Saturday, February 18, 2012

Galaxies and fractals

Now we have a basic chemistry. What do we do with it? Well, we want to distribute atomic particles throughout our model. According to what criteria? We certainly wouldn't want to distribute it evenly, or our little universe would look the same everywhere. We don't want it to be completely random, since the whole idea of Flat Galaxy is to seed our models with order over a large scale. One thing that fulfills this criteria very nicely: fractals.

A fractal is a pattern that is a pattern that exhibits self-similarity. They're seen everywhere in nature, and there's even a competing theory to the Big Bang that matter distribution throughout the universe is based on a fractal pattern. Which means that our own Milky Way galaxy is in some sense a model of the pattern of the entire universe. Fractal patterns are generated by recursively (self-referentially) iterating over complex polynomial functions, taking the output of the previous iteration of the function as input for the next iteration of the function. The classic Julia set function is f(z) = z^2 + c where, basically, z represents a point on the complex plane (which is mapped to the image plane) and c represents a complex number that largely determines the behavior of the fractal. We simply iterate through the function for each point on the screen and if after X number of iterations the output has not diverged (based on some threshold value) to infinity, we draw it on the screen.

Another bonus to fractals is that for now, at least, there is no physics model in Flat Galaxy. Implementing physics would complicate things immensely, even in 2d, and would force me to spend large amounts of time writing collision detection code on a fairly immense scale. Most problematic is that it would eat up processing cycles I would prefer to be dedicated to more important things. We can gloss over the lack of physics in various other ways. Fractals are one of them. We don't have to simulate accretion disks or orbits or anything else. Instead, we can just use a probability map based on a fractal and the amount of energy required to form any given element to determine concentrations of elements. From there, we can narrow our focus to the elements themselves, allowing them to bond depending on their individual properties and proximity.

I grabbed a simple fractal generation algorithm off the internet and made a bunch of modifications. The first modification was to the output functionality so that the images produced would color-code based on the density of the points. The images look best in grayscale, since they look how you'd expect a top down image of a galaxy to look. Unfortunately, the images also take some time to generate (around 2 seconds each on a fairly new computer), so I made a small, simple optimization that gave a 40-50% speed improvement without any degradation of the fractal, provided the threshold value for the fractal in question is of a certain value (the lower the threshold value--which also has a proportional relationship to the "brightness" of the image--the less optimization can be done).

I also added functionality to superimpose two fractals, one on top of another, randomize the complex function used to generate the fractal(s), and to randomize the complex values that represent c.

Here are a few of the images generated by the program (I think these are all one-fractal images. I'll have to generate some decent two-fractal images, since I think I deleted most of them):

You might notice that the "orientation" of these patterns is somewhat similar. This is because the same function was used to generate most of them.

I'm still working on the generation process for the superimposed fractal set, since those ones tend to be far more hit or miss. Randomizing the functions and the c-value also generate a lot of useless images (which are usually automatically filtered out based on a couple threshold values), and my plan is to generate these images continuously on a web server and send them to the client, which can then pick and choose which images to use without having to generate them itself. I also may put some work into smoothing the "webbing" you can see around the areas in between the lighter areas.

Next is taking these images and creating a material density map.

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