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Science

Interview With Math Legend Benoit Mandelbrot 286

Posted by timothy from the strange-bread-loops dept.
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.'"
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Interview With Math Legend Benoit Mandelbrot More

Interview With Math Legend Benoit Mandelbrot

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  • Quote from TFA (Score:5, Insightful)

    by Meostro ( 788797 ) * on Friday November 12, 2004 @02:24PM (#10800957) Homepage Journal
    From TFA, a BRILLIANT! [guinness.com] quote from a fella who apparently enjoys being a crotchety old bastard:
    All my life, I have enjoyed the reputation of being someone who disrupted prevailing ideas. Now that I'm in my 80th year, I can play on my age and provoke people even more.
    I hope to be like him when I get to be that old. In case any of you haven't heard of Mandelbrot [google.com], you should take a look here [google.com].

    • Re:Quote from TFA (Score:5, Informative)

      by legrimpeur ( 594896 ) <legrimpeurNO@SPAMgmail.com> on Friday November 12, 2004 @02:36PM (#10801086)
      then you should loak at this [navi.cx] and this [bates.edu] and this [tky.hut.fi] and ...
    • He probably does enjoy it. I read his book when it first came out in the early 80's. The book was interesting and had beautiful color pictures, but was extremely difficult to read because of the overwhelming arrogance and self aggrandizement. It seemed like every other sentence was something like "We were the first in the world to recognize this" and "All those other smucks never noticed that" and "this would never have been discovered if it weren't for our overarching genius"... I found the mathematics

      • Re:Quote from TFA (Score:3, Interesting)

        by spacey ( 741 )
        Yeah, my uncle used to work with him. In those pretty IBM ads that featured some of the fractal work they were doing, IBM put Benoit in front of a screen with a bunch of pretty work my uncle was doing at the time. My uncle got no credit, of course.

        -Peter

        • Yeah, my uncle used to work with him. In those pretty IBM ads that featured some of the fractal work they were doing, IBM put Benoit in front of a screen with a bunch of pretty work my uncle was doing at the time. My uncle got no credit, of course.

          Yeah? Well, my aunt used to be his maid! She made his breakfast, combed his hair, and gave him all of his ideas. Not only did she teach him math when he was a kid, she walked 8 miles barefoot, in the snow, uphill both ways to do it. And did she get any cre

  • Discovered fractals? (Score:5, Interesting)

    by Superfreaker ( 581067 ) on Friday November 12, 2004 @02:32PM (#10801030) Homepage Journal
    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 think the first fractal discovered should be considered phi, aka the Golden Ratio"

      Do you mean the one in the Kabbalah?

    • A ratio by itself is not a fractal. That ratio can play into a lot of fractals and is seen in many places, but by itself is just a number.
      • A ratio by itself is not a fractal. That ratio can play into a lot of fractals and is seen in many places, but by itself is just a number.

        You can probably consider the shape made by continuing to divide a rectangle using the golden ratio a fractal, as it's definitely self-similar and based on an affine transformation.

        You'd have to do a bit of sleight-of-hand defining the boundary for it to actually meet the definition, though. If you just count the lines added at each iteration, it has a fractal dimensio

    • Self-similar != Fractal (Score:5, Informative)

      by base_chakra ( 230686 ) * on Friday November 12, 2004 @04:20PM (#10802134)
      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 [amazon.com].

  • Fractal compression (Score:3, Interesting)

    by RealProgrammer ( 723725 ) on Friday November 12, 2004 @02:32PM (#10801033) Homepage Journal
    I heard about this a long time ago. Did it ever go anywhere?
    fractal_compress(image) {
    generate fractal equation that 'looks like' portion of image;

    subtract the fractal from the image, leaving remainder;

    return (fractal plus fractal_compress(remainder));
    }

    I guess no one ever learned how to make a fractal equation that looked like a given image on the fly.

    • If I renember, there were sone very innovative things done with fractal immage compression, but it sorta dead-ended because of patent issues. see here [www2.vo.lu]
    • It is called wavelets [wikipedia.org] 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 can't remember where I read it, but I recall the main failure of fractal compression of video wasn't decompression, but in compression. There might be parameters for a fractal algorithm with results which might almost perfectly match the data, but finding parameters would involve a massive trawl through possible inputs until something vaguely suitable turns up.

