Post #1580544
2026-04-09 12:01 UTC
To make a world-like landscapes with an inversive substitution limit set requires the root limit set to be close to spherical, which you get from having the generating spheres placed on the vertices of a Platonic solid. Last post used an octahedral configuration.
This only allows the bumpy tree-tree and cratered shell-shell child sets since the substitution needs to match one sphere and its immediate neighbours (up to a Mobius transformation), leaving only a single sphere for variation.
All the other Platonic solid's give no freedom at all, apart from the icosahedron. Here is the sphere arrangement, using order 2 (right angled) intersections to avoid the 'bald spots' that you get with order 3 (60 degree) intersections.
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Replies (1)
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@TomL@mathstodon.xyz 2026-04-09 12:12
This arrangement retains six spheres for adjusting the style of terrain in each substitutions set. They fall into three categories: The first is either hierarchical hills (tree-tree) or craters (shell-shell). The second is a shell-tree, which is a combination of hierarchical hills and craters. Hills or craters can be inside hills or craters, but they don't overlap each other's edges. I'm not sure the class of the last type, I think it is still a shell-tree. It uses overlapping hills and craters to form a landscape primarily out of connecting ridges and valleys. Examples left, middle and right here.. (2/3)