Download Mandelbrot Zoom !LINK!

I've been working on a C Mandelbrot set program for the past few days and I managed to make it work fine, however, my end goal is to be able to smoothly zoom in the set with my mouse and that's something I haven't yet been able to do yet so I might need a bit of help !

To zoom in, imagine drawing a line from the (mouseRe, mouseIm) zooming centerpoint to each of the corners of the visible area, forming a lopsided X. Based on the zoom amount, find 4 new points a certain fraction of the distance along these lines, these points will give you your new rectangle. For example, if you are zooming in by a factor of 3, find a point 1/3rd of the way from the centerpoint to the corners. This will produce a new rectangle with sides 1/3rd the size of the original and an area 1/9th the size.

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Alternatively, if you want the center of the view to jump to a certain position when you click the mouse, calculate mouseRe and mouseIm as above and then offset the corners of the view rectangle by the difference between these values and the center of the view rectangle. You could store these values at the time the mouse button was first pressed down, and use them to zoom in as long as it is held.

The Mandelbrot set is one of the best known examples of a fractal. It is a structure with an infinite amount of fine detail: you can zoom in on the edge of the fractal forever, and it will continue to reveal ever-smaller details.

This is the idea behind fractals. In a fractal image, you can keep zooming in and the complexity will keep increasing. Most fractals have some sort of self-similarity where you see the same shapes and patterns as you continue to zoom in.

The second one is a fractal tree where more and more branches are added to the ends of previous branches. This is how CGI trees are made in movies and video games. The last image shows the famous Mandelbrot set and demonstrates the characteristic of self-similarity as you zoom into it.

After doing this calculation for every pixel in the image, we can color the diverging pixels based on their iteration count and color the non-diverging pixels black to get the very famous and distinctive shape of the Mandelbrot set as shown here. The red box in these sample images shows where each has been zoomed in to produce the next image. Notice the self-similarity and recursive nature of the patterns.

Because the Mandelbrot fractal is so well known, some names have been given to certain parts of the fractal, and to certain shapes. As you explore the Mandelbrot set, you will start to recognize some of these shapes, and you will see how they repeat over and over again as you continue to zoom in to greater and greater depths.

I want to combine Manipulate with ManbelbrotSetPlot just to get Mathematica to give me a quick and dirty Mandelbrot Zoomer. I want to be able to single/double click on a section, and have it zoom in on that section. All I've been able to get is an adjustable viewing window, but that's really not satisfactory. I imagine a Locator might do the trick, but I'm unsure how to get it to work. How might I accomplish this?

Another way to zoom around is using Manipulate. Here we use a 2D slider to set the position and a regular slider to set the zoom. It scrolls more evenly if you hold down the option key as you move the sliders.

Here is a less quick and dirty version that includes a few more features. To zoom in, you simply click and drag to select a rectangle. Generally, you've got to hit the Generate button to produce the next picture. If you just click on the image, then a picture of the corresponding Julia set will be printed to the notebook, together with the command to generate it.

I do have an idea of how to achieve "infinite" zoom: if you render the picture in software, and use an arbitrary-precision floating point library like GNU MP, you should be able to zoom in a lot more. The drawback is that it would be exteremely slow compared to the OpenGL-based version, although you could combine the two methods: use hardware-accelerated rendering for normal zoom levels and software rendering when high precision is required (not to mention, you can parallelize the software implementation as well as speed it up with SIMD instructions).

A few weeks ago while browsing YouTube for Fractal movies I came across a video that claimed to be (as of its post date of January 26, 2010) the deepest zoom movie on YouTube. The video was of an interesting area, but what really caught my eye was the information that the video took six months to render. Six months on twelve CPU cores.

