[math-fun] 3D Mandelbrot set with "3-Haussdorf-dimensional" boundary?
Just rotate the 2D Mandelbrot set to get a body of revolution with a 3D boundary given that the 2D set has 2D boundary.
Warren's argument (which might or might not be valid -- it sounds good to me but I don't work in that area) reminds me of something I was told (but never understood) 30 years ago: that the fractal dimension of the Cartesian product of two sets need not be equal to the sum of the fractal dimensions of the two sets. Can anyone explain this to me? Jim P.S. After having been publically (albeit gently) scolded on MathOverflow a number of times for the way I've phrased questions, and having seen others being similarly scolded, I've now got an internalized troll in my brain who criticizes the above question ("Can anyone explain this to me?") as follows: "Well of course the answer to your question is YES: that is, there do exist people who can explain this to you. But is that what you really meant to ask? Please try to ask questions in a clearer way." On Friday, May 30, 2014, Warren D Smith <warren.wds@gmail.com> wrote:
Just rotate the 2D Mandelbrot set to get a body of revolution with a 3D boundary given that the 2D set has 2D boundary.
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http://books.google.com/books?id=sKGz7SiCVpEC&pg=PA239&lpg=PA239&dq=hausdorf... Gives an example of two sets of Hausdorff dimension 0 whose product has dimension >= 1. I haven't yet worked through the proof to get an intuitive understanding of it. Andy On Fri, May 30, 2014 at 9:51 PM, James Propp <jamespropp@gmail.com> wrote:
Warren's argument (which might or might not be valid -- it sounds good to me but I don't work in that area) reminds me of something I was told (but never understood) 30 years ago: that the fractal dimension of the Cartesian product of two sets need not be equal to the sum of the fractal dimensions of the two sets.
Can anyone explain this to me?
Jim
P.S. After having been publically (albeit gently) scolded on MathOverflow a number of times for the way I've phrased questions, and having seen others being similarly scolded, I've now got an internalized troll in my brain who criticizes the above question ("Can anyone explain this to me?") as follows: "Well of course the answer to your question is YES: that is, there do exist people who can explain this to you. But is that what you really meant to ask? Please try to ask questions in a clearer way."
On Friday, May 30, 2014, Warren D Smith <warren.wds@gmail.com> wrote:
Just rotate the 2D Mandelbrot set to get a body of revolution with a 3D boundary given that the 2D set has 2D boundary.
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-- Andy.Latto@pobox.com
Which reminds me -- completely irrelevantly -- of receiving by telephone an appointment to meet a gent whose name I took to be "Church", arriving at the appointed time and place, then searching with increasing bafflement up and down the corridor several times before successfully deciphering the plate on his door: Czyż . WFL On 5/31/14, Andy Latto <andy.latto@pobox.com> wrote:
http://books.google.com/books?id=sKGz7SiCVpEC&pg=PA239&lpg=PA239&dq=hausdorf...
Gives an example of two sets of Hausdorff dimension 0 whose product has dimension >= 1. I haven't yet worked through the proof to get an intuitive understanding of it.
Andy
On Fri, May 30, 2014 at 9:51 PM, James Propp <jamespropp@gmail.com> wrote:
Warren's argument (which might or might not be valid -- it sounds good to me but I don't work in that area) reminds me of something I was told (but never understood) 30 years ago: that the fractal dimension of the Cartesian product of two sets need not be equal to the sum of the fractal dimensions of the two sets.
Can anyone explain this to me?
Jim
P.S. After having been publically (albeit gently) scolded on MathOverflow a number of times for the way I've phrased questions, and having seen others being similarly scolded, I've now got an internalized troll in my brain who criticizes the above question ("Can anyone explain this to me?") as follows: "Well of course the answer to your question is YES: that is, there do exist people who can explain this to you. But is that what you really meant to ask? Please try to ask questions in a clearer way."
On Friday, May 30, 2014, Warren D Smith <warren.wds@gmail.com> wrote:
Just rotate the 2D Mandelbrot set to get a body of revolution with a 3D boundary given that the 2D set has 2D boundary.
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-- Andy.Latto@pobox.com
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The Quaternion set in 3D does this but will not have a Hausforff dimension of 3 as in one direction (around the real axis) it is not fractal. On 31 May 2014, at 00:41, Warren D Smith wrote:
Just rotate the 2D Mandelbrot set to get a body of revolution with a 3D boundary given that the 2D set has 2D boundary.
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The meaning and purpose of life is to give life purpose and meaning. The instigation of violence indicates a lack of spirituality.
Argh, I was correct the first time - in the rotattoinal case (quaternion) as I said the surface around the real; axis is not fractal so the points that would make it 3D from the 2D cross-section *are not surface points*. This also answer's James' point I think/ On 31 May 2014, at 00:41, Warren D Smith wrote:
Just rotate the 2D Mandelbrot set to get a body of revolution with a 3D boundary given that the 2D set has 2D boundary.
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The meaning and purpose of life is to give life purpose and meaning. The instigation of violence indicates a lack of spirituality.
Actually that just gave me an idea with respect to quaternions - if one modified their math slightly such that the rotational angle around the real axis itself were a fractal (rather than a smooth circle) this could produce the ultimately "correct" 3D analog of the 2D Mandelbrot maybe even for plain z^2+c. Making the angle fractal is not a approach I've considered previously (or seen elsewhere) ;) On 31 May 2014, at 10:04, David Makin wrote:
Argh, I was correct the first time - in the rotattoinal case (quaternion) as I said the surface around the real; axis is not fractal so the points that would make it 3D from the 2D cross-section *are not surface points*. This also answer's James' point I think/
On 31 May 2014, at 00:41, Warren D Smith wrote:
Just rotate the 2D Mandelbrot set to get a body of revolution with a 3D boundary given that the 2D set has 2D boundary.
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The meaning and purpose of life is to give life purpose and meaning. The instigation of violence indicates a lack of spirituality.
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The meaning and purpose of life is to give life purpose and meaning. The instigation of violence indicates a lack of spirituality.
participants (5)
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Andy Latto -
David Makin -
Fred Lunnon -
James Propp -
Warren D Smith