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I'm trying to model indented circles onto a sphere with clean topology. I think I have accomplished that, but I am struggling to maintain the smoothness of the sphere with the added edge loops. I can't remove these added ridges. I've tried to increase subsurf, subdivide, and manually add in edgeloops to even out the topology density, but this just exaggerated the effect. How can I maintain a smooth spherical look while still having a clean topology for these indents? I created the sphere by subdividing a cube several times, and then applying a subsurf.

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You should try to avoid those extra edge loops. If your sphere has not enough vertices to allow you to create your holes then recreate a new sphere with more vertices, but it looks like you have enough. Also, before the hole extrusion, you can use a Shrinkwrap modifier with a second sphere as Object to make sure that the whole shape is still a sphere.

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This looks like a job for the Cast modifier.

Add it to the end of your modifier stack, and set factor to 1.

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