Tglad
Fractal Molossus
Posts: 703
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« on: January 01, 2014, 11:56:30 AM » |
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I discovered the other day that there are indeed negative power multibrots, here is the z -2 + c one: The trick is to realise that the border of the Mandelbrot set (which is the interesting bit) is the points which don't cycle (or the cycle length tends to infinity). For negative powers we also plot the points that don't cycle (in black)... it matters not that there is no escape to infinity property. Anyway, there are very few images of this fractal... does anyone have any close ups?? I also tested and the negative powers are also universal (an interesting property of the positive power multibrots). It makes me think that the power -2 Mandelbrot above is really just as important as the normal Mandelbrot... or close to it anyway. Also, I didn't find a picture so I made one of z=z -1+c, unlike z=z 1+c it creates a 1d fractal like the cantor set:
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matsoljare
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« Reply #1 on: January 01, 2014, 10:51:09 PM » |
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I think something similar to this was in the Benoit Mandelbrot book.... or maybe some other 80s or early 90s book on fractals?
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xenodreambuie
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« Reply #2 on: January 02, 2014, 02:34:14 AM » |
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As it happens I'm working on finite Mandelbrot rendering now. The z-2 + c Mandelbrot set is a bit unusual in that zooming in doesn't get interesting, nor show more than minor resemblance to the corresponding Julia sets. Here is a zoom near 0.9, 1.56, magnification 537. It doesn't look like there are interesting minibrots to be found, while they are easily found in sets such as 1/(z2 + 1) + c.
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hobold
Fractal Bachius
Posts: 573
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« Reply #3 on: January 02, 2014, 02:41:28 AM » |
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here is the z-2 + c one: Which of the two possible square roots do you pick?
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Tglad
Fractal Molossus
Posts: 703
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« Reply #4 on: January 02, 2014, 02:50:05 AM » |
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Not square root... that would be z 0.5Very interesting xeno, the fact that it doesn't get interesting like the +ve mandelbrot is itself interesting Thanks for the close up. I wonder if it really is a cluster of minibrots like that, or whether there are tendrils between them that we have trouble rendering.
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xenodreambuie
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« Reply #5 on: January 02, 2014, 03:34:52 AM » |
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I wonder if it really is a cluster of minibrots like that, or whether there are tendrils between them that we have trouble rendering.
I can get tendrils near each minibrot by limiting max iterations to about 400. Much more and the set fills them in.
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Tglad
Fractal Molossus
Posts: 703
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« Reply #6 on: January 02, 2014, 04:38:11 AM » |
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Very interesting, and nice job. The little minibrots look a lot like the z 4+c multibrot... how strange. It seems like the black part needs colouring rather than the white part... I guess a simple but slow way would be just supersampling the pixels, so you get a bit of white in lots of the black pixels.
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jdebord
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« Reply #7 on: January 02, 2014, 09:20:05 AM » |
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You can get interesting pictures by adding positive and negative powers : z^p + c * z^(-q) The critical point is z = (c * q / p)^(1 / (p + q)) See for instance Robert Devaney's papers: http://math.bu.edu/people/bob/papers.html
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element90
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« Reply #8 on: January 02, 2014, 01:32:43 PM » |
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« Last Edit: May 26, 2014, 10:41:02 AM by element90, Reason: Replaced file links. »
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youhn
Fractal Molossus
Posts: 696
Shapes only exists in our heads.
