Mitchell Gravity Set Fractals

Gravity Set Fractal

Origins

Back in the 1980’s Fred Mitchell designed a method to visualize Newtonian gravity he called the Mitchell Gravity Set or MGS. See here for his page explaining MGS.

This process of plotting dynamic systems is usually referred to as basins of attraction. See this paper and this paper for other versions of plotting gravitational basins of attraction. Similar methods are used when plotting Root-Finding Fractals and the basins of a Magnetic Pendulum.

Gravity Set Fractal

The 2D images in this post are from when I first experimented with MGS back in 2005.

Gravity Set Fractal

The basics of MGS is that you have a number of gravity objects (stars or gravity wells) in 2D space. Each pixel of a 2D image becomes the starting coordinate for a particle in that 2D space. The particle is attracted to the stars through Newtonian gravity and moves. You follow the path the particle takes for a number of steps and color the pixel depending on how long it took to be flung out of the simulation area.

Gravity Set Fractal

I ended up rewriting/translating my old code into GLSL for speed. Click here to see the shader source code that generates these 2D Gravity Set images.

Gravity Set Fractal

By playing with the various star positions and masses you can get a variety of different images.

The following image shows the current Gravity Set Options dialog in Visions of Chaos showing the various settings that can be tweaked.

Gravity Set Fractal

One thing I have not tried yet that is mentioned in Fred’s YouTube video is that you can use negative gravity constants and negative masses.

Revisiting and Extending Into The 3rd Dimension

I was recently contacted by Fred again and he mentioned he was working on getting a 3D version of MGS going. Getting the 2D code going in 3D is simple enough. Just a few extra lines for adding the Z dimension, so I was inspired to have a go at seeing what the 3D versions could look like. The settings dialog is extended as follows.

Gravity Set 3D Settings Dialog

Similar to the other 3D grid based displays like 3D Cellular Automata I have worked with, displaying a 3D grid of values can be tricky.

The following examples use a 500x500x500 resolution space.

The most obvious is a 3D Grid of little cubes. Each smaller cube is a starting point for the gravity calculations. Color the cubes depending on which gravity well they come closest to during their journey.

3D Gravity Set Fractal

That gives you a general idea of what the shapes of the 3D Gravity Set will be, but the interior is completely hidden.

You can slice an eighth of the cubes away….

3D Gravity Set Fractal

or even slice a half…

3D Gravity Set Fractal

Another option is to only display the cubes for only half of the gravity wells….

3D Gravity Set Fractal

or even for only a single well.

3D Gravity Set Fractal

3D Gravity Set Fractal

Then came the idea for “volumetric rendering”. I already had code for rendering transparent billboard quads so I used that. I am only rendering the edges of each gravity well area. If you render all points the image is just a white blob mess.

3D Gravity Set Fractal

3D Gravity Set Fractal

3D Gravity Set Fractal

Another method. Export the visible cubes as a OBJ point cloud. Import the OBJ into MeshLab. Use MeshLab to generate normals for the points and then marching cubes to generate a mesh from the point cloud. Export the mesh as a PLY format file. Import the PLY into Blender and render it. These are rough results. I am sure someone who knows more about MeshLab and Blender could clean up the mesh and make a much better looking render.

3D Gravity Set Fractal

3D Gravity Set Fractal

Here is a sample movie of rotating gravity sets.

Magnetic Pendulum Alternative to MGS

I also tried using Magnetic Pendulum formulas in 3D to do similar plots. The formula is a 3D extension of the 2D code I used here. Otherwise the rest of the code is the same as for the 3D MGS above.

3D Magnetic Pendulum

3D Magnetic Pendulum

3D Magnetic Pendulum

3D Magnetic Pendulum

3D Magnetic Pendulum

Availability

Both 2D and 3D Gravity Set Fractals are now included in the latest version of Visions of Chaos.

Jason.

More adventures with 3D Gravity simulations and OpenCL

History

3D Gravity simulations are something I have been interested in for many years now. Some worked, some didn’t, some were more realistic than others.

