Polygon averaging

f[pts_, S_] := (
  M = Length[pts];
  new = ConstantArray[0, M];
  vec = pts;
  A = SparseArray[{{M, 1} -> 1, {i_, i_} -> 1, {i_, j_} /; j - i == 1 -> 1}, {M, M}];
  For[s = 1, s <= S, s++,
   new = A.vec/2;
   vec = new;];
  vec = Standardize[vec];
  vec = Append[vec, vec[[1]]];
  ListPlot[{vec, vec}, Joined -> {True, False}, AspectRatio -> 1, Axes -> False])

M = 75;
max = 3 M Sqrt[M];
step = Ceiling[max/200];
pts = RandomReal[1, {M, 2}];
ListAnimate[Table[f[pts, S], {S, 1, max, step}]]