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« Reply #9 on: January 02, 2014, 10:42:32 PM » |
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No access to a Linux computer with gnofract4d since I'm on wintersport in the Alpes right now. But Xaos on Windows can make some pictures too. I would love to see a video in which the bailout condition increases, filling up the whole inside of this fractal. xaos file: ;Position file automatically generated by XaoS 3.5 ; - a realtime interactive fractal zoomer ;Use xaos -load <filename> to display it (initstate) (filter 'anti #t) (palette 3 18081 143) (formula 'user) (usrform "Z^-2+C") (usrformInit "") (maxiter 2400) (bailout 32) (outcoloring 9) (view -1.85053655E 1.93642415E-07 2.38702306E-05 2.38702306E-05) xaos file: ;Position file automatically generated by XaoS 3.5 ; - a realtime interactive fractal zoomer ;Use xaos -load <filename> to display it (initstate) (filter 'anti #t) (filter 'palette #t) (palette 3 21429 61) (formula 'user) (usrform "Z^-2+C") (usrformInit "") (maxiter 2400) (bailout 24) (outcoloring 9) (view -1.812069810 -0.004241906388 3.4602944E-068 3.4602944E-068)
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xenodreambuie
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« Reply #10 on: January 02, 2014, 10:45:57 PM » |
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The little minibrots look a lot like the z4+c multibrot... how strange.
It seems like the black part needs colouring rather than the white part... I guess a simple but slow way would be just supersampling the pixels, so you get a bit of white in lots of the black pixels.
For pure z n+c multibrots the number of lobes of minibrots is the same as the main brot: abs(n-1). But from what little I've seen in general polynomial rationals, the simpler minibrots prevail. Eg in 1/(z 2+1) +c, the minibrots have one lobe, not three. Supersampling would help a little, as in element90's pic. It seems that for rational polynomials of degree less than 2, the Mandelbrot set is much less representative as an atlas for Julia sets. In particular, there are connected Julia sets in areas not near the Mandelbrot set, and non-solid Julia sets in solid areas of the Mandelbrot set. (With multiple critical points there may be more than one Mandelbrot set, but it's also true with only one or two critical points.)
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youhn
Fractal Molossus
Posts: 696
Shapes only exists in our heads.
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« Reply #11 on: January 02, 2014, 11:16:11 PM » |
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Two more: xaos code: ;Position file automatically generated by XaoS 3.5 ; - a realtime interactive fractal zoomer ;Use xaos -load <filename> to display it (initstate) (filter 'anti #t) (filter 'palette #t) (palette 2 6397 44) (formula 'user) (usrform "Z^-2+C") (usrformInit "") (maxiter 3200) (bailout 32) (outcoloring 9) (view 0.854096189440 -1.47934068384 9.4546368911E-07 9.454636877E-077)
xaos code: ;Position file automatically generated by XaoS 3.5 ; - a realtime interactive fractal zoomer ;Use xaos -load <filename> to display it (initstate) (filter 'anti #t) (filter 'palette #t) (palette 3 8016 0) (formula 'user) (usrform "Z^-2+C") (usrformInit "") (maxiter 3200) (bailout 32) (outcoloring 9) (view 0.85409605655- -1.47934100935 1.3358485095E-07 1.3358485075E-07)
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kram1032
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« Reply #12 on: January 02, 2014, 11:21:49 PM » |
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Very nice and interesting patterns there Since only the infinity of the circles matters, what happens for other functions that tend to not give patterns? Do they have such circles? For instance, does the exponential function have any such interesting points? It tends to result in rather boring stuff, usually.
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xenodreambuie
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« Reply #13 on: January 02, 2014, 11:39:29 PM » |
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In youhn's pics of Julia sets, the circles are due to a small bailout radius (32). They disappear (or shrink to dots) with say, 10150 instead, and you get the complete sets.
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« Last Edit: January 03, 2014, 01:52:30 AM by xenodreambuie »
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Tglad
Fractal Molossus
Posts: 703
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« Reply #14 on: January 03, 2014, 12:27:32 AM » |
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Very nice images youhn... I don't understand how you found these when element90's image suggests the z -2+c is filled with a never ending dense cluster of z 4+c-like minibrots. Your images show bendy circles, quite unlike what you get in positive multibrots... or should I take it from zenodreambuie's comment that your images are julia sets which need the bailout to be higher? So does it remain the case that z -2+c looks always like element90's picture, and the pretty patterns only turn up in the julia sets? Also, thanks for the Daveney link, the fractal: z = c(z+1/z) is interesting: it looks very like the z -2+c but the minibrots are all normal mandelbrots (rather than z 4+c) so it seems like a simpler version of the same sort of sponge fractal. Close up below:
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