How to generate a set of K points evenly spaced in unit cube/sphere of dimension D

Question

I'm trying to generate a set of K points that are evenly spread out inside a fixed space, I figured a unit sphere or cube would be easiest. It was easy enough for 2 dimensions, but it is a lot harder as we go up to arbitrary dimension D, unfortunately.

I think this is done in "quasi Monte Carlo methods" but I've been unable to find a formula, or even a statement as to whether this is a tractable problem. Any input would be appreciated.

Answer 1

To see that calculating the best case is "not trivial" look at the paper Dense Packings of Equal Spheres in a Cube which addresses the subcase of points in a 3D cube: It has exact solutions and "best-known" solutions for only upto 28 points.

It does present an algorithm for finding these optimally spaced configurations, which they call the "Stochastic Billiard" procedure. However I do not know whether this can be adapted to spheres, higher dimensions or larger numbers of points.

It also looks like some aspects of the more general case may be covered in the book Finite Packing and Covering - (which I dont have a copy of, so can't verify.)

The 2D case is much more tractable, and you can see further details on wikipedia for the square and the circle.