phi = 0.5 (Sqrt[5] + 1);
m = 500; n = 40 ;
ps = Table[{Sin[\[Theta]], Cos[\[Theta]]}, {\[Theta], 0, 2 Pi, 
    2 Pi/n}];
f = Interpolation[Transpose[{Range[0, n], ps}], 
   InterpolationOrder -> 1];
Graphics[{
  EdgeForm[Directive[RGBColor[0.1, 0.2, 0.25], Opacity[0.1]]], 
  FaceForm[], Polygon[ps],
  Table[Polygon[
    Table[f[Mod[  (i + phi  m t) i, n, 0]], {i, 0, n}]], {t, 0, 1, 
    1/m}]
  }
 , ImageSize -> 1000, Background -> None]