Deep zooming to interesting areas

[simon.snake]

Hi

My deep zooming of the Mandelbrot is mediocre at best.  I can deep zoom, but I'm not sure if I'm choosing the best places to zoom in and I wanted to hear how other people do it.

Some of my images are ok, but I don't seem to get as much variety as others.  Take the flake or tick tock.  Very varied, and not the usual 'circles' of doubling that usually occur the further one zooms in.

Do you prefer to stay close to minibrots, or go as far away from them as is possible?

Is it simply a matter of trial and error?  Do you go in a bit, then have to back track and zoom back out and try another location nearby?

How do you work out how long to zoom in for, or when to zoom back out to try another location?

How do you know if the spiral you are zooming in to is infinite or not?

So many questions...

If anyone has any pointers on what it is they are looking for, how they go about finding these locations, I would be most grateful.

If you can include some visual tips, e.g. show images with indication what you look for, that would be helpful.

Thanks in advance.

[SeryZone]

My deep zooming of the Mandelbrot is mediocre at best.  I can deep zoom, but I'm not sure if I'm choosing the best places to zoom in and I wanted to hear how other people do it.

Don't worry about this. Just zoom. I zoom about 2 years and even NOW sometimes can't choose nice location. Just zoom. Without question 'What's place to choose?'!!!

Some of my images are ok, but I don't seem to get as much variety as others.  Take the flake or tick tock.  Very varied, and not the usual 'circles' of doubling that usually occur the further one zooms in.

I can't explain, how to morph Julia, you must understand this yourself. But I can show to you very interesting methods, how to zoom.

Do you prefer to stay close to minibrots, or go as far away from them as is possible?

This is custom taste. I rarely make shots near minibrots, oftenly - morph Julia. All is possible!!! I, Karl Runmo and Dinkydau very good demonstrate this:
https://www.youtube.com/channel/UCt0uOdnSb3tDxW35xt4JCYg - Karl Runmo
https://www.youtube.com/channel/UCGdUeXGPqqcjdaVovbLp02A - Dinkydau Linteum
https://www.youtube.com/channel/UCsRbcv18VcWJVtfAtbKV1Vw - I (Sergey Pechonyy or SeryZone)

Is it simply a matter of trial and error?  Do you go in a bit, then have to back track and zoom back out and try another location nearby?

I prefer deep and hard places. Yes, it is matter of trial and error. But if you like non-denser figures, you can zoom far deeper! If location is denser - I need lot of time for rendering, but I don't try and not advice to zoom very deep.

How do you work out how long to zoom in for, or when to zoom back out to try another location?

My patience is not limited. For some locations I need hours, days and even weeks (for 10^8000 and deeper, for example). But for tests - 5 minutes, no more! hehee

How do you know if the spiral you are zooming in to is infinite or not?

Spiral is infinite, but limited by iterations and precision. If you want to zoom to spiral - use perturbation, it use approximation - e.g. if your location need from 500,000,001 to 500,010,485 iterations (for example), perturbation 'Skip' 500,000,000 iterations, therefore works very fast.

So many questions...

If anyone has any pointers on what it is they are looking for, how they go about finding these locations, I would be most grateful.

If you can include some visual tips, e.g. show images with indication what you look for, that would be helpful.

Hehehheee, I alone by life. I never had 'teachers', I learn all by myself, but now I can teach you from begginning to end how to zoom Mandelbrot Set =) Visual tips is impossible, because needs calculations unknown coordinates. programms can't find unknown interesting places, because don't recognize visual shapes.

Add me at Skype: mstrkrft_xxl
Please, answer, all questions are welcome =))  

Good luck in exploring Fractal Universe!!!

[ellarien]

I'm only a beginner at this myself (I've only explored a couple of hundred locations to 1e100 or more), but I have a few tips. I'm sure the real experts know a lot more.

Going close to a minibrot results in very dense julias deeper down, whereas going far away makes a lot of spidery lines and generally fewer iterations at a given depth.

