Footprints
Make a footprint pattern using only reflections.
Age
16 to 18
Challenge level
Problem
Image
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.
Getting Started
Investige combinations of two reflections in parallel and in intersecting mirror lines.
Suppose you have an infinite frieze pattern of footsteps. Investigate a combination of reflections that will give a glide reflection mapping any footstep in the pattern to the next one. Where would the mirror lines have to be? Show that, with the mirror lines you have chosen, the third step maps to the fourth, the fourth to the fifth and so on...Teachers' Resources
To find the formula for this combinaton of three reflections requires very little knowledge of complex numbers. All you have to do is to combine the complex maps as you combine functions.. The example shows the power of complex numbers as a tool for working with transformations in the plane.
In the problem Complex Rotations you see another examle of using complex numbers for work with transformations.
The problems Rots and Refs and Reflect Again are examples of the use of matrices for work with transformations.
This problem is about combinations of reflections. The problem Reflect Again uses matrices to show that the combination of two reflections in intersecting mirror lines gives a rotation.