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Author Topic: Shortcuts in juliasets  (Read 3011 times)
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TheRedshiftRider
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« on: June 19, 2015, 08:31:33 AM »

Yesterday I posted an image of a dense julia-set (http://www.fractalforums.com/kalles-fraktaler-gallery/simple-density). I'm going to explain how I did this.

Morphing is frequently discussed on this forum and the rules are clear about how it works:
Quote
The pattern is doubled and wrapped around the point, and also the area nearest the point is magnified.

But not just the pattern of the juliaset is doubled but also the points you have zoomed into in previous stages of the morph. Look at the 4th image here to see what I mean:
http://imgur.com/a/ShV0U


 These points contain the previous stages of the morph. And can be used as a shortcut to a next julia-set. In general the shortcuts that are closer to the center of the juliaset are shorter. Look at the 4th image again. The center is of course the fastest but it only doubles. The point close to the outside of the juliaset contains the julia-set of image 1. The one closer to the center contains the one of the second images. And the one closest to the center contains julia-set 3.

Basically what I did with the image I made yesterday was I just zoomed into the shortcut which was close to the center. I did this several times and the julia-set became more complexe and it did not need a lot of zooming. And I think the rendertime is also greatly reduced compared to deep complex morphs.

The location of the image is in the file in as attachment.

* lost.rar (0.95 KB - downloaded 268 times.)
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Chillheimer
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« Reply #1 on: June 19, 2015, 09:13:20 AM »

I can confirm your finding. complexity raises really fast this way
I have to say that I already did this, but more intuitivley than in the analytic way you did it. Fascinating what happens when you really repeat this very often.

I'm still trying to find out how and why this works.
Unsure if I use the mathematical correct words to desribe this:
I guess it's simple periodic doubling (but of an already complex shape).
═f you doulbe an already very complex part of the julia set (instead of e.g. a simple spiral at the beginning) you double that complexity. its like 128▓ compared to 2▓.
and if you repeat this process all over the complexity must rise much faster than if you take the long route.
though of course you will reach 128 too when starting with 2▓. but at much deeper zoom depth.
it might not "speed up" the distance to the next minibrot, but the minibrot is much more complex than without.

the only "downside" to this is that you will only repeat the simple patterns you used in the very beginning, the shapestacking phase.
with the long route you can still change details in your zoom path at any time.
details that of course show up later. but those will be subtle details in complex deep julia-sets like your example from yesterday. so it doesn'T really matter in that context.

I guess you could call it level-2-shapestacking. shapestacking of julia-sets. smiley

very interesting. nice share. smiley
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TheRedshiftRider
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« Reply #2 on: June 19, 2015, 09:53:53 AM »

I can confirm your finding. complexity raises really fast this way
I have to say that I already did this, but more intuitivley than in the analytic way you did it. Fascinating what happens when you really repeat this very often.
Unsure if I use the mathematical correct words to desribe this:
I guess it's simple periodic doubling (but of an already complex shape).
═f you doulbe an already very complex part of the julia set (instead of e.g. a simple spiral at the beginning) you double that complexity. its like 128▓ compared to 2▓.
and if you repeat this process all over the complexity must rise much faster than if you take the long route.
though of course you will reach 128 too when starting with 2▓. but at much deeper zoom depth.
it might not "speed up" the distance to the next minibrot, but the minibrot is much more complex than without.

the only "downside" to this is that you will only repeat the simple patterns you used in the very beginning, the shapestacking phase.
with the long route you can still change details in your zoom path at any time.
details that of course show up later. but those will be subtle details in complex deep julia-sets like your example from yesterday. so it doesn'T really matter in that context.

I guess you could call it level-2-shapestacking. shapestacking of julia-sets. smiley

very interesting. nice share. smiley
Thanks for your reply. What you say is correct I think. Though, I think it is possible to create a bit more complex shapes by changing which shortcut you use at every stage of the morph, this will take more depth but the shapes will be more complex.
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TheRedshiftRider
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« Reply #3 on: June 19, 2015, 01:28:49 PM »

I tried to find a more complicated location:


For this one I used varied shotcuts.