The sharp end of a minibrot gives you needle-shaped julias; the blunt end gives apples; off to the side gives diamonds/dragons. Try mixing up which way you go from successive minibrots if you want a variety of shapes.

To get interesting shapes, go for the corners or off-center islands-within-islands, more than once. The prettiest patterns come from being systematic about that.

To get non-infinite spirals, ride an infinite one a few turns down, then jump into an island.

As a rule of thumb, whatever kind of shape you zoom into will show up again as you zoom further in.

As a visual example, here's a fairly shallow 'diamond' island:

<a href="https://www.flickr.com/photos/ellarien/11119593525" title="fractal1127_004 by ellarien, on Flickr"><img src="https://farm8.staticflickr.com/7423/11119593525_9b873f457e.jpg" width="500" height="313" alt="fractal1127_004"></a>

The big spirals separating the sections are infinite. All the little twiddly bits around the border are also infinite spirals, and the more you zoom in on them, the more elongated the little islands inside become. If you go for a few changes of direction while doing that, you'll get branches.

Here's an example from a fairly random exploration with Kalles Frakaler, showing funky branches and spiral arms.
<a href="https://www.flickr.com/photos/ellarien/13937247634" title="apr19_01 by ellarien, on Flickr"><img src="https://farm8.staticflickr.com/7207/13937247634_175facec95.jpg" width="500" height="281" alt="apr19_01"></a>

And here's the location. You can try zooming out from there to see how I got there, or in to see how the features look once the doubling gets going.

Re: -0.16065657415302501867320352028064469250209645754318964636979132166883104121582980647958037851678063533139798420579570750032848111059531978615606365384845096179245335865229690254827844119540917062487699488456283258919024391941
Im:
-1.03710888947698724903979678931627246302501079891436898575464615209743918324026979781948044222422096610585519718149824317984915540963659290952992354218974953270221940138320539201753422997267866657453016074638500570715339362434

Zoom: 8.21896234715E208

(This needs about 60K iterations at that level. I would not recommend trying it without the perturbation method to speed things up!)


I hope that helps, and apologize if I'm only pointing out the stuff you already know! I see SeryZone, who is one of the real experts, has also posted while I was writing this, but I'll let it stand.

[simon.snake]

Thanks for the replies.

I agree there is a lot of information to be gained by viewing deep zoom videos, but they only tell one small part of the story - the successful location.

Questions are, for example:

I wonder how many times the person who created the zoom movie had to zoom in and out in order to find that path?
What were they looking for?
How did they select the location they did?

Is it purely trial and error, or does having an understanding of the maths behind the fractal give an advantage?

Thanks again

Simon

[Kalles Fraktaler]

Simon, I don't agree with you, I think you have made beautiful and interesting Mandelbrot images.

Anyway, we are almost just zooming in random, thinking that it may be a cool result when going of the center at some point, because the shapes at this point will reappear in double, but without any clear sense of the result, without any real control.

But I think it would be possible, to control the zoom path to get a desired shape, if we only could systematize the result of going off the center at different locations for different shapes.

I'll show what I mean with some images.
First I want a basic shape and then stretch it further to create something interesting. Going close to a near minibrot gives a julia that can be shaped further, and in the elephant valley of a minibrot this julia is round, which make it a good starting point.
The round julia is first stretched and then zoomed in the center so that it becomes a cross, this will be the basic shape.


We can then zoom in somewhere in this shape and make it be doubled and stretched around this point.
So here is a list of some different stretching/doubling points:


Going into the (infinite) spiral for some time and then centering on a julia will make the cross be doubled around a big spiral


Going into a small julia in one of the arms makes the arm be twisted 90 degrees


Going into a more centered small julia in one of the arms and the twisted arm is a little thinner


Going in a small julia in the edge makes the result less symetric


Going in to the far edge of an arm and the result is a double cross


Going along the edge for a while and then going in to a julia makes the result look almost like a triangle

And then there are of course infinite possibilities to stretch the basic shape and continue stretching the result shape into more complex shapes...

