Newton-Raphson zooming and Evolution zoom method

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DinkydauSet's avatar
For more than a year I didn't have any inspiration. Actually I think my last two submissions were even pretty boring. Now I have many new ideas again. I have at least 5 more renders planned right now, deeper and with more iterations than ever before.

I have so much to say related to my latest image submission "Trees revisited" that I decided to write a journal about it. I hope to clarify what I mean by the term evolution. Information about what it is is spread out over comment sections and deviation descriptions. Also there's been a breakthrough in computer-assisted zooming, which is what's helping me to zoom this deep.

This is "Trees Revisited":
Trees Revisited by DinkydauSet

Evolution zoom method

Maybe the title of "Trees Revisited" is misleading because it's not really about the trees. It's the same old trees again. Instead this is a variation of what I have come to call the evolution zoom method. In general, given some shape that lies somewhere in the Mandelbrot set, evolution can be described as:
1. double the shape
2. morph one of the two copies
3. repeat by treating the morphing of step 2 as the new shape

julia_morphing.png (1024×533)
In the left image, 2 points are labeled 1 and 2 respectively. Zooming in on the point labeled 1, which is outside of the "shape" yields the middle image, a doubling. Zooming in on the point labeled 2 yields the right image, a morphing.

Doubling a shape can be done by zooming to 3/4 (as a good rule of thumb - it's a little more complex than this) of the depth of a minibrot of choice outside of the shape. The exact result depends on the choice of the minibrot. A doubling leads to two copies of the same shape next to each other. That's step one. Morphing one of them involves choosing a minibrot INSIDE that shape, so we choose one, but that means it's not inside the second copy of the shape, so the second copy gets doubled, causing both the morphed shape and two copies of the original shape to be present in the result, which is a set of shapes. By iterating the steps, the original shape and every morphing tied to an iteration of the steps are present in the result and all visible at once. That allows one to see how the original shape evolved, iteration by iteration of the steps, into the final morphing. That's why I call the result an evolution set.

Here's what's new: So at each iteration of the steps we have a morphing and two copies of the previous stage. The way I used to do step 1 in pretty much every previous render where I mentioned the word "evolution" was to morph one of those two copies, but I realized many other ways could be used to double. The only requirement is that the chosen minibrot is outside of the shape to be doubled. I tried a few things and this is the most interesting one I was able to find, at least thus far.

Automated zooming

There is also a lot to be said about the computer assisted zooming I have used to get to this shape. Claude on found an algorithm to determine the location and depth of the nearest minibrot inside a bounded region, involving the Newton-Raphson method. Because doubling and morphing shapes is equivalent to choosing a minibrot and zooming to 3/4 of the depth, knowing where the minibrot is and how deep it is allows one to find the coordinate and the depth of the morphing immediately. The coordinate is the same. The depth (the exponent in the magnification factor required for the minibrot to fill the screen) needs to be multiplied by 3/4. All you need to do is do a few zooms manually to make sure the algorithm searches for the correct minibrot and the computer can do the rest. Kalles Fraktaler has this algorithm implemented and I've been using it a lot. Some links to information about how it works can be found here:…

This is revolutionary. I think we can call it the best invention since the perturbation and series approximation thing. Zooming manually takes A LOT of time. I have spent days to several weeks just zooming for one image. Once the desired path has been chosen, it's a very simple and boring process of zooming in on the center until the required depth is reached. Note that this is not what the algorithm does. It doesn't need to render any pixels or use any visual reference whatsoever. It's a solid mathematics-based method and it works if you give it an "accurate enough" guess of where the minibrot is. Note also that it doesn't help in choosing a location to zoom to. You really just tell it "zoom into this center" and it finds the minibrot inside it for you, saving a lot of work.

It's pretty fast generally, usually faster than manual zooming, especially in locations with few iterations. Based on my experience with the Newton-Raphson zooming in Kalles Fraktaler thus far, I think it's actually a lot slower than manual zooming for locations with a high iteration count. Usually that's still more than made up for. You can work, sleep, study and (most importantly, of course) explore other parts of the mandelbrot set while the computer works for you, 24/7. If you have a processor with many cores you can let it zoom to several locations at once. Effectively that makes it faster in almost every situation.

The evolution zoom method involves a number of iterations of a few steps and I have found that generally it holds that the more steps taken, the better the result. The way the result looks like converges to a limit as the number of steps goes to infinity. The Newton-Raphson zooming allows me to perform more such iterations without as much effort as before. I always want to push the limits of what's possible, so I will perform those extra iterations, meaning I will be zooming a lot deeper. It will lead to shapes that are even more refined with even more symmetries and patterns.
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timemit's avatar
I do love these deep areas.. keep on exploring  :D
interesting! thanks for sharing. I wasn't aware of the implications of the newton-raphson method.
And the "trees revisited" is a beauty.