Interpolation: The original render calculated every frame, whereas Fractal eXtreme calculates key frames, separated by a magnification increase of 2.0. The original render ran at 30 frames per second for 312 seconds, for a total of 9,360 frames. Some of them are stationary frames at the beginning, but because the zoom speed drops at the end a higher percentage of the frames are towards the end where rendering takes longer, so the ratio of 9,360:916 is probably pretty reasonable and FX does about ten times less work. FX speedup: 10:1

Being able to calculate the same movie (same maximum iterations, depth, and target location) in 18 hours is nice, but we can do better. And, to be honest, when you interpolate key frames that have aliasing you do lose some quality. However if you antialias the key frames before interpolation then you actually get a higher quality movie, with more of the subtle detail visible, even after interpolation. And if you antialias and render at a higher resolution, you get an even higher quality movie. And, due to the nature of the Fractal eXtreme zoom movie interpolation system, you can play the movie back at a higher resolution than it was rendered and actually increase the quality even more.

Same location, same maximum iterations, slightly higher maximum zoom depth (to finish all the way into the final Mandelbrot set), 960540 render resolution (up from 640480), 33 antialiasing, and a final output resolution of 1280720.

The higher resolution and antialiasing make a stunning improvement to the video quality. The moir patterns and flickering pixels are gone, subtle ribbons of color are consistently visible even when they are less than a pixel wide, and the boundaries between bands are smooth. Because the interpolation is done after the time-consuming render it was possible to experiment with different zoom speeds, which allows the movie to go fast when there is little detail, and slow down whenever a mini-brot or other tourist attraction appears.

I looked through the comments on that movie and and the creator says that he actually calculates only ~700 pixels per frame, for a total of 12,500,000 pixels calculated. The rest is filled in with digital-zooming/interpolation, which explains the blurriness.

I just checked and my movie required calculating 1.1 billion pixels (after pixel reuse and guessing), which is why my movie is sharper. He said that at 300 digits precision Mathematica was giving him about 20,000 iterations per core per GHz per second. At 1,000 zooms (1,088 bits precision, 327 digits) FX gives me about 300,000 iterations per core per GHz per second (8.9 million iterations on my 2.2 GHz 8-core Sandybridge with Turboboost going up to 3.2 GHz with guessing disabled).

Awesome work!! These kind of optimizations are so important and I believe mandelbrot will increase in popularity as we are able to examine it more and more in real time. I am running a Mandelbrot zoom on my 7900K right now. I imagine my code is closer to the original video you are comparing to than yours. 1600 1080p frames is taking about 3 days with a max iteration of 5000. Could you give some details about your max iteration profile along the zoom?

At a glance it may seem completely hopeless that we could ever do such a thing. Just for a start, it's not obvious that the Mandelbrot set consists of one single piece ("connectedness" is the relevant mathematical property). Maybe we could start drawing it, go around one of the islands making up the set, return to our starting point, and find that we'd left out a lot of interesting details. It turns out that that is not a problem - it actually is a single piece, so there is a single boundary we could trace that would outline it perfectly. But the boundary is of infinite length! So if we want to draw it, proceeding at a finite speed in terms of length covered per unit time, it will take forever. Even if there's a way around that, the shape is also, as implied by the exercise of zooming in on pictures of it, an extremely complicated curve. It's not obvious that we can compute this entire curve in any useful way even with a lot of computer assistance.

I was really into fractals in college (I know...) when I was making rave flyers (I know!) for a friend's parties in Iowa (I know! I know! Shut up already!). Anyway, the thing that I really used to love doing with this fractal application that I had on my computer was zooming in to different parts of the familiar Mandelbrot set as far as I could. I never got very far...between 5 or 6 zooms in, my Packard Bell 486/66 (running Windows 3.11) would buckle under the computational pressure and hang. Therefore, I absolutely love this extremely deep HD zoom into the Mandelbrot set:

The final magnification is e.214. Want some perspective? a magnification of e.12 would increase the size of a particle to the same as the earths orbit! e.21 would make a particle look the same size as the milky way and e.42 would be equal to the universe. This zoom smashes all of them all away. If you were "actually" traveling into the fractal your speed would be faster than the speed of light.

The magnification factor is so much less in the video above but that one's more fun/artistic. And 10^10000 is such an absurdly large number1 that there's no way to think about it in physical terms...the zoom factor from the size of the universe to the smallest measurable distance (the Planck length) is only about 10^60. 2351a5e196