Complex numbers can be used to represent isometries. We write the conjugate of $z = x + iy$ as $\bar z = x- iy$.
A reflection in the imaginary axis $x=0$ is given by $\alpha (z) = -\bar z$. A reflection in the line $x=1$ is given by $\beta(z) = 2 - \bar z$. A reflection in the real axis $y=0$ is given by $\gamma (z) = \bar z$.
Find the formula for the transformation $\gamma \beta \alpha (z)$ and explain how this transformation generates the footprint frieze pattern shown in the diagram.