[youhn]

Am I right in thinking this process is called julia morphing?

I'll try to describe some of the rules of thumb that I use in exploring the Mandelbrot set. Zooming into a particular kind of shape, gives more of the same (slight variation though). When 2 distinct shapes are seen, you can zoom halfway between those to get a mix of the two. This way will keep the different shapes visible in the same level of zoom, if that makes sense. Compare this to the common deep zooms into the "light" antenna areas, in which a sequence of different shapes becomes visible by zooming through lots of different levels. I like to combine things for contrast. Not only between dark and light, but also between dense areas and less dense. First zoom near the border of the bigger bulbs, then after a while try to find less dense areas. Results can look like this:



The following relationship gives hints about interesting location. Counting from the biggest bulb on the right to the smaller ones on the left, you can see the onset of chaos in the bifurcation diagram of logistics map. This makes me think that just left of the fourth biggest bulb are lots of interesting locations:



Also note the node in the bifurbication diagram, on the same spot where the antenna of the Mandelbrot set has a node.

[simon.snake]

Thanks for those images Karl, along with your words of wisdom.

It has really helped give me a fresh perspective on fractal zooming.

I hadn't really got a feel for what to expect from different areas within one location, so probably kept going around in 'circles'.

[Kalles Fraktaler]

Am I right in thinking this process is called julia morphing?

Yes I think so too. Maybe I should point out that I am only speaking for myself when I say I have no clear sense of how the result will turn out when selecting a point of stretching/doubling. Others here, Dinkydau for example, have done many excellent Julia morphings.
But then.... How do I do an Easter bunny? Don't tell me it's impossible!

[aleph0]

Robert Munafo's Mu-Ency web site is well worth looking at for anyone not aware of it. This page is very relevant for zooming guidance into embedded Julias:
- http://mrob.com/users/jwm/20101213.html

Many years ago, Jonathan Leavitt did some trailblazing deep zooming exploration of the M-Set using Fractint. His work is hard to find now on the web, but he achieved amazing results, given the limitations of Fractint and typical hardware available at that time. Given the variety and unique nature of the images he produced, he must have determined certain methods to guide his exploration, though I have never seen them described anywhere. Use this Mu-Ency page as a starting point to find his work:
- http://mrob.com/pub/muency/leavittjonathan.html

More pages from Mu-Ency:
- http://www.mrob.com/pub/muency/2foldembeddedjuliaset.html
- http://www.mrob.com/pub/muency/secondorderembeddedjuliase.html

At one time, Eric Bazan's web site had some excellent tips for finding interesting areas, but I can no longer find any way to link to them in the current web site, assuming they are still there:
- http://www.fractalfreak.com

This page is also worth a visit:
- http://www.miqel.com/fractals_math_patterns/mandelbrot_fractal_guide.html

And these (posted by the author of Fracton):
- http://www.fracton.org/fmlposts/bristling_energy_p1.html
- http://www.fracton.org/fmlposts/bristling_energy_p2.html
- http://www.fracton.org/fmlposts/two_left_turns.html
- http://www.fracton.org/fmlposts/shut_eye.html
- http://www.fracton.org/fmlposts/line_method.html

Edit: Added a few more links.

[aleph0]

And this, posted by Ingvar Kullberg in 2003 on the sci.fractals newsgroup, in response to questions from David Knaack...

Hi David,

Below I've recycle a response I wrote for a somewhat
similar question on news:alt.fractals for a while ago.
Hope you will find take at least some advantage of it.
Thus:

"If you zoom against the centers of stars and spirals,
so called Misiurewizs points, that is spots were there
are infinite emptiness, nothing seems to change. If
you step aside to another star center the same thing
happens. Now the filaments of the M set are constituted
by the play between infinite emptiness and infinite individual
fullness, that is the spots were the minibrots reside. If you
don' t see any minibrot in the filament, these spots yet
revile themselves by sending out first two filaments, then
4, 8, 16 filaments etc. Now a suggestion how to find
more and more complicated atterns in the M set.

