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## 'Footprints' printed from http://nrich.maths.org/

The isometries in the plane (reflections, rotations, translations
and glide reflections) are transformations that preserve distances
and angles.

Draw diagrams to show that all the isometries can be made up of
combinations of reflections.

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.