Sierpinski

pinski[0] = {  { {0, 0}, {1, 0}, {1/2, Sqrt[3]/2} } };
f[m_] := 1/2 (-1 + 3^(1 + m));
subtriangs[triang_] := {
   {triang[[1]], (triang[[2]] + triang[[1]])/
     2, (triang[[3]] + triang[[1]])/2},
   {(triang[[1]] + triang[[2]])/2,
    triang[[2]], (triang[[3]] + triang[[2]])/2},
   {(triang[[1]] + triang[[3]])/2, (triang[[2]] + triang[[3]])/2,
    triang[[3]]}
   };
pinski[n_] := pinski[n] = Module[{prev = pinski[n - 1], new},
    new =
     Join[prev,
      Flatten[Map[subtriangs, prev[[f[n - 2] + 1 ;; f[n - 1]]]], 1]]
    ];
Manipulate[Graphics[
  Table[Join[{ColorData["AlpineColors"][(i + 1)/(n + 1)]},
    Map[Polygon, pinski[n][[f[i] + 1 ;; f[i + 1]]]]]
   , {i, -1, n - 1}]
  ]
 , {n, 0, 5, 1}]