runner and the speech synthesizer LOL
intellectual property law is philosophically incoherent. it is your moral duty to ignore it or sabotage it
God damn, this is turning into an accursed homework assignment entirely too quickly for my liking. What's next, compare and contrast?
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.
<my guess>
Space has dimensionality; a plane has 2 dimensions, a cube exists in 3, hypercube 4... the numbers here are positive. Mandelbrot said he was using negative dimensions to measure "emptiness". He mentions that only one set is considered "empty" (I presume the null set). My guess (and I only minored in math so don't go betting on this) is that a negative dimension is to a positive dimension what a negative number is to a positive one. I'm thinking that if an object existed in -2 dimensions, it would be capable of having negative area. If you could add that object to an object with positive area, you'd reduce the second object's area.
</my guess>
Here's Mandelbrot's homepage at Yale.
Here's more links.
"A witty saying proves nothing." ~Voltaire
"d'Oh!" ~Homer
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?
Play Command HQ online
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
Costner = Bobcat * $1,000,000 to sign for a movie.
Which is really not normalized very well since Costner measures several dozen Costners himself.
Play Command HQ online
Could most kids today get their PS2 to draw a mandelbrot set? Does Windows XP provide the tools to acquire and use this knowledge? No.
Like tinyurl, but one letter less! http://qurl.co.uk/
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
I'm still in therapy...
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.
No article about fractals could be complete without mentioning Elenas excellent ZonXplorer fractal package for AmigaOS 3.5+ and MorphOS (running on the Pegasos PPC). Check out her stuning pictures in her gallery.
I hope her webpage can handle the load, it's sure enough worth a visit.
At UC Berkeley, back in 1990, you told a great story of you and your wife attending a movie premiere which used a fractal landscape effect they'd hired you to produce. (please forgive my repeating old family gossip, especially if I've misremembered the details :) As I recall, it took longer to generate than the producer's patience lasted, so they cropped it rather than wait for its last triangle to completely render. Your wife hadn't heard about the "shortcut", but when your effect came onscreen, she gave you a big pinch. After the movie ended, you asked her what was wrong, and she said, in effect, "That's not a fractal!" - apparently she could recognize even partial fractals as incomplete, therefore nonfractal.
Have you learned more about any other fractal recognition, either people or artificial (eg. software)? Identifying fractals, fractal metrics, noniterative predictions, comparisons without analysis... Have you heard about the recently published African Fractals, a scientific investigation of fractal "sensibility" in traditional African designs, both unconscious and explicit? Do you think human fractal recognition and execution can inform our computer science investigations of this geometry? Perhaps the popularization of fractals in European-rooted design might influence our modern global culture as deeply as it seems to have influenced culture in Africa?
--
make install -not war