So far I've been using a cube as the basis for the tesselation. The cube has square sides. This is very convenient from the point of view of coordinate system but has some pretty bad deformations at the corners:
This picture is not a 3D view of a coordinate system; it's what the actual 2D coordinate system looks like in the locally flat surface of the sphere at a corner of the underlying cube. Ideally the terrain fractal wouldn't show the underlying grid, but in reality it sometimes does (due to inadequacies in the fractal) and so the corners or edges can be visible.
I was prepared to live with this in order to get the nice Cartesian coordinate system, but I ran into another much worse problem: random networks on a square grid are MUCH more complicated than they are on a triangular grid. I'm planning on using these random networks to generate rivers, mountains, and other extended structures that can't be easily generated by a hierarchical function. I'll call them "hierarchical networks" to indicate that they're generated the same way as the hierarchical functions are. Here's an example of some expansions of a hierarchical network on a square mesh:
I define a legitimate expansion as one that leaves the network (drawn in yellow) exits (drawn in red) from the coarse-level cell (in white, on the left) on the same sides in the fine-level cell (on the right), and such that the network remains connected, and such that the network in a finest-level cell is entirely defined by which sides it passes. The problem is that there are a tremendous number of legitimate expansions for each network cell. After studying these networks a bit, I concluded that I gain no meaningful flexibility by using square cells, and so I am going to switch to a basic expansion of equilateral triangles. This network has an as-simple-as-possible expansion. Here's a network expansion on such a mesh: