More examples

The Koch snowflake

c1 = {{{1/3, 0}, {0, 1/3}}, {0, 0}}; 
c2 = {rotate[Pi/3]/3, {1/3, 0}}; 
c3 = {rotate[-Pi/3]/3, {1/2, Sqrt[3]/6}}; 
c4 = {{{1/3, 0}, {0, 1/3}}, {2/3, 0}}; 
f1 = {{{1, 0}, {0, -1}}, {0, 0}}; 
f2 = {rotate[Pi/3], {0, 0}}; 
f3 = {rotate[-Pi/3], {1/2, Sqrt[3]/2}}; 
kochFlake = {{{c1, c2, c3, c4}, {}}, 
    {{f1, f2, f3}, {}}}; 
ShowDigraphFractals[kochFlake, 7, 
    Prolog -> {AbsolutePointSize[0.4]}]; 

The Heighway Dragon

f1 = {rotate[Pi/4]/Sqrt[2], {0, 0}}; 
f2 = {-rotate[-Pi/4]/Sqrt[2], {0, 1}}; 
g1 = {{{1/2, 0}, {0, 1/2}}, {0, 0}}; 
g2 = {{{-1/2, 0}, {0, -1/2}}, {0, 1}}; 
heighwayDragon = {{{f1}, {f2}}, {{g1, g2}, {}}}; 
lineSegment = Line[{{0, 0}, {0, 1}}]; 
heighwayBoundary = ShowDigraphFractals[heighwayDragon, 14, 
    Initiators -> {{lineSegment}, {lineSegment}}, 
    DisplayFunction -> Identity]; 
Show[GraphicsArray[heighwayBoundary]]; 

Show[heighwayBoundary, 
    DisplayFunction -> $DisplayFunction];

More spirals

a1 = {.985 rotate[goldenAngle], {0.,0.}};
b1 = {.95 rotate[goldenAngle - 0.05], {0.,0.}};
a2 = b2 = {{{.1,0},{0,.1}}, {1,0.}};
spirals = {{{a1},{a2}},{{b2},{b1}}};
Show[GraphicsArray[
    ShowDigraphFractalsStochastic[spirals,30000,
        DisplayFunction -> Identity]]];