This question arises apropos of an earlier question I asked that was (***VERY!!!***) helpfully answered by Anton Petrunin:

http://mathoverflow.net/questions/85624/fitting-a-mesh-to-a-density-function

The picture below is the image of a regular equilateral triangular lattice in the complex plane under the map $z \mapsto z^2$:

![enter image description here][1]


  [1]: https://i.sstatic.net/v0VyA.png

(for whatever reason, I seem to be missing a few links in the picture above, but I hope this is sufficient).  This picture has two nice visual properties, namely that the points are distributed in a "regular" or "predictable" way, and that the density of points becomes more concentrated as we move towards the origin.  My question is, are there configurations of points in 3 dimensions that also have these two visual properties?  Owing to Liouville's theorem, I realize that I won't be nearly as fortunate in constructing a nice and easy conformal mapping, so I'm just looking for any possible way to place points in $\mathbb{R}^3$ where I could get some kind of behavior similar to that pictured above.  The best I can think of would be some sort of concentric geodesic buckyball-type structures, or possibly the cartesian image of a regular lattice in spherical coordinates, but I'm wondering if there's anything that's more interesting.