After figuring out why the equation in the last post described a tetrahedron, I was able to apply the same technique to generate the four other Platonic solids. By popular demand, here they are!
The rounded cube has equation
and the rounded octahedron is
I won’t post the icosahedron and dodecahedron’s formulas because they are pretty big and ugly (they involve the square root of 5, and respectively have degrees 12 and 9).