      Basically, with no nicely predictable way of converting input into algorithm parameters, there wasn't much chance of encoding video in any vaguely
    • Here's your problem:

      error: `generate' undeclared (first use in this function)


      I think your C is a little shaky.

  • Tried to read it (Score:5, Funny)

    by HarveyBirdman ( 627248 ) on Friday November 12, 2004 @02:32PM (#10801038) Journal
    The interview was very complex, so I broke it down into sentences, but the sentences were as complex as the overall article. How could that be? So I broke it down into words, but still I found more complexity. Analyzing single characters simply brought out more detail. I zoomed into the pixels and whole worlds were unveiled. Where does it end?

    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. :-(

    • Re:Tried to read it (Score:4, Interesting)

      by qwijibo ( 101731 ) on Friday November 12, 2004 @02:38PM (#10801110)
      I remember typing that program in from one of the Antic magazines. Those were the good ol days. Between 1-2 days to generate each picture. Now we can do it in a matter of seconds on the average PC. Takes all the pride of accomplishment out of it when it's that simple.

      • Re:Tried to read it (Score:3, Interesting)

        by Ford Prefect ( 8777 )
        I remember typing that program in from one of the Antic magazines. Those were the good ol days. Between 1-2 days to generate each picture. Now we can do it in a matter of seconds on the average PC.

        Why not do it in real time [theory.org]? A fairly old program, with smooth zooming into various fractals. Worked well on an old Pentium, looks bloody amazing on a modern machine!

        Does various tricks to avoid calculating too much, and is rather clever about it...
      • Try aa-xaos -- now you can view fractals as 80x25 ASCII art. The past is now! Geezers unite!

    • Re:Tried to read it (Score:3, Interesting)

      by selderrr ( 523988 )
      I wrote my first fractal in 8-bit color, sucker ! On a MacII no less. In Lightspeed (what's in a name...) C (or was it MPW ? don't remember).

      The average calculation time was 15min per pixel if i recall correctly. I just left it running the whole weekend and then on monday had to abort it cause someone needed to print and the damd mac couldn't multitask properly (Finder 1.x or so... not even multifinder in those days)

      Damd those were the days... I recall spending a whole day trying to find a way to optimi
      • I wrote my first fractal in 8-bit color, sucker ! On a MacII no less.

        As long as we're having a pissing contest, I have code for Mandelbrot rendering on a TI-81 calculator kicking around ;). Took a couple of hours to render a 96x64 image taken to 32 iterations, if memory serves.

        I've been meaning to dust off that calculator for quite a while, now. Main problem is that it eats batteries for breakfast.
        • Yeah, same here. But I screwed up the code at first, and instead of getting the Mandelbrot set, I got the bifurcation diagram. Since I'd read Gleick's _Chaos_, I was still pretty excited, and screwed around w/ the bifurcation program for a while until coding the thing better.

          It used up batteries like a bitch, though (whatever that means), so I collected a bunch of old batteries and wired them in series, and just ran the calc off of those overnight. One screen per night was pretty cool.

          Loved coding f
        • As long as we're having a pissing contest, I have code for Mandelbrot rendering on a TI-81 calculator kicking around ;). Took a couple of hours to render a 96x64 image taken to 32 iterations, if memory serves.


          I've been meaning to dust off that calculator for quite a while, now. Main problem is that it eats batteries for breakfast.


          Well, of course. When you have a steam-driven accumulator it's going to take some energy to operate.

    • Re:Tried to read it (Score:4, Funny)

      by coupland ( 160334 ) * <dchase@ho[ ]il.com ['tma' in gap]> on Friday November 12, 2004 @02:46PM (#10801198) Journal

      Haha, I love it. When I read the first paragraph of your post I couldn't help but picture Calvin [calvinandhobbes.com] on one of his voyages of discovery while daydreaming in class. Tumbling through space as words zoom in on him and resolve into letters, then pixels, then photons...

    • The interview was very complex, so I broke it down into sentences, but the sentences were as complex as the overall article. How could that be?

      Yes, but was it complex, or merely complicated?

      I recently rewrote a quotation for why some work would cost a client more using a similar line of thought, swapping the word 'complicated' for 'complex', because it sounds so much more... Complex.

      It really brought a smile to my face when I saw a certain Mr. B. Mandelbrot essentially agreeing with my use of the Englis
    • I remember writing a Mandelbrot set generator on an 80386. All of the arithmetic was done on the 80387, using extended floats, and using the whole 8-value stack to avoid pulling and pushing from the FPU each loop. This made for extremely accurate fractals, but it was far from fast, even though I'd used fairly tight assembly.

