Very nice.

In fact, with a little more algebraic work (in my case, I used GroebnerBasis[] to help me), one can redo this to use Disk[] instead of Polygon[]:

With[{cols = RGBColor /@ {"#e5e7de", "#f54123", "#0b3536"}}, 
     Manipulate[Graphics[{EdgeForm[], cols[[1]], Disk[{2, 0}, 2],
                          cols[[-1]], Disk[{1, 0}, 1], cols[[2]], 
                          Table[Disk[{6, -4 (s + y)}/(2 + (s + y)^2), 
                                     2/(2 + (s + y)^2)], {y, -100, 100}]},
                         ImageSize -> 540, Background -> cols[[-1]],
                         PlotRange -> {{-.6, 4.6}, {-2.6, 2.6}}], {s, 0, 1}]]

Beautiful art, @Clayton, as usual, thanks for sharing! I will look into the emoji situation, thanks for the note.