This wasn't exactly research, but I have a couple animations I made using a modified version of Melinda Green's Buddhabrot method to render the Mandelbrot set, and what came out was definitely unexpected and pretty shocking to me. I don't think I've ever seen this particular method anywhere else. I've been hoping to get some proper mathematicians to look at the process and give me some insight into why such wild objects seem to form.

This is the first one I made.

Then I tried to make a higher definition animation with different inputs.

You can turn up the quality to see the detail a bit better before watching them. It defaults to 480p, but can be changed to 720p.

To create these, I first began with Melinda's method, which is still explained at her site. It's basically a heat map of how many points in each pixel escaped to "infinity" under the action of the complex seed function. To create motion I decided I would take the coefficients of the function, which was a generalized Mandelbrot-type equation like this:

$$ z(w) = aw^3 + bw^2 + cw + C $$

Where $w$ is the complex conjugate of the previous value of the function.

And I would treat those coefficients like a 3-vector (a, b, c). To create motion, I rotated that vector just as if it was a spatial vector rotating through space. The animations are built up of individual images created by slightly transforming the coefficients little by little.

I would really enjoy hearing any insights people have as to why such incredible structures seem to come alive in these visualizations. It is almost eerie. You can see there is a smoke-like effect that gathers around the extended "arms" of the object as it moves, and it almost acts like it is responding to some kind of force (which is mystifying considering what we're looking at). It also has these little three-pointed sparks that fly off the tips, but eventually look like creases in fabric rather than little stars. There are even biological looking structures that appear when the sparks come together and seem to annihilate each other.

On a simpler level, it shocked me that it actually looks like a very distorted physical rotation of some object rotating in higher dimensions, even though it is only a rotation in coefficient space, and not a an actual rotation of spatial coordinates. About halfway through each video, you can see that it really is a rotational transformation, because it comes back around and repeats the entire rotation once more as the vector comes back through its initial position, which was something like (1, 0, 0). In fact, in the first video you can see the exact moment it repeats because the numbers didn't come back around exactly right due to rounding errors that I fixed in the second video.

This wasn't exactly research, but I have a couple animations I made using a modified version of Melinda Green's Buddhabrot method to render the Mandelbrot set, and what came out was definitely unexpected and pretty shocking to me. I don't think I've ever seen this particular method anywhere else. I've been hoping to get some proper mathematicians to look at the process and give me some insight into why such wild objects seem to form.

This is the first one I made.

Then I tried to make a higher definition animation with different inputs.

You can turn up the quality to see the detail a bit better before watching them. It defaults to 480p, but can be changed to 720p.

To create these, I first began with Melinda's method, which is still explained at her site. It's basically a heat map of how many points in each pixel escaped to "infinity" under the action of the complex seed function. To create motion I decided I would take the coefficients of the function, which was a generalized Mandelbrot-type equation like this:

$$ z(w) = aw^3 + bw^2 + cw + C $$

Where $w$ is the complex conjugate of the previous value of the function.

And I would treat those coefficients like a 3-vector (a, b, c). To create motion, I rotated that vector just as if it was a spatial vector rotating through space. The animations are built up of individual images created by slightly transforming the coefficients little by little.

I would really enjoy hearing any insights people have as to why such incredible structures seem to come alive in these visualizations. It is almost eerie. You can see there is a smoke-like effect that gathers around the extended "arms" of the object as it moves, and it almost acts like it is responding to some kind of attractive force (which is mystifying considering what we're looking at). It also has these little three-pointed sparks that fly off the tips, but eventually look like creases in fabric rather than little stars. There are even biological looking structures that appear when the sparks come together and seem to annihilate each other.

On a simpler level, it shocked me that it actually looks like a very distorted physical rotation of some object rotating in higher dimensions, even though it is only a rotation in coefficient space, and not a an actual rotation of spatial coordinates. About halfway through each video, you can see that it really is a rotational transformation, because it comes back around and repeats the entire rotation once more as the vector comes back through its initial position, which was something like (1, 0, 0). In fact, in the first video you can see the exact moment it repeats because the numbers didn't come back around exactly right due to rounding errors that I fixed in the second video.