* simple density v2.rar (0.51 KB - downloaded 260 times.)
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Chillheimer
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« Reply #4 on: June 19, 2015, 01:50:01 PM »

hehe, I know that guy!
http://chillheimer.deviantart.com/art/I-see-you-500674399
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TheRedshiftRider
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« Reply #5 on: June 19, 2015, 02:12:02 PM »

laugh Nice one.

One other thing I found which some may consider good or bad is that the set contains in this context ''eyes''. I think it looks cool in most cases but you probably have those with very deep morphs with the ''normal'' method.
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TheRedshiftRider
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« Reply #6 on: June 30, 2015, 01:55:50 PM »

it might not "speed up" the distance to the next minibrot, but the minibrot is much more complex than without.

I'm still not sure about this. I did some test and some had a shorter distance to the minibrot. (I used a method of just zooming, so not very accurate) I think it depends on whether a shortcut is taken in a juliaset as last part of the morph before it starts doubling. And maybe also if in the morph other minibrots have been passed or not.
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TheRedshiftRider
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« Reply #7 on: June 30, 2015, 10:44:14 PM »

I found out that fractal universe made a video with this technique a year ago.
<a href="http://www.youtube.com/v/4Yg1SeFjOik&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/4Yg1SeFjOik&rel=1&fs=1&hd=1</a>
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Adam Majewski
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« Reply #8 on: July 01, 2015, 03:17:30 PM »

Is it similar techique as in :

http://mathr.co.uk/blog/2015-05-18_two_spirals_out.html

?
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TheRedshiftRider
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« Reply #9 on: July 01, 2015, 03:26:49 PM »


Almost. The main thing im talking about is the shortcut near the center of the julia set. The main ''eye'' near the center as I call it.
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Dinkydau
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« Reply #10 on: July 28, 2015, 03:55:52 PM »

If you deviate from your path you can basically "continue" the old path from that shortcut you mention, if I understand it correctly. It saves a ridiculous amount of zoom depth. It can be used to achieve more complex results than otherwise possible but you have to sacrifice some levels of choice. In this image I used the method to make an S that contains shapes while still being at only relatively low zoom depth (in perturbation era understanding at least).


"Evolution of trees" is a more complex version of this method, but it's still the same kind of zoom method. It works by going into the shortcuts and morphing the julia sets there.
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TheRedshiftRider
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« Reply #11 on: July 30, 2015, 01:35:20 PM »

Thanks.
I think that morphing for images is more appropriate without shortcuts. But for movies the shortcuts are great.
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claude
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« Reply #12 on: July 30, 2015, 02:30:03 PM »

Binary decomposition colouring can help to find the shortcuts (including the central minibrot) when you have very thin filaments - follow the lines that start further out, they end on the tips of filaments surrounding a shortcut.  For these images I used an edge detection filter on integer iteration count and whether the final angle was positive (these create the grey grid), the filaments are coloured with distance estimate.

First 2 images:
Code:
real=-1.9963810738192679122181468389365117014665310349376e+00
imag=7.4685300102574615047563223151397682359866402430498e-06
radius=1.2325951644078309e-32
Second 2 images, zoomed in slightly below left of center:
Code:
real=-1.99638107381926791221814683893651199888802191225853e+00
imag=7.46853001025746150475632231492495186779320342460899e-06
radius=7.7037197775489434e-34


* 2015-07-30_shortcut_decomposition_1a.jpg (76.04 KB, 640x360 - viewed 628 times.)

* 2015-07-30_shortcut_decomposition_1b.jpg (86.51 KB, 640x360 - viewed 582 times.)

* 2015-07-30_shortcut_decomposition_2a.jpg (4.85 KB, 640x360 - viewed 579 times.)

* 2015-07-30_shortcut_decomposition_2b.jpg (16.86 KB, 640x360 - viewed 618 times.)
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quaz0r
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« Reply #13 on: May 15, 2016, 10:30:37 PM »

have you guys posted any videos which clearly depict what it is exactly you are talking about here?
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TheRedshiftRider
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« Reply #14 on: May 15, 2016, 10:32:55 PM »

Of course:

<a href="https://www.youtube.com/v/x2g_dlsX5k4&rel=1&fs=1&hd=1" target="_blank">https://www.youtube.com/v/x2g_dlsX5k4&rel=1&fs=1&hd=1</a>
« Last Edit: May 16, 2016, 09:46:02 AM by TheRedshiftRider » Logged

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