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Author Topic: Fractally animating fractals  (Read 8085 times)
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Tglad
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« Reply #15 on: February 20, 2011, 12:45:18 AM »

Spot on Kali, so here are the answers-
1. In all of these videos, how does the speed of the small parts compare to that of the large parts?
The average speed is the same at all sizes.
2. How does the fractal 'self-connectivity' change from the first to the most recent videos?
They get more connected. The first (growth) video is not even always connected to itself over time, since there are gaps when the disks don't exist.
3. Which is the odd one out?
They all show two forms, e.g. cluster+tree, tree+sponge, apart from the orbitting planets which is just a cluster. (The growth fractal is void+cluster).

No, it isn't a contradiction that you can imagine a motion by using a space dimension, usually looks like these http://www.fractalforums.com/ifs-iterated-function-systems/antisierpinski/msg15366/#msg15366 which are not isotropic, though they are still conformal. Isotropic fractals generally don't look like anything natural when seen as a moving slice, if you have counter-examples then go ahead and post them in a new topic.

So back to these questions, if these dynamic fractals can have two forms, then can they have 3 forms, or 4 forms? How many forms can a dynamic fractal have?
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Kali
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« Reply #16 on: February 21, 2011, 05:12:35 PM »

Did I won any prize?  cheesy

About the "contradiction": If you look back, and as I understood (keep in mind I'm not so good at english), Hobold's idea is basically what you are pointing at: making time (or call it movement) an space dimension and draw the evolution through time along this dimension.
He didn't talk about a formula-generated 3D fractal and then taking 2D slices and animating it... he clearly mention taking each 2D frame and making a 3D graph out of it, just like the link you have given.
Maybe you are confusing it with The Rev's post?
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Kali
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« Reply #17 on: February 21, 2011, 11:00:00 PM »

Ok, he mentioned first a 3D spheres fractal whose moving 2D slices should look like your boiling waters dynamic fractal (indeed, if they are fractally sized and spaced, they will)

Then he said:
Another way to see this special quality of the growing circles fractal is the above construction backwards: take all animation frames and stack them on top of each other as slices of volume data. Some other examples will result in whipped cream, some in fractals with substantial directional bias, and only one example will result in an isotropic three dimensional fractal.

Isn't what you are talking about?

Sorry for bothering you with this, but I'm really interested so I want to see if I'm getting it well.
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Tglad
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« Reply #18 on: February 22, 2011, 12:51:56 AM »

Oh ok... well maybe you're right, I can't speak for hobold, but this is what I think:

- You can view an animating fractal by stacking each frame along a space dimension
Agree, yes it can be a helpful way to think about the animation
- The most interesting animating 2d fractals are the ones that when stacked are isotropic 3d fractals
Disagree. By the way, the growing circles fractal is not isotropic when stacked... the circles would need to grow then shrink.. and if you wanted growing squares or growing mushrooms it wouldn't be isotropic.

Back to the question.. how many forms can a dynamic fractal have?

There are 7 main classes of 3d fractal (void, cluster, tree, sponge, shell, foam, solid), but overall there is a 7x7 triangular table (http://sites.google.com/site/mandelbox/what-class-of-fractal), which means 28 forms. This means a total of 228 different classes of dynamic fractal (http://en.wikipedia.org/wiki/Combination#Number_of_k-combinations_for_all_k), which is 268 million classes... which doesn't make for a very good classification.
So instead we can classify these dynamic fractals by which triangular sub-table they range within. For example the 'void cluster + void sponge' dynamic fractal in the previous video ranges from void cluster up to void sponge, so includes dynamic fractals that also show a void tree form... this is natural because generally dynamic fractals that cover this range will show tree forms when going from a cluster to a sponge. The video is just a special case.
So in this classification, the most number of different forms a 3d dynamic fractal can change between is still 28, which is 7 main forms, and is called a 7-form void solid (it covers the full range from void to solid). The total number of different classes of dynamic fractal is then just 84, forming a 7x7x7 pyramid table.

What would such a maximum variation dynamic fractal look like? I'm not sure... it would obviously be massively shape shifting. The closest I could come up with is this boiling mud theme in 2d, after all boiling is a process of going from solid/liquid to gas (which is almost void). Most of the variation happens close to the surface:
<a href="http://vimeo.com/moogaloop.swf?clip_id=20052421&amp;server=vimeo.com&amp;fullscreen=1&amp;show_title=1&amp;show_byline=1&amp;show_portrait=0&amp;color=01AAEA" target="_blank">http://vimeo.com/moogaloop.swf?clip_id=20052421&amp;server=vimeo.com&amp;fullscreen=1&amp;show_title=1&amp;show_byline=1&amp;show_portrait=0&amp;color=01AAEA</a>
But this is a barely correct example, it would be interesting to see a real example.
I'm also not sure if the classification is the best, its too early to tell as we have so few examples. It would be great if the 3d fractal programs generated dynamic fractals, not just animating the parameters.
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PDN777
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« Reply #19 on: March 28, 2015, 12:56:40 AM »


Very cool work!

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Chillheimer
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« Reply #20 on: March 28, 2015, 11:31:27 AM »

thx for posting in this old thread - hadn't seen it before, very interesting! smiley
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hobold
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« Reply #21 on: April 01, 2015, 02:45:45 AM »

That reminds me I still have to explain with a picture rather than with a thousand words. Here is a sequence of adjacent parallel slices through a randomized Apollonius fractal.


The overall volume is a cube filled with spheres, with their radii following the usual fractal rules:


This shows the effect that I had in mind: large blotches are changing slowly, but are surrounded by smaller bubbles that change more quickly. Those in turn are surrounded by smaller, faster changing, boiling, bubbling droplets.
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