      The graphics was done via super vga, poking values directly onto the card. The only "acceleration" I coded in was to use the mirroring across the y=0 axis on the Mandelbrot set. The se

  • sqrt(-1) (Score:5, Funny)

    by phyruxus ( 72649 ) <jumpandlink@yaho[ ]om ['o.c' in gap]> on Friday November 12, 2004 @02:34PM (#10801059) Homepage Journal
    ith post!

    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).

    • Re:sqrt(-1) (Score:4, Insightful)

      by MustardMan ( 52102 ) on Friday November 12, 2004 @02:40PM (#10801129)
      You know it really says something about the slashdot moderation system that you had to explain this joke, in fear that mods-on-crack without a clue would mod you down as offtopic or some other such nonsense. I have mod points right now, but decided to comment on the abysmal state of the mod system instead.

      • .

        I have mod points right now, but decided to comment on the abysmal state of the mod system instead.

        That's not necessarily "abysmalness". The mod system is simply an implementation of a rules based system that gains participation from unpaid participants to create community.

        The results of this system?

        Democracy, feedback, self-articulation

        ... there are many nouns that can be applied to the results. Abysmalness is yours.

        What changes would you make to the mod system?

        What would be the resu

      • I'm lmao at the score of my post right now.

        It started off at 2, +1 funny, -1 overrated. Current score 1. 2+1-1=1? :)
        Must be slashcode's way of dinging me for mentioning the root of a negative number :)

      • what if you could only spend mod points in stories in sections you had earned karma in?

        Eg, if you reaped a load of karma in IT, but none in politics, you could mod in IT, but not in politics?

        Alternatively, what if moderation wasn't anonymous, and your moderation showed up in your user page, as well as in the comment?

        I know I've wished for that on numerous occasions. (the second thing) I think either of these changes would make the moderation system hella better. Although I do like being able to moderate i

        • Non-anonymous moderation by people that have earned mod points in particular subsections sounds good. But with that system it might be best it was not completely left up to the person with a fist full of mod points to select the posts that they moderate.

          I think that a known mod system would devolve into clusters of moderatinf furballs, for better or worse. The reason being is that people will quite naturally pay attention to the people that are paying attention to them, rather than the discussion at large.


    • A basic axiom of a joke is that, if you have to explain it, it's not.

  • Julia (Score:5, Insightful)

    by Ann Coulter ( 614889 ) on Friday November 12, 2004 @02:34PM (#10801070)
    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.

    • Book (Score:5, Informative)

      by bsd4me ( 759597 ) on Friday November 12, 2004 @02:48PM (#10801227)

      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.

    • Re:Julia (Score:5, Informative)

      by jdcook ( 96434 ) on Friday November 12, 2004 @03:03PM (#10801367)
      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."

    • Re:Julia (Score:3, Informative)

      by Koyaanisqatsi ( 581196 )
      Mandelbrot gives Gaston Julia proper attribution in TFA. But it took this extraordinary man to bring new life to this field.
  • 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.

  • Seeing it (Score:5, Interesting)

    by wombatmobile ( 623057 ) on Friday November 12, 2004 @02:36PM (#10801083)

    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?

    • Re:Seeing it (Score:5, Informative)

      by zunis ( 830494 ) on Friday November 12, 2004 @02:45PM (#10801186)
      The first version of the Mandlebrot set was printed on a flat bed plotter in the 60's, if I remember my history correctly.

    • Re:Seeing it (Score:3, Informative)

      by HarveyBirdman ( 627248 )
      It was a paper printout. The tiny satellite Mandelbrot sets showed up as little dots, and were initially dismissed as dirt from an unclean printhead. This was in the 1970's, actually.

      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 [uniovi.es]

      • Re:Seeing it (Score:3, Interesting)

        by Beautyon ( 214567 )
        but I don't recall which one

        It was in:

        "The Beauty of Fractals", H. O. Peitgen P. H. Richter, Springer -Verlag Berlin, page 152

        the diagram says 1980.

    • Re:Seeing it (Score:2, Informative)

      by SMQ ( 241278 )
      Printed out on a teletype terminal at 132x66 if I remember correctly from the SciAm article.

