4.7 Dragons

In the second chapter we learned about the Heighway dragon and the Pythagorean tree. It turns out that both of these fractals belong to a richer family that can be parametrized by a single number. Such a pair can be defined for each \(T \in [0, 1]\).

\[ f_1(x, y) = \frac{1}{\sqrt{2}} \begin{bmatrix} \cos(\pi/4) & -\sin(\pi/4) \\ \sin(\pi/4) & \cos(\pi/4) \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} \] \[ f_2(x, y) = \frac{1}{\sqrt{2}} \begin{bmatrix} \cos\left(\frac 34 \pi - \frac{T}2 \pi\right) & -\sin\left(\frac 34 \pi - \frac{T}2 \pi\right) \\ \sin\left(\frac 34 \pi - \frac{T}2 \pi\right) & \cos\left(\frac 34 \pi - \frac{T}2 \pi\right) \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} 1 - T/2 \\ T/2 \end{bmatrix} \]

Six fractals from this family are shown below.