Title: Negative multibrots
Post by: Tglad on January 01, 2014, 11:56:30 AM
I discovered the other day that there are indeed negative power multibrots, here is the z-2 + c one:
(http://upload.wikimedia.org/wikipedia/en/e/ef/Juliasetsdkpict55.jpg)
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=z1+c it creates a 1d fractal like the cantor set:
(http://upload.wikimedia.org/wikipedia/en/e/ef/Juliasetsdkpict55.jpg)
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=z1+c it creates a 1d fractal like the cantor set:
Title: Re: Negative multibrots
Post by: matsoljare on January 01, 2014, 10:51:09 PM
I think something similar to this was in the Benoit Mandelbrot book.... or maybe some other 80s or early 90s book on fractals?
Title: Re: Negative multibrots
Post by: xenodreambuie on January 02, 2014, 02:34:14 AM
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.
Title: Re: Negative multibrots
Post by: hobold on January 02, 2014, 02:41:28 AM
here is the z-2 + c one:
Which of the two possible square roots do you pick?Title: Re: Negative multibrots
Post by: Tglad on January 02, 2014, 02:50:05 AM
Not square root... that would be z0.5
Very 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.
Very 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.
Title: Re: Negative multibrots
Post by: xenodreambuie on January 02, 2014, 03:34:52 AM
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.
Title: Re: Negative multibrots
Post by: Tglad on January 02, 2014, 04:38:11 AM
Very interesting, and nice job. :beer:
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.
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.
Title: Re: Negative multibrots
Post by: jdebord on January 02, 2014, 09:20:05 AM
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 (http://math.bu.edu/people/bob/papers.html)
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 (http://math.bu.edu/people/bob/papers.html)
Title: Re: Negative multibrots
Post by: element90 on January 02, 2014, 01:32:43 PM
Here is detail from the z = z^-2 + c Mandelbrot using convergence bailout of 0.001. This image was produced using gnofract4d with continuous potential as the colouring method, it automatically increased the number of iterations to over 54000 for the final image. The colouring consists of a myriad of roughly circular blobs of varying sizes, the larger the bailout value the larger the blobs. Original size 5250x3200 scaled to 1000x625.
(https://copy.com/wZQZJfgVPnZb)
Fractals of this type can also be produced using the bailout condition used for the standard Mandelbrot, the resulting pictures are different. Here are two examples produced using Saturn and Titan, the parameters are shown using Titan's summary tab. Original size 6000x6000 scaled to 1000x1000.
(https://copy.com/qFtXFvFwNbGU)
(https://copy.com/dFpuanIDBINa)
(https://copy.com/bNIUPDUS2x3I)
(https://copy.com/FhmYDAuN9psn)
Saturn and Titan can currently only produce divergent bailout versions of these fractals using a generalised formula for complex powers called Zcpac (z (Z) to complex power (cp) add constant (ac)).
(https://copy.com/wZQZJfgVPnZb)
Fractals of this type can also be produced using the bailout condition used for the standard Mandelbrot, the resulting pictures are different. Here are two examples produced using Saturn and Titan, the parameters are shown using Titan's summary tab. Original size 6000x6000 scaled to 1000x1000.
(https://copy.com/qFtXFvFwNbGU)
(https://copy.com/dFpuanIDBINa)
(https://copy.com/bNIUPDUS2x3I)
(https://copy.com/FhmYDAuN9psn)
Saturn and Titan can currently only produce divergent bailout versions of these fractals using a generalised formula for complex powers called Zcpac (z (Z) to complex power (cp) add constant (ac)).
Title: Re: Negative multibrots
Post by: youhn on January 02, 2014, 10:42:32 PM
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.
(http://imagizer.imageshack.us/v2/xq90/849/0cj5.png) (https://imageshack.com/i/nl0cj5p)
xaos file:
(http://imagizer.imageshack.us/v2/xq90/841/8jnj.png) (https://imageshack.com/i/nd8jnjp)
xaos file:
(http://imagizer.imageshack.us/v2/xq90/849/0cj5.png) (https://imageshack.com/i/nl0cj5p)
xaos file:
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)
(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)
(http://imagizer.imageshack.us/v2/xq90/841/8jnj.png) (https://imageshack.com/i/nd8jnjp)
xaos file:
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 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)
Title: Re: Negative multibrots
Post by: xenodreambuie on January 02, 2014, 10:45:57 PM
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 zn+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/(z2+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.)
Title: Re: Negative multibrots
Post by: youhn on January 02, 2014, 11:16:11 PM
Two more:
(http://imagizer.imageshack.us/v2/xq90/31/n2vd.png) (https://imageshack.com/i/0vn2vdp)
xaos code:
(http://imagizer.imageshack.us/v2/xq90/89/e038.png) (https://imageshack.com/i/2he038p)
xaos code:
(http://imagizer.imageshack.us/v2/xq90/31/n2vd.png) (https://imageshack.com/i/0vn2vdp)
xaos code:
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)
(http://imagizer.imageshack.us/v2/xq90/89/e038.png) (https://imageshack.com/i/2he038p)
xaos code:
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)
Title: Re: Negative multibrots
Post by: kram1032 on January 02, 2014, 11:21:49 PM
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.
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.
Title: Re: Negative multibrots
Post by: xenodreambuie on January 02, 2014, 11:39:29 PM
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.