1) zoom in a minibrot on a filament.

2) Near the minibrot, zoom in to the fine structure
(I call these structures "secondary decorations" as
they are not part of the minibrot).

3) Where you expect to find a new minibrot you
will instead find a new phenomena called "embedded
Julia sets", virtual Julia sets" (I call it "Julia-like barriers").
Oh yeah, the new minibrot is there far deep inside
these barriers. More of this can be read at:

http://www.mrob.com/pub/muency.html

4) A useable principle is: The longer you zoom besides
the minibrots, the more interesting patterns you will
find when you finally "dive" into the spot of a minibrot.

Another method to find more and more complicated
patterns is to zoom into a sub-sub-sub- sub-etc-bud.

In other word. The border of the Mandelbrot set IS
more and more complicated as the magnification
increases. I have myself made many zooms far
beyond double precision using fractal eXtreme
and found it well worth the effort. However
because the calculations becomes very slow when
the math-chips no longer is used most people
find it more fun to apply a lot of filters, kaleidoscope
transformations etc on low magnification-motives
(at high magnifications most filters are of no use)."

------------------------------
Regards
Ingvar Kullberg


Ingvar produced this PDF for one of his zoom journeys:
- http://klippan.seths.se/lists/Diverse/Blomkalsfort.pdf
- It's in Swedish, but he has some description of the zooming method from page 18 onwards if you feed the text into Google Translate.


Edit: added PDF link.

[Dinkydau]

I think it's important that an up-to-date and simply accessable tutorial is made. Right now it's difficult to get started, even for people who would want to. I used to make fractals with apophysis and I thought the mandelbrot set wasn't really interesting, until I got inspired by stardust4ever who was making rows of X's. We haven't heard much of him anymore, but he's still the person who inspired me. I decided to continue where he stopped and extend the zooming technique to do more. By doing so, I have found the mandelbrot S and evolution of trees. Those shapes have convinced myself that there is still a lot of stuff hiding in this set. There are more unique things to be found! Except... I'm out of ideas. For now this is what I have been able to contribute, and maybe someone else has to continue, like I continued where stardust4ever stopped.

In the past I have recorded two video tutorials, but I couldn't manage to keep them in any way short and I found them not clear enough. In the tutorials I tried to explain the following points about the mandelbrot set:
1. minibrots
2. julia sets
3. symmetry
4. repetition
5. morphing
6. shape-stacking

Those seem to be 6 separate points, but what makes explaining them really difficult is that all of them happen together. In the end they're all just properties of one single phenomenon that I don't fully understand either.

Another problem is that explaining the mandelbrot set is not the same as teaching how to explore. In order to work with the software and explore efficiently, quite some understanding about math and software is required. I always have a calculator open together with fractal extreme.

What is also needed is a good computer. You can still find nice things with an outdated computer, but the faster your computer is, the easier it becomes to explore the mandelbrot set.


The golden rule behind most of my images is the following. Its aspects can be used to do pretty much everything that you find in my gallery:
Given a location with a zoom level n, moving away from the center to a different center has the following effect:
The shape at zoom level n is doubled at zoom level 1,5n in such a way that the rotational symmetry becomes 2-fold.
At 1,75n the symmetry becomes 4-fold.
At 1,875n the symmetry becomes 8-fold.
...
In general: the zoom level increases in steps of 2^-1, 2^-2, 2^-3, ... and goes on forever. The symmetry increases by a factor 2 for every extra step. The limit of the sum of all of those steps 2^-1 trough 2^-n where n goes to infinity is 1, so after infinitely many steps we arrive at a finite zoom level. Indeed, at a depth of 2n, twice as deep as where we went off center, there is a small mandelbrot set, where the symmetry goes to infinity.
The rule itself has not been proven as far as I know and there are endless exceptions where it is not exact. Sometimes shapes appear a little earlier than the rule would predict, but the small mandelbrot set will never occur FURTHER than 2n. (I think I know what the inaccuracy is, by the way.)

