Voronoi 3d Explorations

While most of the time Voronoi diagrams are based on (at least seemingly) random set of point, I started to explore some regular patterns.

If you place input points in a regular square grid, the Voronoi diagram is a grid as well. Well, that's not a huge surprise.

If you slide each row of points in a grid for half a distance between the points, you get a honeycomb pattern. That's interesting.

What's the purpose of these games? To understand the relation between input (set of points) and output (Voronoi diagram) better. In my applications, to put it very simply, the former would represent genotype and latter phenotype. Of course, in the evolutionary process I would not control the input points directly, but I somehow sense that it would be beneficial if I understood the Voronoi diagrams as much as possible.