    • Re:Seeing it (Score:3, Interesting)

      by CausticPuppy ( 82139 )
      There's an image of the actual first printout in James Gleick's book [i]Chaos: Making a New Science.[/i]

      It didn't have the neato color shading, it basically looked like the cardiod shaped main blob with a bunch of "noise" around the perimeter.
      He later figured out that the black dots were actually connected-- the entire set is connected.

  • A simple equation... (Score:4, Interesting)

    by badfrog ( 45310 ) on Friday November 12, 2004 @02:38PM (#10801104)
    It is so simple that most children can program their home computers to produce the Mandelbrot set.
    That's exactly what I did when I was about 12, on my Tandy Color Computer 3. Took about 24 hours to make one ~320x190 screen.

    • Re:A simple equation... (Score:3, Interesting)

      by kzinti ( 9651 )
      I did the same with my CoCo - I was about 19 at the time. Hating waiting on its slow BASIC interpreter. Fortunately, I knew its assembly language and even had the Macro-assembler cartridge. I thought about how to program it it assembler, but didn't want to attempt writing floating point routines, or trying to call the floating poing routines in the ROM. Eventually, I realized that you could calculate a Mandelbrot set using fixed-point math. The 6809's MUL instruction made it a snap - you just shift the deci
      • 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
        • Too cool, dude.

          Yeah, I did a Life program too, in assembly. Also wrote an assembly version of a one-dimensional cellular automaton. I remember optimizing the heck out of the screen-scrolling routine, which was the limiting factor on how fast it could run. I eventually hit upon the idea of turning off hardware interrupts, then using the stack pointers, together with a software interrupt, to move bigger chunks of memory in fewer instructions - because the interrupt-servicing instructions could load or save m
      • The term you're looking for is "radix point."

        HTH

        HAND

        ;)

  • BRILLIANT (Score:5, Insightful)

    by scribblej ( 195445 ) on Friday November 12, 2004 @02:42PM (#10801147)
    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!

  • Everything old is new again! (Score:2, Insightful)

    by Anonymous Coward
    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 [wired.com]

    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 al

  • Negative space? (Score:3, Interesting)

    by TrentL ( 761772 ) on Friday November 12, 2004 @02:43PM (#10801169) Homepage
    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?

    • Re:Negative space? (Score:5, Insightful)

      by TCM ( 130219 ) on Friday November 12, 2004 @02:59PM (#10801326)
      ^H^H

      • lol (Score:2)

        by phyruxus ( 72649 )
        that's a better, and funnier, description than mine below. :)

        Give the man a +1 (either funny or insightful)

    • You asked about negative space... in art, that's the area which isn't filled by the subject. Some of Escher's works use interlocking positive and negative space that fills the whole area. In TFA though, Mandelbrot mentioned negative dimensions... and I don't know what those are; but since I'm blabbering away already, I'll take a stab at it from what he said in TFA.

      <my guess>
      Space has dimensionality; a plane has 2 dimensions, a cube exists in 3, hypercube 4... the numbers here are positive. Mandelbro

  • Did anyone else feel disapointed that every third leter wasn't missing?

    Bwhahahhahahhaha....*sob*...no, it was funny, trust me...

  • micro-mandelbrot (Score:4, Funny)

    by jeffmock ( 188913 ) on Friday November 12, 2004 @02:44PM (#10801180)
    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
  • As someone mentioned above, and I second: Thanks for the brainfood, Mandelbrot!

    For years, I have been using Fractint, and generating fractals on my PC, usually for print. I prefer it's zebra pattern, and it's appeal when printed very large [zhrodague.net] -- especially when you can take a magnifying glass to the resulting printed image for more fractal fun!
    • Incidentally, the Fractint software is available here [triumf.ca] and is entirely free including source code. Both beer and speech freedom.

      ^_^ That program was a massive source of entertainment to me as a child.

    • Ah, thanks for the link. I've always loved Fractint, and the creative ways the developers put together lots of math to draw pretty pictures. Also note, that their development version supports individual images above 2048x2048 pixels -- great for printing large-format.

  • New Scientist: What's the mystery?

    Mandelbrot: It relates to a rather subtle mathematical property. In simple terms, there are two ways to define the Mandelbrot set. It is rather like proving that 3+1 and 2+2 give the same result. I have always thought that the two definitions were equivalent. But one is easy to study whereas the other is extremely difficult. So far, the proof has defeated many people. The fact that my conjecture is so simple to state, yet baffles everybody, makes it attractive to mathe

  • Source [sourceforge.net] Output [sourceforge.net] It is an imagemap, so you can click anywhere to zoom in.

