Inversive Global Landscapes

One of the nice things about inversive limit sets is that they typically are deviations from a sphere (e.g. here) so it is tempting to think they could be used to define a global landscape such as mountains, craters and hills on a planet or moon. To get the variety you can use the inversive substitution rules approach of my recent post, which happened to be based on an octahedral configuration of six spheres. You then start with a spherical limit set and switch to varied configurations. There are a wide variety of configuration but in this post I'll consider only distortions, meaning the same connectivity but different placement and size of the spheres.

The problem with the octahedral arrangement is that there is only one degree of freedom in distorting from a sphere. This gives you a cratered shell-shell all the way to the lumpy tree-tree seen in the recent post, but no others. Even here the degree of freedom is only when the octahedron is perfectly regular.

For the tetrahedral spherical limit set it is worse, there are no degrees of freedom. You must stay as a sphere.

For the cubic arrangement there are also no degrees of freedom. The faces must stay planar.

For the dodecahedral arrangement there is no connectivity diagram possible that covers the whole face, which means it does not generate a surface.

That leaves only the icosahedral arrangement among the regular spherical limit sets. Its limit set is shown on the right:

Fortunately this does allow at least four degrees of freedom in transitioning from spherical. In these examples the top six spheres remain fixed to allow the transition. I'll give the degrees of freedom parameter names:

Peak: this controls how large the bottom sphere is. 

above right shows the bottom six spheres viewed from below, with red being the enlarged one

Below positive peak value is a tree-tree (left), negative is a shell-shell.

Mid: controls how large the bottom ring of five spheres are.

because the sphere at the bottom is still small it creates a shell-tree (craters and domes). Positive (left) and negative (right):

Ridge: enlarges a ridge of three spheres along the bottom:

positive(left) and negative (right). These seem to generate more ridges and valleys than above.

 

Offset: enlarges a cluster of three spheres at the bottom but offset from centred:

Quite similar quality to the ridge parameter. The ridges and valleys cross each other.

 

From these four degrees of freedom we can create any weighted combination of peak, mid, ridge and offset parameters to describe the landscape one level down from the outer sphere. For instance, here is the globe with the top sphere substituting for the positive ridge set:

Or the top sphere one level down substituting for the ridge set. You can just about make out some indentations here. Which would of course be huge on the zoomed in planet.

Here is a combination of mid times 1.4 and ridge times 1.7:

which seems to be a little more ridgey and less cratery, here it is on the north polar region:

and here I'm making it more of a snowy mountain white and combining with a green hilly variant (peak 3.8, mid 0.5) and a golden shell-tree variant (peak 1.3, mid 1.6):

So there's more variety to be had with just distortions of the icosahedron. There are also a few other distortion types than just the four shown, but I showed the main ones.

It is interesting to see that there is a class of surfaces that is different from the standard shell-tree. Its craters and domes cross over each other to generate a surface of ridges, valleys, and saddles where they meet. This is closer to realistic landscapes where ridges and valleys are common.