Re: [math-fun] Mandelbrot's Conjecture (as mentioned in this video interview)
Suddenly two definitions of the Mandelbrot set come to mind, the first one I learned (I), and the much more common one (II): For any c in C, define f_c(z) as z^2 + c. I. The set of c in C for which the Julia set of f_c is connected. (See < http://en.wikipedia.org/wiki/Julia_set >.) II. the set of c in C for which the orbit of 0, under (forward) iteration of f_c, is bounded. I've never seen a proof that these definitions are equivalent, though I haven't looked very hard, either. (In II, it seems to me that considering the orbits of 0 rather than of any other point is somewhat arbitrary.) --Dan Sometimes the brain has a mind of its own.
I believe this is what Mandelbrot was referring to. Here's another interview with him in which he talks very specifically about it: http://www.webofstories.com/play/10523 Tom Dan Asimov writes:
Suddenly two definitions of the Mandelbrot set come to mind, the first one I learned (I), and the much more common one (II):
For any c in C, define f_c(z) as z^2 + c.
I. The set of c in C for which the Julia set of f_c is connected. (See < http://en.wikipedia.org/wiki/Julia_set >.)
II. the set of c in C for which the orbit of 0, under (forward) iteration of f_c, is bounded.
I've never seen a proof that these definitions are equivalent, though I haven't looked very hard, either. (In II, it seems to me that considering the orbits of 0 rather than of any other point is somewhat arbitrary.)
--Dan
Sometimes the brain has a mind of its own.
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On Thu, Jul 28, 2011 at 10:28 AM, Tom Karzes <karzes@sonic.net> wrote:
I believe this is what Mandelbrot was referring to. Here's another interview with him in which he talks very specifically about it:
I believe the conjecture he states there, and claims both that he originated and that no-one would have thought of without computer graphics, is the Fatou conjecture. I can't find a statement of the Fatou conjecture (which appears to be different from the "Fatou conjecture on wandering domains", since the former is unsolved and the latter proven). But the Real Fatou Conjecture (which has now been proved) appears to be exactly the same conjecture restricted to real values of c. Andy
Tom
Dan Asimov writes: > Suddenly two definitions of the Mandelbrot set come to mind, > the first one I learned (I), and the much more common one (II): > > For any c in C, define f_c(z) as z^2 + c. > > I. The set of c in C for which the Julia set of f_c is connected. > (See < http://en.wikipedia.org/wiki/Julia_set >.) > > II. the set of c in C for which the orbit of 0, under (forward) > iteration of f_c, is bounded. > > I've never seen a proof that these definitions are equivalent, > though I haven't looked very hard, either. (In II, it seems > to me that considering the orbits of 0 rather than of any > other point is somewhat arbitrary.) > > --Dan > > Sometimes the brain has a mind of its own. > > _______________________________________________ > math-fun mailing list > math-fun@mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >
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Yeah, that must be it. Thanks for the link to that interview. *I put as a conjecture, and the conjecture is that the first set which I was considering - call it M0 - if you add it to all its limit points, make it the limit points plus the set, you obtain the set M which is obtained by connectedness of the Julia set. This conjecture looked extremely simple. It can be explained to a good high school student, because there was no complicated concept involved. The concept of a limit cycle is straightforward. The concept of connectedness is straightforward and intuitive. The relation between the two was postulated by a very simple identity, that M equals M0 plus its limit points. Well, believe it or not, this conjecture is still open and what is so striking about the study of iteration of Z squared plus C is that the first serious difficulty I encountered in its study has remained, after now eighteen, nineteen years of study, totally baffling.* - Benoit Mandelbrot (May 1998) Even simpler than the lemniscate thing, and clearly an easy thing to assume given how Mandelbrot defines and discusses the separator set (page 183 of his 1982 book[1]). Odd, I thought Fatou had proven that one, but I was wrong (-: [1] Benoit B. Mandelbrot. The Fractal Geometry of Nature. New York: W. H. Freeman and Company, 1983. ISBN 0-7167-1186-9. On Thu, Jul 28, 2011 at 10:28, Tom Karzes <karzes@sonic.net> wrote:
I believe this is what Mandelbrot was referring to. Here's another interview with him in which he talks very specifically about it:
http://www.webofstories.com/play/10523
Tom
Dan Asimov writes:
Suddenly two definitions of the Mandelbrot set come to mind, the first one I learned (I), and the much more common one (II):
For any c in C, define f_c(z) as z^2 + c.
I. The set of c in C for which the Julia set of f_c is connected. (See < http://en.wikipedia.org/wiki/Julia_set >.)
II. the set of c in C for which the orbit of 0, under (forward) iteration of f_c, is bounded.
I've never seen a proof that these definitions are equivalent, though I haven't looked very hard, either. (In II, it seems to me that considering the orbits of 0 rather than of any other point is somewhat arbitrary.)
-- Robert Munafo -- mrob.com Follow me at: gplus.to/mrob - fb.com/mrob27 - twitter.com/mrob_27 - mrob27.wordpress.com - youtube.com/user/mrob143 - rilybot.blogspot.com
Thanks Tom, I confess I didn't think he was that confused and I was convinced that there must be something somewhere explaining things in more detail. On 28 Jul 2011, at 15:28, Tom Karzes wrote:
I believe this is what Mandelbrot was referring to. Here's another interview with him in which he talks very specifically about it:
http://www.webofstories.com/play/10523
Tom
Dan Asimov writes:
Suddenly two definitions of the Mandelbrot set come to mind, the first one I learned (I), and the much more common one (II):
For any c in C, define f_c(z) as z^2 + c.
I. The set of c in C for which the Julia set of f_c is connected. (See < http://en.wikipedia.org/wiki/Julia_set >.)
II. the set of c in C for which the orbit of 0, under (forward) iteration of f_c, is bounded.
I've never seen a proof that these definitions are equivalent, though I haven't looked very hard, either. (In II, it seems to me that considering the orbits of 0 rather than of any other point is somewhat arbitrary.)
--Dan
Sometimes the brain has a mind of its own.
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On 7/28/11, David Makin <makinmagic@tiscali.co.uk> wrote:
Thanks Tom, I confess I didn't think he was that confused and I was convinced that there must be something somewhere explaining things in more detail.
It strikes me that there's the meat for an interesting Math. Intelligencer style article in this little story --- a bit of detective work, various experts pitching in, opportunity for some pretty pictures, recently deceased prophet /genius / riddler, the occasional doubter sniping from the sidelines ... and the maths, of course. WFL
Was that the sound of you offering to write something, Fred? :-) --Michael On Thu, Jul 28, 2011 at 3:35 PM, Fred lunnon <fred.lunnon@gmail.com> wrote:
On 7/28/11, David Makin <makinmagic@tiscali.co.uk> wrote:
Thanks Tom, I confess I didn't think he was that confused and I was convinced that there must be something somewhere explaining things in more detail.
It strikes me that there's the meat for an interesting Math. Intelligencer style article in this little story --- a bit of detective work, various experts pitching in, opportunity for some pretty pictures, recently deceased prophet /genius / riddler, the occasional doubter sniping from the sidelines ... and the maths, of course.
WFL
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Er, no actually --- it was the sound of me trying to con David into writing something! WFL On 7/28/11, Michael Kleber <michael.kleber@gmail.com> wrote:
Was that the sound of you offering to write something, Fred? :-)
--Michael
On Thu, Jul 28, 2011 at 3:35 PM, Fred lunnon <fred.lunnon@gmail.com> wrote:
On 7/28/11, David Makin <makinmagic@tiscali.co.uk> wrote:
Thanks Tom, I confess I didn't think he was that confused and I was convinced that there must be something somewhere explaining things in more detail.
It strikes me that there's the meat for an interesting Math. Intelligencer style article in this little story --- a bit of detective work, various experts pitching in, opportunity for some pretty pictures, recently deceased prophet /genius / riddler, the occasional doubter sniping from the sidelines ... and the maths, of course.
WFL
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On Thu, Jul 28, 2011 at 5:40 PM, Fred lunnon <fred.lunnon@gmail.com> wrote:
Er, no actually --- it was the sound of me trying to con David into writing something!
An excellent suggestion! --Michael
WFL
On 7/28/11, Michael Kleber <michael.kleber@gmail.com> wrote:
Was that the sound of you offering to write something, Fred? :-)
--Michael
On Thu, Jul 28, 2011 at 3:35 PM, Fred lunnon <fred.lunnon@gmail.com> wrote:
On 7/28/11, David Makin <makinmagic@tiscali.co.uk> wrote:
Thanks Tom, I confess I didn't think he was that confused and I was convinced that there must be something somewhere explaining things in more detail.
It strikes me that there's the meat for an interesting Math. Intelligencer style article in this little story --- a bit of detective work, various experts pitching in, opportunity for some pretty pictures, recently deceased prophet /genius / riddler, the occasional doubter sniping from the sidelines ... and the maths, of course.
WFL
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-- Forewarned is worth an octopus in the bush. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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participants (7)
-
Andy Latto -
Dan Asimov -
David Makin -
Fred lunnon -
Michael Kleber -
Robert Munafo -
Tom Karzes