The first 3D Gravity simulation movie I still have on my YouTube channel is this one from way back in May of 2007. Low res with only a bunch of blurry objects.

Since then I have increased the details and object counts. I also started experimenting with OpenCL for big speedups that allowed many more objects to be simulated in a reasonable time frame.

Moving forward to now

For this latest post I went back and rewrote my code and the OpenCL kernel code to correctly compare every object to all other objects in the gravity calculations. The simulation is using Newton’s law of universal gravitation.

Newtonian Gravity

Every point mass attracts every single other point mass by a force acting along the line intersecting both points. The force is proportional to the product of the two masses and inversely proportional to the square of the distance between them.

How this simulation works

I am using software OpenGL on the CPU for all the rendering of the visuals and CPU and OpenCL for the gravity calculations. OpenCL code runs on your graphics card GPU and GPUs are great at running lots of small bits of code fast at the same time. The gravity formula maths is perfect for multi threading. Every objects velocity and acceleration can be calculated at the same time as the other objects.

The basics of using OpenCL is you fill arrays with the information you want the OpenCL code to use (for 3D gravity I am passing position, velocity, acceleration and mass of the objects), pass it to OpenCL, run the code on the GPU, and then read back the results from the GPU when it is done.

This is the current OpenCL code I am using for these latest simulations. Each of the arrays passed (posx, posy, etc) contain all the current objects, ie for a 1 million object simulation the posx array has 1,000,000 floating point values to cover every object’s X position in 3D space.


__kernel void Gravity3DKernel( __global float * posx,
		               __global float * posy
		               __global float * posz, 
		               __global float * velx, 
		               __global float * vely, 
		               __global float * velz, 
		               __global float * accx, 
		               __global float * accy, 
		               __global float * accz, 
		               __global float * mass, 
		               __global float * mingravdist)
{	
	int index=get_global_id(0);
	float dx,dy,dz,distance,force;
	float positionx=posx[index];
	float positiony=posy[index];
	float positionz=posz[index];
	float mingravdistsqr=mingravdist[index]*mingravdist[index];
	float accelerationx=0;
	float accelerationy=0;
	float accelerationz=0;
	float thismass=mass[index];
	for(int a=0; a<get_local_size(0); a++) {
		if (a!=index) {
			dx=posx[a]-positionx;
			dy=posy[a]-positiony;
			dz=posz[a]-positionz;
			distance=sqrt(dx*dx+dy*dy+dz*dz);
			dx=dx/distance;
			dy=dy/distance;
			dz=dz/distance;
			//old method - all objects are assumed to have the same mass
			//force=1/(distance*distance+mingravdistsqr);
			//new method - allows objects to have different masses
			force=(thismass*mass[a])/(distance*distance+mingravdistsqr);
			accelerationx+=dx*force;
			accelerationy+=dy*force;
			accelerationz+=dz*force;
		}
	}
	velx[index]+=accelerationx;
	vely[index]+=accelerationy;
	velz[index]+=accelerationz;
	accx[index]=accelerationx;
	accy[index]=accelerationy;
	accz[index]=accelerationz;
}

The kernel code loops through every object and calculates the forces against every other object. This is the naive unoptimized O(n2) version of the algorithm. Once all the loops are finished the new object velocity and acceleration values are read back from the GPU memory into local memory and then the CPU can access the results. All of the object positions are then updated using the new velocity and acceleration values and then displayed. For displaying the objects I am using the old software only OpenGL billboard quads. A billboard quad is a texture on a quad (rectangle) that always faces the “camera” in OpenGL. If you put a nicely shaded and transparent “blob” as the texture it looks like a simple star and blends in with other stars.

Calculation Times

Using the above code allows me to process millions of objects in a reasonable time frame. For these simulations the slowest part is always the display and CPU calculations. The OpenCL is always the fastest part of the simulations.