  • Dear Mandelbrot (Score:4, Funny)

    by bludstone ( 103539 ) on Friday November 12, 2004 @02:55PM (#10801291)
    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*

  • Mandelbrot's ideas... (Score:5, Informative)

    by jd ( 1658 ) <`imipak' `at' `yahoo.com'> on Friday November 12, 2004 @02:58PM (#10801318) Homepage Journal
    Some of Mandelbrot's work borrowed off the research of others, but failed to give proper credit. Well, that happens a lot in science, unfortunately.


    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.

    • In the interview, he says that a lot of interesting mathematics is stuff that's been done by people already, but where the original discoverer didn't go far enough or didn't publish everything. He advocates looking at things that were worked on 150 years ago and then dropped.

      Fractals are generally random. They show self-similarity, but the way in which they are not identical but similar is often unpredictable. (E.g., in a period of noise, there will be periods of signal with a certain distribution, but the
      • The Feigenbaum number is a bit like the normal distribution, in that is something about how statistics behave in the aggregate rather than depending on the system.


        ObTrivia: Because populations, behaviour, social systems, etc, are chaotic, the Feigenbaum Number would be a logical first-step to Isaac Asimov's "Psychohistory", and indeed the last couple of Foundation novels migrated towards Psychohistory being a branch of chaos theory.

    • Re:Mandelbrot's ideas... (Score:4, Interesting)

      by div_B ( 781086 ) on Friday November 12, 2004 @05:43PM (#10802998)
      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 dude's name was actually Mitchell Feigenbaum. He was working at LANL at the time. A good read if anyone is interested in the (convoluted) chronology of chaos theory and non-linear dynamics is Chaos: Making a New Science by James Gleick. It gives a feel for how the seperate contributions of people like Lorenz, Julia, Feigenbaum, Mandelbrot, Serpiensky, etc, came together, and the battle Chaos theory fought to be recognized as a legitimate field of mathematics in the 20th century.

  • Mandelbrot in Postscript (Score:4, Interesting)

    by 3770 ( 560838 ) on Friday November 12, 2004 @03:02PM (#10801355) Homepage
    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.

  • NOT the inventor of fractals! (Score:5, Informative)

    by Mark_in_Brazil ( 537925 ) on Friday November 12, 2004 @03:24PM (#10801588)
    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 [wikipedia.org], inventor of the Hausdorff dimension [wikipedia.org], Georg Cantor [wikipedia.org], inventor of the fractal Cantor "middle thirds" Set [wikipedia.org], and Gaston Julia [wikipedia.org], who discovered/invented the Julia Set [wikipedia.org], 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

  • Most children... (Score:4, Insightful)

    by terrencefw ( 605681 ) <slashdot@@@jamesholden...net> on Friday November 12, 2004 @03:33PM (#10801696) Homepage
    (From TFA...) It is so simple that most children can program their home computers to produce the Mandelbrot set.
    Well, yes, I suspect most of us could and most likely did on our ZX81's, C64's, BBC B's etc etc.

    /puts old man hat on

    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.

  • There are tons of fractal generators out there, my favorite is Chaos Pro [chaospro.de]. It's windows only, but it's free (as in beer). It supports a ton of advanced features like transparent layering/blending, and generating AVI's, and the author claims it's 100% feature-compatible with Ultra Fractal [ultrafractal.com] (a commercial package).

    It's also compatible with formula and parameter files from other fractal programs (including the legendary FractInt).

    Anyway, if you have a decent photo printer, and any fractal program that can do
  • No article about fractals could be complete without mentioning Elenas [elena-fractals.it] excellent ZonXplorer [elena-fractals.it] fractal package for AmigaOS 3.5+ [amiga.com] and MorphOS [morphos.de] (running on the Pegasos PPC [pegasosppc.com]). Check out her stuning pictures in her gallery [elena-fractals.it].
    I hope her webpage can handle the load, it's sure enough worth a visit.

  • Fractal wetware? (Score:3, Interesting)

    by Doc Ruby ( 173196 ) on Friday November 12, 2004 @05:50PM (#10803068) Homepage Journal
    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 [rpi.edu], 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?

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