(* We will try to construct the snowflake *)

first[{a_,b_},{c_,d_}] := {a,b} + 1/3( {c,d}-{a,b});


last[{a_,b_},{c_,d_}] := {a,b} + 2/3 ({c,d}-{a,b});


middle[{a_,b_},{c_,d_}] :=
1/2 {a+c,b+d} + 1/3 Sqrt[3]/2 Sqrt[(c-a)^2+(d-b)^2] {+(d-b),-(c-a)}/Sqrt[(d-b)^2+(c-a)^2];


A = Table[{0,0},{i,1,4}];

A[[1]] = {Cos[  0],Sin[  0]};
A[[2]] = {Cos[2 Pi/3],Sin[2 Pi/3]};
A[[3]] = {Cos[4 Pi/3],Sin[4 Pi/3]}; 
A[[4]] = {Cos[6 Pi/3],Sin[6 Pi/3]};

a1 = ListPlot[A,  PlotJoined->True,  AspectRatio->1];

a2 = ParametricPlot[{Cos[t],Sin[t]},{t,0,2 Pi}];

Show[a1,a2];

look := (
B = Table[{0,0},{i,1,4*(Length[A]-1)+1}];
Do[
   B[[4*(i-1)+1]] = A[[i]];
   B[[4*(i-1)+2]] = first[A[[i]],A[[i+1]]]; 
   B[[4*(i-1)+3]] = middle[A[[i]],A[[i+1]]]; 
   B[[4*(i-1)+4]] = last[A[[i]],A[[i+1]]]; 
   B[[4*(i-1)+5]] = A[[i+1]],{i,1,Length[A]-1}];

A = B;
a1 = ListPlot[B,PlotJoined->True, AspectRatio->1];
ans = Show[a1,a2]);