Is there a 3d equivalent of this picture?

Question

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

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 $z \mapsto z^2$:

(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.

I should also mention that I don't necessary need the configuration to be "completely regular". That is, I'm happy to tolerate occasional points that are inconsistent with the rest of the configuration; there should just be some kind of visual consistency that one would readily observe by inspection.

Answer 1

The restriction to conformal maps is a natural one, as it means that there is no affine distortion in the neighbourhood of a point. Specifically, the Voronoi cells of the points will not be oblated or prolated, which is not the case for non-conformal maps. Once we insist on that restriction, Liouville's theorem insists that the only possibilities are Möbius transformations (compositions of Euclidean transformations and geometric inversions).

It seems like a good idea to invert a lattice, and the dense face-centred cubic lattice is more aesthetically pleasing than the ordinary cubic lattice. Geometric inversion of the face-centred cubic lattice ($A_3 \cong D_3$ in Sphere Packings, Lattices and Groups) yields an arrangement of points, all of which are contained within the unit ball, and the density diverges to infinity as you approach the origin. Specifically, we take the set of points:

$$\{ \dfrac{(x,y,z)}{x^2+y^2+z^2} : x,y,z \in \mathbb{Z}, \enspace x+y+z \equiv 1 \mod 2\}$$

Here's a three-dimensional rotating view of the configuration, where I've made the points semitransparent:

Is this what you're looking for? Unlike your two-dimensional configuration, this is bounded and infinitely dense in any open ball containing the origin.

Answer 2

As we can see in the picture, here are the rules.

1. The pattern has the symmetries of a triangle. I made with blue three rays starting at the origin, and making equal angles.
2. For each blue ray, there is an array of parabolas which are symmetric with respect to that ray.
3. The parabolas symmetric with respect to the same line are distributed so that the distance between the apexes of two consecutive parabolas is larger, when the distance to the center is larger. This follows from the complex map used.
4. Also, the parabolas are larger as the distance to the center increases.
5. The dots are at the intersections between pairs of parabolas.

Now, that we know the rules, it is pretty straightforward to generalize to $\mathbb R^3$.

1. Take a regular tetrahedron, with the center in the origin.
2. Draw four blue rays, starting from origin, and going through the four vertices of the tetrahedron.
3. Build sequences of paraboloids which have as revolution axes the blue rays.
4. Make sure the paraboloids are at the same distances, and have the same ratios, as in the planar case.
5. Draw the dots at the intersections made by triples of paraboloids.