Here are a few stats for how quick (slow) these simulations are on a PC with an i7 CPU and GTX 1080 GPU rendering 4K resolution frames;

250,000 objects – 856 ms per frame – 31 ms OpenCL time
500,000 objects – 1606 ms per frame – 57 ms OpenCL time
1,000,000 objects – 3124 ms per frame – 91 ms OpenCL time
5,000,000 objects – 15336 ms per frame – 430 ms OpenCL time
10,000,000 objects – 30638 ms per frame – 864 ms OpenCL time

The reason the per frame time is so much longer than the OpenCL is that I am still using software OpenGL to render all the particles. Software OpenGL falls back to the older v1.1 OpenGL DLLs provided by Microsoft in Windows and has no benefits of hardware acceleration. On my to do list is getting the OpenGL code up to date to use hardware acceleration, then the above times should come way down. Rendering the objects as pixels rather than billboard quads is roughly 1/3 the time, but the billboard quads give a much nicer blended display.

Results

Here is a new sample 4K resolution 3D Gravity movie.

After the last movie I went back and improved the color shading code and added the option for a “black hole”, which in this case is only a single object with a larger mass than the others. The black hole has a mass of around 100 to 500 times the other stars. Any higher and all the stars are flung out of the simulation area too quickly. Here are are some of the latest results.

The spiral galaxy like results are mostly a fluke. I started the simulation with a disk or oblate spheroid (squished sphere) of particles rotating around the origin (Y axis) with a central black hole and let it run.

After a bit more tweaking of the code and coloring algorithms the next movie was ready.

Try It Yourself

The latest 3D Gravity code is now updated and included in Visions of Chaos. For now I am finally happy that at least the basic Newtonian gravity is working and using all objects for the calculations. Future changes will be converting the “old” OpenGL v1.1 display into “modern” OpenGL v3.3+ so the display time can be reduced and get closer to realtime.

Jason.

3D Gravity using OpenCL

Note: This is an older post and the gravity simulation code has since been updated. See here for the latest.

This post did have a bunch of benchmarks and times that were incorrect and have been removed. I was not comparing every particle to every other particle in the gravity calculations. That was the cause of the impossibly fast speeds I had listed.

What is left of this post is still correct. The object counts are correct and there are millions of particles being shown. It is just the actual gravity calculations that are not complete.

Previous Results

Since I originally posted the following 3D gravity movies to YouTube…

…there were questions and some skepticism in the comments so hopefully this blog post helps clarify things. That skepticism about how I could possibly calculate these millions of objects at once was well founded.

Latest Results

Here is a movie with 3,000,000 particles.

Unfortunately YouTube’s compression ruined the movie quality a bit. Mostly due to the noisy/static like nature of the millions of particles. Not enough I frames and too many P frames. Here are a series of screenshots showing what the frames looked like before the compression.

3D Gravity Simulation

3D Gravity Simulation

3D Gravity Simulation

3D Gravity Simulation

3D Gravity Simulation

3D Gravity Simulation

3D Gravity Simulation

To push it further, here is a rotating disk of 5,000,000 objects.

Again, here are some uncompressed screenshots.

3D Gravity Simulation

3D Gravity Simulation

3D Gravity Simulation

3D Gravity Simulation

3D Gravity Simulation

3D Gravity Simulation

3D Gravity Simulation

3D Gravity Simulation

3D Gravity Simulation

Try It Yourself

If use Windows you can download Visions of Chaos and see the simulations run yourself. Here are a few quick steps to get you going;

1. Open Visions of Chaos
2. Select Mode->Gravity->3D Gravity to change into the 3D Gravity mode
3. The 3D Gravity Settings dialog will appear
4. Change the number of objects to 1,000,000
5. Change the Time step to 0.02
6. Change the Size of objects to 0.2
7. Check Create AVI frames if you want to create an AVI movie from the simulation
8. Click OK

Make sure you have the latest video card drivers so the OpenCL code runs as optimal as possible. For NVidia go here and for AMD go here.

Jason.