Heighway dragon and twindragon relation

Question

The Heighway dragon F is defined as the limit set for the iterated function system $\begin{cases}f_1(z)=\frac{1+i}2 z\\f_2(z)=1-\frac{1-i}2z\end{cases}\quad$ starting from the two points 0 and 1.

The symmetric G of the Heighway dragon by $z\mapsto1-z$ is defined by the IFS: $\begin{cases}g_1(z)=\frac{1-i}2+\frac{1+i}2z\\g_2(z)=\frac{1-i}2-\frac{1-i}2z\end{cases}\quad$ starting from 0 and 1.

The rotated twindragon H is defined by the IFS: $\begin{cases}h_1(z)=\frac{1+i}2z\\h_2(z)=\frac{1-i}2+\frac{1+i}2z\end{cases}\quad$

I have put a little program to display $F,G,H$ on the same graphic. I can see that $H=F\cup G$. My problem is that I have not been able to prove it.

$h_1=f_1$ and $h_2=g_1$ so $h_1(F)\cup h_2(G)$ is already of one half of $H$. It remains to be proved that the other half of $H$ is $h_1(G)\cup h_2(F)$.

Can anyone help me solve this ?

I have already consulted to no avail

Knuth "the art of computer programming volume 2" page 206

Barnsley "fractals everywhere" page 306,310

G.Edgar "utm measure topology and fractals geometry" page 34 in particular

Thanks in advance!

Answer 1

The standard (and quite wonderful) reference is:

Chandler Davis & Donald J. Knuth, "Number Representations and Dragon Curves".
Part I: J. Rec. Math. 3 (1970) 66--81;
Part II: J. Rec. Math. 3 (1970) 133--149.

To work on proving it yourself: Determine the IFS for each of the two Heighway dragons, and for the twindragon. Use that to show that the union of the two Heighway dragons satisfies the IFS for the twindragon.