[Kalles Fraktaler]

And here is the frisky Easter bunny

One of the links from aleph0 led me to this site with lovely images (unfortunately without any location parameters)
http://www.fractalfoundation.org/gallery.html

[simon.snake]

Thanks for all the replies guys.

This thread is turning into a very useful resource and source of valuable information.

Going to take me a while to get my head around it.

Simon

[ellarien]

Indeed, at a depth of 2n, twice as deep as where we went off center, there is a small mandelbrot set, where the symmetry goes to infinity.

That's a very useful rule, particularly for those of us with limited hardware and patience. I think I've been using it subconsciously, but it's nice to see it written down. Do you have a feel (or a formula) for how much the maximum iterations will increase between the last off-center location and the final minibrot? My impression has been that if I want to end up with a recognizable minibrot image at 10^6 I need to start the final zoom-in at around 50-100K, but I haven't made a systematic study of it.

One thing I've wondered about, not that it's particularly relevant to interestingness of locations, is whether there's any way to predict the orientation of the final minibrot.

[aleph0]

Something to consider when searching for interesting deep zooms in the M-Set, is that symmetries and period doubling (bifurcation)/period halving (reverse bifurcation) patterns are key to finding the embedded Julias and M-sets. Working with a complex colouring scheme can work against this by obscuring the symmetries and patterns. It's usually more effective, in my opinion, to work with a simpler colouring scheme and distance estimation. I've written a colouring formula for UltraFractal that replicates Robert Munafo's colouring approach seen in the Mu-Ency site, which combines DE, dwell and binary decomposition. It's very effective for exploration, but UltraFractal becomes too slow for this lifetime when zooming deep. It would be great if the programs that have implemented the perturbation/series approximation technique could also implement a distance estimation option. I see it's planned for Mandel Machine, but I'm not sure whether it's planned for the other programs.

Robert has a description of his colouring approach here:
- http://mrob.com/pub/muency/color.html
- He scales the distance estimate to brightness, so that points near the set are dark. Robert uses a fixed scaling, but I made the scaling configurable in my UF formula.
- Dropping the binary decomposition and combining just dwell colouring and DE is also very effective. Again, I made that configurable in my formula.

Paul Derbyshire (pauldelbrot) has also done excellent work with distance estimation and hybrid colouring schemes:
- http://www.fractalforums.com/images-showcase-(rate-my-fractal)/de-test-i
- http://www.fractalforums.com/images-showcase-(rate-my-fractal)/weave
- http://www.fractalforums.com/images-showcase-(rate-my-fractal)/the-core
- http://www.fractalforums.com/images-showcase-(rate-my-fractal)/protist
- The image links are, unfortunately, all broken. Paul/moderators - any chance of getting them fixed? They were stunning, beautiful images and inspired my own investigations into distance estimation.

I often use a pure black and white DE rendering when exploring; it's actually greyscale, with a very 'sharp' profile to black near the set (using similar techniques to Paul and another fractal software author, Duncan Champney).

Once an interesting area is found, one can revert to a more complex colouring scheme to refine the visual appeal of the image.

Some more links to explore:
- http://mrob.com/pub/muency/silverrollo.html Sadly, Rollo is no longer with us, but he was one of the fractal imaging and exploration pioneers. Someone posted a few issues of his Amygdala newsletter on the forum a while ago.
- http://mathr.co.uk/blog/2013-02-01_navigating_by_spokes_in_the_mandelbrot_set.html From Claude Heiland-Allen's blog (mathr), well worth a more general browse. In this post, check out the black and white renderings. They may be too plain visually for some, but I love them; they fully reveal the beauty of the symmetries and patterns.

One fascinating aspect of the embedded Julias, as pointed out by Robert Munafo, is that although they appear connected they are actually Cantor dusts, i.e. disconnected. This becomes very apparent if you spend any significant time zooming with a DE-based rendering. Structures which appear connected at a certain zoom level continually 'melt away' as you zoom ever deeper. Again, this can be very difficult to spot when using a complex colouring scheme.


Edit: Added Cantor dust comment.