prem@geomag.fsu.edu (Prem Subrahmanyam) (11/29/89)
Here's an observation I have made that seems contrary to (at least some) popular belief concerning the Mandelbrot set. I read in one of the Computer Recreations articles in Scientific American that the Mandelbrot set is not self-replicating at different levels. Yet I have clearly seen identical reproductions of it in many places. If you look at the "line" protruding from the "head" of M, you will see that there is a tiny replica of the big part of M. Magnifying in on this reveals another tinier replica coming off of this head, and so on ad infinitum. Albeit, the "domains" are not identical around the replicas at every level, but the shape of the black part is the same. I haven't tried to zoom in on tiny areas of the tiny replica to see if the fine detail is the same, but in general, it seems that M, as well as all J sets, is self-similar to a high degree. Any other opinions, observations about this? ---Prem Subrahmanyam (prem@geomag.gly.fsu.edu)
afoiani@nmsu.EDU (Anthony Foiani) (11/29/89)
In an earlier article, prem@geomag.fsu.edu (Prem Subrahmanyam) writes: >Here's an observation I have made that seems contrary to (at least >some) popular belief concerning the Mandelbrot set. I read in one of >the Computer Recreations articles in Scientific American that the >Mandelbrot set is not self-replicating at different levels. Yet I >have clearly seen identical reproductions of it in many places. ^^^^^^^^^^^^^^^^^^^^^^^ > >If you look at the "line" protruding from the "head" of M, you will >see that there is a tiny replica of the big part of M. Magnifying in >on this reveals another tinier replica coming off of this head, and so >on ad infinitum. Albeit, the "domains" are not identical around the >replicas at every level, but the shape of the black part is the same. >I haven't tried to zoom in on tiny areas of the tiny replica to see if >the fine detail is the same, but in general, it seems that M, as well >as all J sets, is self-similar to a high degree. That line at the end of paragraph 1 is where you get into trouble. True, the 'baby' Mandelbrot sets centered about the real axis from about Re(-1.75) to Re(-2.0) bear a significant resemlance to the main set; but this is only to appearances. The 'line' you mention is indeed a real line, connecting all these Baby Mandelbrots to the main body. Someone proved a theorem [someone mentioned the reference a few days back, but I forgot to keep it] which states that all points in the Mandelbrot Set are connected. If these Baby Mandelbrots are connected, they must have a line of points going from their 'tail' to the 'head' of the next Baby on the right. You will notice that there is no line of points proceeding to the right of the main Mandelbrot Set. Thus, these are not 'identical' reproductions. Tony. -- tony foiani (afoiani@nmsu.edu) "And remember...don't lose your a.k.a. Tkil (mcsajf@nmsuvm1.bitnet) head..." -Ramirez, HIGHLANDER