Footprints
Make a footprint pattern using only reflections.
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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.
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...
Thank you Hutch from Park College and Andrei from Tudor Vianu
National College, Bucharest for your solutions to this
problem.
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Reflection in two parallel
axes gives a translation Reflection in line A maps foot 1 to foot 2. Reflection in line B maps foot 2 to foot 3. The combination of these 2 reflections is a translation perpendicular to the mirror lines by twice the distance between the two mirror lines. Reflection in two intersecting axes gives a rotation Reflection in line D maps foot 4 to foot 5. Reflection in line C maps foot 5 to foot 6. The combination of these 2 reflections is a rotation about the point of intersection of the two mirror lines by twice the angle between the mirror lines. |
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Consider the three reflections in the order they are given,
performed on the foot shape at the far left. These three
reflections give a glide
reflection
First the reflection $\alpha (z) = -\bar z$ maps $z = x + iy$
to $-(x-iy)=-x + iy$. This gives a reflection in the imaginary axis
$x = 0$ resulting in a foot going toe-to-toe with the original,
above the x-axis, with big toe near the origin, but pointing back
in the opposite $(-x)$ direction.
Now perform the reflection in the axis $x = 1$ given by $\beta
(z) = 2 - \bar{z}$ so that we have $\beta\alpha(z)$ maps $z = x +
iy$ to $-x + iy$ then to $2 - (-x - iy) = 2 + x + iy$. This gives a
foot pointing in the same direction as our original and immediately
in front of it.
Now do the reflection in the axis y = 0 given by $\gamma (z) =
\bar z$ so that the three reflections map $z = x + iy$ to $-x + iy$
then to $2 + x + iy$ then to $2 + x - iy.$ You end with a foot
below the x-axis, pointing in the same direction as the original,
with heel 2 units immediately in front of the toe of the original
giving a combination of translation and reflection known as a glide
reflection.
Repeating the reflections with the big toe of the new foot
becoming the new origin each time gives the sequence of alternating
feet walking left-to-right.
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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.