Hybrid fractals occur when you alternate between fractal formulas during the inside iteration loop of generating a fractal. The idea came from posts on the old Fractal Forums site.
For example, rather than the usual Mandelbrot Fractal iteration, you can modify the inside loop to alternate between the Mandelbrot, then the Burning Ship, then the Mandelbrot, and the Mandelbrot again. You then repeat the sequence for as many iterations as that pixel needs.
This gives a result like the following
The code to create this (and other hybrid formulas) is included with Visions of Chaos, but you can also see the OpenGL shader code for it here.
I have also used the same hybrid technique for 3D Mandelbulb Fractals as shown here.
The Burning Ship Fractal is a slight variant on the Mandelbrot Set Fractal.
The basic Mandelbrot Fractal formula is z=z^2+c. The Burning Ship Fractal formula is z=abs(z)^2+c.
The following image is the standard power 2 Burning Ship Fractal rendered using CPM smooth coloring.
Zooming in to the right antenna part of the fractal shows why it was named the Burning Ship.
The next 3 images change the exponent 2 in z=abs(z)^2+c to 3, 4 and 5.
The same power 2 through power 5 Burning Ships but this time using Triangle Inequality Average (TIA) coloring
The next 4K resolution movie shows a series of zooms into Burning Ship Fractals between power 2 and power 5 colored using CPM coloring
and finally another 4K movie showing more Burning Ship zooms colored using TIA coloring
All of the above images and movies were created with Visions of Chaos.
In the video game industry older games get remastered all the time. Hardware power improves so older games are adapted to have better graphics and sound etc. So for a nostalgic trip into history (at least for me) it was time for some Visions of Chaos movie remasters.
After recently adding the new Custom Formula Editor into Visions of Chaos I decided to go back and re-render some sample Mandelbrot movies at 4K 60fps. Most of these Mandelbrot movie scripts go back to the earliest days of Visions of Chaos even before I first released it on the Internet. The only changes made have been to add more sub frames so they render more smoothly and last longer.
Thanks to the new formula editor using OpenGL shading language and a recent NVidia GPU all the movie parts rendered overnight. Every frame was 2×2 supsampled so the equivalent of rendering an image 4 times as large and then downsampling to increase the quality. The original movie was a 46 GB XVID encoded AVI before YouTube put it through all of its internal recoding steps.
It took a lot longer than I wanted, but the new Visions Of Chaos for Android v1.0 is now available for download. It is 100% free and ad free.
The main new feature is support for running OpenGL ES shaders on your Android device. There are over 1000 sample shaders included with the app.
So, if you own one of the 85% market share Android devices download the app from the Visions Of Chaos for Android web page or directly from Google Play now.
After reading this page from Paul Bourke I had a go at creating fractal planets by one of the methods he mentions;
1. Take a sphere of points (best to use a geodesic sphere so the points are evenly spread over the surface of the sphere).
2. Cut the sphere in half with a plane that goes through the origin. See the section on Great Circles here.
3. All points on one side of the plane are moved outwards and all points on the other are moved in.
4. Repeat this process a few hundred times and you get a fractal planet.
And here is a movie of the creation process.
Fractal planets are now included in the latest version of Visions Of Chaos.
After seeing Paul Bourke’s page about Thorn Fractals I had to have a play and add them into Visions Of Chaos.
Sample C source code is available on Paul’s page and the inner formula comes down to CX and CY parameters in the main iteration loop for each pixel;
a = ir;
b = ii;
ir = a / cos(b) + cr;
ii = b / sin(a) + ci;
Here are some sample images using the above formula. The CX and XY in this case can be anywhere between -10 and +10.
I also experimented with changing the formulas and discovered the following three variations.
For this variation the CX and CY should stay within the range of -1 and +1. Values outside this range will lead to very noisy images.
Again CX and CY should stay within the range of -1 and +1.
When rendering these sorts of fractal images you will get lots of aliasing, noise and moire effects. These can be overcome by supersampling (rendering the image to 10 or 20 times the actual size and downsampling). Most of the images in my sample gallery used at least 10 times supersampling, so every pixel is actually the average of 100 sub-pixel calculations. Even at 20 times supersampling there are still visible areas that show aliasing in some renders.
Dane Vandeputte (mentioned on Paul’s page) uses Contrast Limited Adaptive Histogram Equalization to bring out some really nice details in these sort of fractals. I will have to try adding CLAHE to Visions Of Chaos in the future.