(* SierpinskiTriangle and SierpinskiPicture *)



SierpinskiTriange[1]={Polygon[{{-1,0},{1,0},{0,Sqrt[3]}}]};



SierpinskiTriange[n_Integer]:=

  SierpinskiTriange[n]= 

    N[Flatten[

        SierpinskiTriange[

            n-1] /. {Polygon[{a_,b_, 

                  c_}] \[Rule] {Polygon[{a, (a+b)/2, (a+c)/2}], 

                Polygon[{(a+b)/2, b, (b+c)/2}], 

                Polygon[{(a+c)/2, (b+c)/2, c}]}}]]



SierpinskiPicture[n_, m_]:= 

  Show[GraphicsArray[

      Graphics[SierpinskiTriange[#], PlotRange\[Rule]All, 

            PlotLabel \[Rule] 

              StyleForm["Sierpinski  " <> ToString[#], "SubSubsection", 

                FontSize \[Rule] 8], AspectRatio \[Rule] Automatic]& /@ {n,

          m}]]



(* ex1 *)

SierpinskiPicture[1,2];SierpinskiPicture[3,4];

SierpinskiPicture[5,6]; SierpinskiPicture[7,8];



Show[Table[Graphics[{Hue[Sin[n]], SierpinskiTriange[n]}], {n,8}], 

    PlotRange\[Rule] All, AspectRatio \[Rule] Automatic];



(* ex2 *)

Show[% /. 

      Polygon[l_] \[RuleDelayed] (* invert *) 

        Polygon[#/#.#& /@ (# - {0, Sqrt[3.]/3}& /@ l)]];