Beautiful! Heres the one example from post:
xcoord[k_]:=N[(2 k/4000)+(1/28) Sin[(42 Pi k/4000)]+
(1/9) Sin[(21 Pi k/4000)]^8+(1/4) Sin[(21 Pi k/4000)]^6 Sin[(2 \[Pi]/5) (k/4000)^12]]
ycoord[k_]:=N[(1/4) (k/4000)^2+(1/4) (Sin[(21 Pi k/4000)]^5+
(1/28) Sin[(42 Pi k/4000)]) Cos[(Pi/2) (k/4000)^12]]
rad[k_]:=N[1/170-1/67 (-1+Cos[(21 k \[Pi])/4000]^4) Sin[(21 k \[Pi])/2000]^2]
Graphics[{Opacity[.3],RGBColor[0,0.16,0],Table[Circle[{xcoord[i],ycoord[i]},rad[i]],{i,4000}]}]
and with animate
circles = Table[Circle[{xcoord[i], ycoord[i]}, rad[i]], {i, 4000}]; Animate[
Graphics[{Opacity[.3], RGBColor[0, 0.16, 0], circles[[;; Clip[i, {1, 4000}]]]},
PlotRange -> {{-0.005, 2.28}, {-0.26, 0.44}}, ImageSize -> 600], {i, 1, 4500, 1},
AnimationRate -> 300]
