High-dimensional sphere packing

[beero1000]

Cool stuff.

 

[Kharakov]

Cool. I was just talking to someone about Sphere Packing and Jos Leys work came up. Like the video.

Semi-pertinent:

http://www.josleys.com/show_gallery.php?galid=225

http://www.josleys.com/articles/Sphere_inversion_Article.pdf

 

[ryan]

I am going to attempt to at least chip away at the problem of getting a visual sense of optimally packing 4d spheres in a 4d space. But I have a question too for anyone interested.

Let's use 1 meter as a radius. So if we think of time as a 4th dimension and then measure the "distance" of one "meter" of time, we get
1m/(3*10^8m/s) = 3*10^(-8)s = 1 "time meter", using the speed of light as a measure.

She said in the video that a sort of "scan" of a 4d sphere would look like an increasing sphere and then a decreasing sphere, which is kind of what we would see during 1 4d sphere in our 3d spatial world.

Now I will try to construct an optimal 4d box for only one 4d sphere. Let's use a 2m by 2m by 2m by 2m box, (x,y,z,t) = (2,2,2,2). A 1m radius 4d sphere inside the 4d box would seem to appear, like what was said in the video, as a sphere increasing and then decreasing over the span of the spatial dimensions and the duration of the temporal dimension of a quick 2*3*10^(-8)s.

I am starting to think that we would maximize the number of 4d spheres intuitively by creating a simple computer program. Couldn't we just make an outline of a 3d box and see how many spheres can increase and decrease in the proportionally "cubic" time frame with a temporal distance of r*3*10^(-8)s?

Then we can easily see, at least intuitively, that we can fit two 4d spheres tightly together as one sphere decreases another increases, which would at least appear to be optimal for two 4d spheres in a (4,2,2,3) box, where t = 3 because of the offsetting balls overlapping in the space to time dimension.

Then we can just imagine building much larger boxes and seeing what kind of arrangement allows the most 4d spheres. Then like they intuitively knew how to maximize 3d spheres in a 3d box, maybe the necessary equation will become more obvious for the 4d spheres.

 

[beero1000]

I am going to attempt to at least chip away at the problem of getting a visual sense of optimally packing 4d spheres in a 4d space. But I have a question too for anyone interested.

Let's use 1 meter as a radius. So if we think of time as a 4th dimension and then measure the "distance" of one "meter" of time, we get
1m/(3*10^8m/s) = 3*10^(-8)s = 1 "time meter", using the speed of light as a measure.

She said in the video that a sort of "scan" of a 4d sphere would look like an increasing sphere and then a decreasing sphere, which is kind of what we would see during 1 4d sphere in our 3d spatial world.

Now I will try to construct an optimal 4d box for only one 4d sphere. Let's use a 2m by 2m by 2m by 2m box, (x,y,z,t) = (2,2,2,2). A 1m radius 4d sphere inside the 4d box would seem to appear, like what was said in the video, as a sphere increasing and then decreasing over the span of the spatial dimensions and the duration of the temporal dimension of a quick 2*3*10^(-8)s.

I am starting to think that we would maximize the number of 4d spheres intuitively by creating a simple computer program. Couldn't we just make an outline of a 3d box and see how many spheres can increase and decrease in the proportionally "cubic" time frame with a temporal distance of r*3*10^(-8)s?

Then we can easily see, at least intuitively, that we can fit two 4d spheres tightly together as one sphere decreases another increases, which would at least appear to be optimal for two 4d spheres in a (4,2,2,3) box, where t = 3 because of the offsetting balls overlapping in the space to time dimension.

Then we can just imagine building much larger boxes and seeing what kind of arrangement allows the most 4d spheres. Then like they intuitively knew how to maximize 3d spheres in a 3d box, maybe the necessary equation will become more obvious for the 4d spheres.

Your intuition is failing you, you need to do some actual calculations. Choose centers for your hyperspheres and compute actual distances, i.e. \( d(p_1,p_2) = \sqrt{(x_{1} - x_{2})^2 + (y_{1}-y_{2})^2 + (z_{1}-z_{2})^2 + (t_{1} - t_{2})^2}\).

 

[ryan]

Crap, I was too tired last night. It's just a little over 3.4 actually 2+(2)^(1/2). And now I just realized that I should have made the 4d spheres diagonal since we are using a 4d cube. The x and y = 3.4 and z = 2 with t still = 3.4; I think. This should probably optimally fit two 4d spheres in a minimal 4d cube with the spheres diagonal.

As for proving it with the equation, I have no idea.

Seeing this in person, it should at least look tight. As one sphere grows to maximum and begins to shrink, the next will start growing so that they are constantly touching each other diagonally.

 

[Jayjay]

This will surely come handy the next time I move.

 

[beero1000]

This will surely come handy the next time I move.

Nah, you're much better off packing with high-dimensional cubes.