Implementing Dragon Curve in Mathematica

The algorithm for getting a folding paper fractal AKA Dragon Curve is straightforward. We start with a single bend, let's say C (for clockwise). After we fold the paper in the same way once again, and then unfold it back, we can see that the number of bends doubled, so that now it is C-C-A. The next step gives us CCA-C-CAA pattern, then CCACCAA-C-CCAACAA, and so on. One can derive the following algorithm:

1. Let's say we have some pattern P (at the beginning P = {C})
2. Let Q be a reversed and inverted P (e.g. if P = {CACC}, then Q = {AACA})
3. Obtain new value for P by concatenation: P = PCQ
4. Repeat until the required depth is reached

This is exactly what CalculateCurve function does:



It takes two arguments - depth (which is self-explanatory), and folding, which is a function returning the folding direction (1 or -1) on the given step / depth.

To understand the code better, let's extract f = #1~Join~{folding[#2]}~Join~Reverse[-#1]&;
Clearly f performs steps 2 and 3 of the algorithm described. Then Fold[f, {folding[0]}, Range[0, depth]] is equivalent to f[f[f[{folding[0]}, 0], 1], 2] and so on.

For the standard Dragon fractal the paper is always folded in the same direction, so that folding function is just a constant, e.g. -1. In this case a sample output of CalculateCurve with depth 2 looks like {-1, -1, 1, -1, -1, 1, 1, -1, -1, -1, 1, 1, -1, 1, 1}. Now in order to get the graphical representation of this sequence, we need to implement some kind of the simple automata.



Again, for simplicity let's define f = (a += #; {x, y} += {Sin[a angle], Cos[a angle]})&; It is easy to see that this function increments the current "direction angle" by 1 or -1, and then it appends the increment in the current direction to the {x, y} coordinates (increment angle is most often PI / 2, but other values sometimes give interesting results as well). Note that the current {x, y} value is returned as a result. Applying this function to the folding sequence (f /@ CalculateCurve[depth, folding]) yields a list of bend point coordinates. It is then interpolated with B-splines of the 2nd degree, and output as Graphics.

Together with controls for changing fractal depth and bending angle the code can be made as short as this:



and in this form it can be considered as a shorter alternative to the code from the Paperfolding Dragon Curve demo.

Some other examples of the growing Dragon Curves:



You can download complete notebook here.