Title: Re: Negative multibrots
Post by: Tglad on January 03, 2014, 12:27:32 AM
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 z4+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:
(http://math.bu.edu/people/bob/papers/figs/blow-up.gif)
it looks very like the z-2+c but the minibrots are all normal mandelbrots (rather than z4+c) so it seems like a simpler version of the same sort of sponge fractal. Close up below:
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:
(http://math.bu.edu/people/bob/papers/figs/blow-up.gif)
it looks very like the z-2+c but the minibrots are all normal mandelbrots (rather than z4+c) so it seems like a simpler version of the same sort of sponge fractal. Close up below:
Title: Re: Negative multibrots
Post by: xenodreambuie on January 03, 2014, 02:17:59 AM
... or should I take it from xenodreambuie's comment that your [youhn's] 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?
Yes, that is exactly right.
On the Devaney stuff:
I've just been looking at z+1/z+c instead (because I already have the capability without writing any new code). Very slow to render and hard to get good images. Here is most of the whole Mandelbrot set and a zoom into the valley of the smaller bud near the main bud.
It has good Julia sets too.
Title: Re: Negative multibrots
Post by: element90 on January 03, 2014, 02:13:54 PM
Quote
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 z4+c-like minibrots.
Judging by the pictures it looks like the divergent bailout is used (the same bailout method as the standard Mandelbrot set), the picture with M4 (z^4) Mandelbrots was produced using convergent bailout.
Here are some more pictures using the divergent bailout and several colouring methods.
(https://copy.com/2kZ0m9guLV6I)
(https://copy.com/E3ZTTeo9HNvT)
Changing the method to "absolute log of standard deviation of angle, scale = 1000".
(https://copy.com/SSLiZLpqB9ST)
Changing the method to "absolute log of exponential sum of ratio, smaller(real(z), imag(z)) divided by larger(real(z), imag(z)), scale = 100".
(https://copy.com/ESquswZenHOe)
The patterns in these fractals are dependent on the bailout value used, the next three pictures show the effect of increasing the bailout value. Values used are 16, 32 and 64.
(https://copy.com/VFR1Gu4do7Bu)
(https://copy.com/lFpwfLpECl8r)
(https://copy.com/IIbKo6oDPbqJ)
The final version uses a different method of selecting between inner and outer colouring methods. Titan's summary tab has a bug where "Colour selection" is shown as "unknown" so here is the colour tab from Saturn:
(https://copy.com/TGRh1RH8APNF)
(https://copy.com/9TU7rD9mZSqm)
Adding transforms to this fractal produces some familiar features:
(https://copy.com/r9XsraEEY2AZ)
(https://copy.com/i8Mlaygb2v9K)
(https://copy.com/F1PVG4vD2h2o)
(https://copy.com/CVvuKkUpUEqW)
Title: Re: Negative multibrots
Post by: Roquen on January 03, 2014, 02:18:17 PM
Which of the two possible square roots do you pick?
There's only one "principle" square root.Title: Re: Negative multibrots
Post by: youhn on January 03, 2014, 06:08:27 PM
Judging by the pictures it looks like the divergent bailout is used (the same bailout method as the standard Mandelbrot set),...
True. XaoS uses the standard Mandelbrot settings as default. I only added a minus sign as Tglad suggested and played around with the number of iterations and bailouts. Download XaoS for free and try for yourself. The program is pretty limited but has some cool features I did not find in other fractal software (real time zooming and color cycling).
This evening I'll try to create a video to show increasing bailout condition on the negative mandelbrot.
Title: Re: Negative multibrots
Post by: youhn on January 05, 2014, 01:29:46 AM
Not so exciting as i thought, but here the short HD video of the increasing bailout condition:
http://www.youtube.com/watch?v=2IvEctTlPU8 (http://www.youtube.com/watch?v=2IvEctTlPU8)
Made using XaoS and ffmpeg.
http://www.youtube.com/watch?v=2IvEctTlPU8 (http://www.youtube.com/watch?v=2IvEctTlPU8)
Made using XaoS and ffmpeg.
Title: Re: Negative multibrots
Post by: claude on April 30, 2015, 05:36:56 PM
Using Lyapunov exponents for colouring negative multibrots seems to work well, even able to get "atom domains" similar to the regular Mandelbrot set by seeing when the derivative (as opposed to the iterate itself) gets really small (Fragmentarium source attached, each atom domain accumulated with cumulative additive blending, final Lyapunov exponent is used to colour the minibrots black):
(http://mathr.co.uk/mandelbrot/2015-04-30_Mandelbrot_set_for_z%5E-2+c_coloured_by_Lyapunov_atom_domains.jpg)
(http://mathr.co.uk/mandelbrot/2015-04-30_Mandelbrot_set_for_z%5E-2+c_coloured_by_Lyapunov_atom_domains.jpg)
Title: Re: Negative multibrots
Post by: lkmitch on May 01, 2015, 05:42:08 PM
I find that using a high bailout and few iterations is good for clearing up some of the visual noise with negative exponents. At least, it'll allow you to see where the structure is. This one is with an exponent of -2, 250 iterations, and a bailout of 1E12.
Title: Re: Negative multibrots
Post by: knighty on May 01, 2015, 08:06:34 PM
That's cool!
Thank you claude for the fragmentarium script. :)
Thank you claude for the fragmentarium script. :)