Post #1757932

2026-04-22 11:50 UTC

This image shows what's happening with the substitutions. The top sphere of the parent set (left) substitutes in the green (right) set, and the top sphere of the green set (right) substitutes in the golden coloured parent set (left). It is of course that large bottom green sphere that makes the green limit set so hilly. Variations include substituting a different child sphere, or a grandchild sphere, or substituting in a third set. (2/4)

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@TomL@mathstodon.xyz 2026-04-22 12:05

The second problem is that the terrain patches are all separated disks which is also unnatural. That's because you can't apply a substitution to two neighbouring spheres, or it causes discontinuities. This is solvable but it is a bit tricky. You need to create an inversive limit set specifically for the overlap region between the neighbouring spheres. You then need to constrain the overlap limit set spheres in a specific way. If set A constrains sphere ids 𝑎ᵢ to its parent and child set B constraints sphere ids 𝑏ᵢ to its parent, then the overlap set needs to contrain its sphere ids 𝑎ᵢ to those on B and ids 𝑏ᵢ to those on A. Here is a basic flat icosahedral limit set with a green mountainous set on top and a brown hilly set on the left. The subtitution spheres are neighbours so the patches overlap and you can see the discontinuity between them. This happend regardless of whether you switch to the brown or green set at the overlap. (3/4)

Open ##1757933