I observed a weird behaviour plotting the twin dragon for different scaling factors $\alpha$. I used this mathematica code to generate them
M = {{1/2, -1/2}, {1/2, 1/2}};
twindragon = {
{M, {-1, 0}},
{M, {1, 0}}
};
This corresponds to the IFS:
$ \begin{pmatrix} 1/2 & -1/2 \\ 1/2 & 1/2 \end{pmatrix} (\cdot)+ \begin{pmatrix} -1 \\ 0 \end{pmatrix} $
$ \begin{pmatrix} 1/2 & -1/2 \\ 1/2 & 1/2 \end{pmatrix} (\cdot)+ \begin{pmatrix} 1 \\ 0 \end{pmatrix} $
For $\alpha = 1/2$ one gets a usual twin dragon:
However, for $0 < \alpha <1/2$, I get separated pieces:
Why does this happen?
Hypothesis: the images are disconnected and therefore the entire attractor or fractal is totally disconnected.
Totally disconnected IFS
IFS $\{f_i\}$ is totally disconnected $\iff \forall i,j = 1,\ldots,N,i \neq j. \overline{f_i}(\mathcal{A}) \cap \overline{f_j}(\mathcal{A}) = \emptyset$.
Totally disconnected fractal set (these notes)
The set A is totally disconnected if the connected component of each point consists of just that point
Perhaps these concepts can help to see what is exactly happening here.