# How do I built a zoomable Koch curve?

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, 1}], {Sqrt, 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, 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, 1}, {0, 2}, {Sqrt/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]

• Nothing on this site helps? Your question is not a good question for this site. There are so many things that haven't been implemented yet, but we are not a coding service here to make that happen. We answer specific questions that people have come across in the process of them trying to solve a problem. If you are a beginner there are many, many examples already that you can start with. You don't need this particular example. – C. E. May 24 '14 at 4:18
• @Pickett: thank you for the link. I've edited the question. – Leo May 24 '14 at 5:00
• What do you mean by zoomable? In the case of this linked Java application this is only about changing plot range while iteration level is constant. If this is really what you need, take a look at PlotRange, EvenHandler etc. In particular, this topic may be helpful: manipulating 2D plots – Kuba May 24 '14 at 7:26
• I request this question be reopened because it wasn't too broad for bill s to provide an answer worth preserving. – m_goldberg May 24 '14 at 22:25

## 1 Answer

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, 1}], {Sqrt, 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, 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, 1}, {0, 2}, {Sqrt/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]

• Many thanks! I have a few problems, this one, for example: i.imgur.com/y8aFygq.gif – Leo May 25 '14 at 23:25