Snowfall[n_, a_, b_, x_, 
        y_] := 
Block[{l, s, t, u, v, p, 
          q}, 
If[
          n == 
            0, 
l = {{x, 
                  y}};
, 
s = (2  a + 
                  x)/3; 
          t = (2  b + y)/3; 
u = (a + 2  x)/3; 
          v = (b + 2  y)/3; 
p = 
            s + 1/2 (u - s) - 
              Sqrt[3]/2 (v - t); 
q = 
            t + Sqrt[3]/2 (u - s) + 
              1/2 (v - t); 
l = 
            Snowfall[n - 1, a, b, s, t]; 
l = 
            Flatten[{l, Snowfall[n - 1, s, t, p, q]}, 
              1]; 
l = 
            Flatten[{l, Snowfall[n - 1, p, q, u, v]}, 
              1]; 
l = 
            Flatten[{l, Snowfall[n - 1, u, v, x, y]}, 
              1];
]; 
Return[
          l];
]

    lev = 5;

    a = -1/2; b = -Sqrt[3]/6;

    l = 
        Flatten[{{{a, b}}, 
            Snowfall[lev, 0 + a, 0 + b, 1/2 + a, Sqrt[3]/2 + b]}, 
          1];

    l = 
        Flatten[{l, Snowfall[lev, 1/2 + a, Sqrt[3]/2 + b, 1 + a, 0 + b]}, 
          1];

    l = 
        Flatten[{l, Snowfall[lev, 1 + a, 0 + b, 0 + a, 0 + b]}, 
          1];

    ListPlot[l, PlotJoined -> True, 
      AspectRatio -> 1]