How do I built a zoomable Koch curve?

Question

I'm new to Mathematica and my goal is to write a simple program in order to demonstrate self-similarity of the Koch curve by zooming in. Here is a good example of what I mean (it's a Java applet). I was going to start learning the built-in powers of Mathematica for a long time and it seems to be a good opportunity. Given the simplicity of the program and popularity of fractals I was sure I'd find many working examples online, yet it turned out to be not the case.

In particular, how can one make this zoomable:

f[form_, {a_, b_}] := 
AffineTransform[{{b - a, ({{0, -1}, {1, 0}}).(b - a)}\[Transpose], 
a}][1/Norm[
 Last[form] - First[form]] TranslationTransform[-First[form]][
form]]

g[form_, points_] := 
Flatten[Map[f[form, #] &, Partition[points, 2, 1]], 1]

Manipulate[form = Append[Prepend[pts, {-Sqrt[3], 1}], {Sqrt[3], 1}];
base = Nest[g[form, #] &, form, refinements];
If[maketriangle, 
triangle = 
Join[base, RotationTransform[4 \[Pi]/3.][base], 
RotationTransform[2 \[Pi]/3.][base]]];
plaatje = 
Graphics[{If[
maketriangle, {ColorData[1][1], Polygon[triangle]}, {}], 
AbsoluteThickness[1.3], Line[If[maketriangle, triangle, base]], 
If[refinements == 0, {Thick, Line[form]}, {}]}, 
PlotRange -> {{-3.5, 3.5}, {-2.3, 2.3}}, 
AspectRatio -> 
Automatic], {{pts, {{-Sqrt[3]/3, 1}, {0, 2}, {Sqrt[3]/3, 1}}}, 
Locator, LocatorAutoCreate -> True, 
ContinuousAction -> 
If[refinements > 2, False, True]}, {{refinements, 0, 
"Refinements"}, 0, 6, 1, 
SetterBar}, {{maketriangle, True, "Make triangle"}, {True, False}}, 
SynchronousUpdating -> False, SaveDefinitions -> True]

Source: Create Alternative Koch Snowflakes

Answer 1

I'm sure you can make it slicker, but one way to approach this is to change the plotrange dynamically. Here I've added four sliders to change the x and y scaling and the x and y offset. As in your original code, the amount of detail in the curve is given by the refinement variable.

f[form_, {a_, b_}] := 
 AffineTransform[{{b - a, ({{0, -1}, {1, 0}}).(b - a)}\[Transpose], 
    a}][1/Norm[
     Last[form] - First[form]] TranslationTransform[-First[form]][
    form]]

g[form_, points_] := 
 Flatten[Map[f[form, #] &, Partition[points, 2, 1]], 1]

Manipulate[form = Append[Prepend[pts, {-Sqrt[3], 1}], {Sqrt[3], 1}];
 base = Nest[g[form, #] &, form, refinements];
 If[maketriangle, 
  triangle = 
   Join[base, RotationTransform[4 \[Pi]/3.][base], 
    RotationTransform[2 \[Pi]/3.][base]]];
 plaatje = 
  Graphics[{If[
     maketriangle, {ColorData[1][1], Polygon[triangle]}, {}], 
    AbsoluteThickness[1.3], Line[If[maketriangle, triangle, base]], 
    If[refinements == 0, {Thick, Line[form]}, {}]}, 
   PlotRange -> 
    Dynamic[{{-3.5, 3.5} xzoom + xoff, {-2.3, 2.3} yzoom + yoff}], 
   AspectRatio -> 
    Automatic], {{pts, {{-Sqrt[3]/3, 1}, {0, 2}, {Sqrt[3]/3, 1}}}, 
  Locator, LocatorAutoCreate -> True, 
  ContinuousAction -> 
   If[refinements > 2, False, True]}, {{refinements, 0, 
   "Refinements"}, 0, 6, 1, 
  SetterBar}, {{maketriangle, True, "Make triangle"}, {True, 
   False}}, {xzoom, 1, 0}, {yzoom, 1, 0}, {xoff, -1, 1}, {yoff, -1, 1},
 SynchronousUpdating -> False, SaveDefinitions -> True]