This is odd. I've been studying the Mandelbrot set off and on for a frighteningly long time (26 years). I listened to the interview and read the transcript (which has many errors, but is at bigthink.com/ideas/19207 ) and did some Google searches in case maybe I was missing something. Just to clarify the question, Mandelbrot said the following (I have corrected several errors in the transcript): *The conjecture itself consists in two definitions of Mandelbrot set - two alternative definitions which are too technical to describe without a blackboard, but which are both very simple and which I assumed naively to be equivalent. Why did I assume so? Because in the pictures I could not see any difference. Obtaining pictures in one way or another way, I couldn't tell them apart. Therefore, I assumed they were identical and I went on studying this beast. I found that, again, many interesting observations of which most were very confirmed by many other very, very skilled mathematicians. But the idea that these two conditions, definitions, are identical is still open. So there are two definitions of Mandelbrot set, the usual one and another one, and they may theoretically be different. People are getting close, but have not proven it completely.* The best I can do at connecting this statement to truth is as follows: The only thing that has been worked on extensively by "many very skilled mathematicians" who are "getting closer, but have not proven it completely" is the MLC conjecture, namely that the Mandelbrot set is locally connected. The Mandelbrot set is normally defined by the "z=0, iterate z'=z^2+c, see if it remains bounded" definition which can be implemented as a computer program. This is done by defining a c value for each pixel on the screen and computing the z values numerically. Using a modern programming model (like OpenCL), each pixel performs its own iteration, all pixels iterate in parallel, and if you have infinite patience (-: all pixels destined to "escape" will eventually do so. The Mandelbrot set is also sometimes defined as (using the wording on the Wikipedia page) "the limit set of a sequence of plane algebraic curves, the Mandelbrot curves, of the general type known as polynomial lemniscates". These curves are: L1: |c|=2, L2: |c^2+c|=2, L3: |(c^2+c)^2+c|=2, and so on. They form a set of "contour lines" which are seen in many early computer images, including some in Mandelbrot's 1986 book. They first few are illustrated (with formulas) here: http://en.wikipedia.org/wiki/File:Lemniscates5.png A Mandelbrot set image can be produced by plotting these curves, which needs to be done not by scanning pixels but by "walking" along the curve, using a numerical method similar to root-finding. So my guess at making sense of Mandelbrot's "conjecture" statement is that, the "limit of lemniscates" definition might imply MLC. In other words, if it can be shown that the lemniscates L1, L2, L3, ... converge on the boundary of the Mandelbrot set, then perhaps such a proof would also prove MLC. If so, then the MLC conjecture is equivalent to a conjecture that the two ways of drawing the Mandelbrot set (pixel scanning and tracing lemniscates) yield the same image, and Mandelbrot's statements would make sense. It's a stretch, but it's the best I can do. - Robert Munafo On Tue, Jul 26, 2011 at 21:18, David Makin <makinmagic@tiscali.co.uk> wrote:
In this interview:
http://bigthink.com/benoitmandelbrot
He mentions two forms for the Mandelbrot that are believed to be equivalent but are not proved to be so. He doesn't give the details - can anyone either explain in more detail or point me at the details ? I've tried looking on the web but failed to find the conjecture the he mentions...
bye Dave
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