Maximize Points in n-D Space with Distance Threshold

[EvoG]

I have a problem and I'm not sure if it has been formulated and/or solved before.

Suppose you have a finite n-dimensional space in which each dimension is constrained to [0,1]. I need the maximum number of points p that can be put in this space such that no two points are closer than some threshold t.

For instance, if n = 2 and t = 1, then p = 4. Think of the corners of a square with sides of length 1.
If n = 2 and t = 1.4, then p = 2, since opposite corners of a square with sides of length 1 are > 1.4 distance apart.

In general, my n is going to be large (>10000) and my t will certainly be greater than 1. I don't need a perfect formula. An approximation that gets me to within a factor of about 2 of the actual solution would be good enough.

I suspect this problem may be reducible to the sphere packing problem which has only been solved for up to 8 dimensions, but I thought I'd check with the experts to see.

 

[mathman]

It does look like a sphere packing problem with radius = t/2.

 

[Mark44]

It does look like a sphere packing problem with radius = t/2.

Except for at the corners and edges of the enclosing "box".

 

[HallsofIvy]

But since the "t" is minimum possible distance, the corners not really relevant. Using spheres of radius t/2 should give the same result as using cubes with side length t.

 

[mathman]

But since the "t" is minimum possible distance, the corners not really relevant. Using spheres of radius t/2 should give the same result as using cubes with side length t.

I don't think so. Using cubes will fill up the box neatly. Spheres will pack more tightly.