Rotating mirror

You can do interesting things with mirrors. You can find this out for yourself at the exhibits “rotating mirror” and “mirror funnel”. Mirrors have fascinated mankind since the Stone Age. But what is the mathematics behind them?

And now … the mathematics of it:

Mathematically, a reflection on a plane simply consists of inverting an axis of a rectangular coordinate system. For example, a reflection on the $xy$ -plane (for example, a water surface) is completely represented by the mapping $s$ , which is given by equation

$s\begin{pmatrix} x\\ y\ z\end{pmatrix}=\begin{pmatrix} x\\ y\\ -z\end{pmatrix}=\begin{pmatrix} 1 & 0 & 0\ 0 & 1 & 0\ 0 & 0 & -1\end{pmatrix}\begin{pmatrix} x\ y\ z\end{pmatrix},$

is described. The original image (for example, the sky) is thereby mapped onto its mirror image (the sky suddenly appears to be below the water surface). What is remarkable is that the orientation is reversed: If you hold your right hand in the mirror, for example, your reflection raises the left hand.

But what happens if we move and rotate the mirror plane $E$ in space? Let’s say it passes through the point $\mathbf{p}$ and has the normal vector $\mathbf{n}$ . We now want to determine how the associated reflection $s_{\mathbf{p},\mathbf{n}}$ maps any point $\mathbf{x}$ . To do this, we first form the perpendicular of $\mathbf{x}$ to $E$ (i.e., the line whose one vertex is $\mathbf{x}$ and whose other vertex lies in $E$ and is perpendicular to $E$ ). This has exactly the length $l=\langle\mathbf x-\mathbf p,\mathbf n\rangle$ , where $\langle\mathbf u,\mathbf v\rangle$ denotes the dot product of the vectors $\mathbf u$ and $\mathbf v$ . This is namely exactly the projection of the line $\overline{\mathbf x\mathbf p}$ onto the normal vector $\mathbf n$ . The mirror image of the point $\mathbf x$ on the plane $E$ is now exactly the corner point of the mirrored perpendicular not lying in $E$ . So we have to subtract the distance $l$ twice in the direction of the normal vector of $\mathbf x$ . So we get the equation

$s_{\mathbf p,\mathbf n}(\mathbf x)=\mathbf x-2\langle\mathbf x-\mathbf p,\mathbf n\rangle\mathbf n.$

About the exhibit “Revolving mirror”

But now we will turn our attention to the “rotating mirror” exhibit. You probably know the case that there is only one mirror in front of you from everyday life. You simply see yourself standing in front of it as a mirror image. If you turn the mirror, nothing changes at all, because the mirror plane remains the same.

But now let’s start from the more interesting case where there are two mirror planes $E_1$ and $E_2$ which have the normal vectors $\mathbf n_1$ and $\mathbf n_2$ and intersect at the point $\mathbf p$ . Now what happens when you look into such a construction? We can easily deduce (“calculate”) this with the above considerations: let $s_1$ and $s_2$ be the reflections on the plane $E_1$ and $E_2$ , respectively. We get

(1) $\begin{gather*}s_1(s_2(\mathbf x))=s_1(\mathbf x-2\langle\mathbf x-\mathbf p,\mathbf n_2\rangle\mathbf n_2)=x-2\langle\mathbf x-\mathbf p, \mathbf n_2\rangle\mathbf n_2-2\langle x-2\langle\mathbf x-\mathbf p,\mathbf n_2\rangle\mathbf n_2-\mathbf p, \mathbf n_1\rangle\mathbf n_1\=\mathbf x-2(\langle\mathbf x-\mathbf p,\mathbf n_1\rangle\mathbf n_1+\langle\mathbf x-\mathbf p, \mathbf n_2\rangle\mathbf n_2)+4\langle\mathbf x-\mathbf p,\mathbf n_2\rangle\mathbf n_2,\mathbf n_1\rangle\mathbf n_1\quad(\ast). \end{gather*}$

Here we used the bilinearity of the scalar product. A number of things can be seen from this expression: For example, it is in general not symmetric in $\mathbf n_1$ and $\mathbf n_2$ , i.e., does not coincide with the double mirror image $s_2(s_1(\mathbf x))$ . Symmetry (and thus equality of the two expressions) occurs exactly when the last summand in the above equation $(\ast)$ becomes zero, i.e. $\langle\mathbf n_1,\mathbf n_2\rangle=0$ , i.e. $E_1$ and $E_2$ are perpendicular to each other (intersection angle $\alpha=90^\circ$ ). So in this case it doesn’t matter which of the two mirrors you look into — you don’t see a break at the intersection line. You can check this yourself on the exhibit: At the rotating mirror, where both mirror planes meet perpendicularly, there is no “break” at the intersection line. With the other one, however, there is. This corresponds exactly to the above observation.

But what actually happens at the two mirrors? Let $g=E_1\cap E_2$ be the intersection line of the mirror planes $E_1$ and $E_2$ . We look once from “above” on the whole construction, thus along $g$ . To do this, we rotate our coordinate system so that $E_1$ just becomes the $xz$ plane and $g$ becomes the $z$ axis. Then the projection of $E_2$ along the straight line $g$ onto the $xy$ -plane becomes exactly an origin line intersecting the positive $x$ -axis at the intersection angle $\alpha=\arccos(-\langle\mathbf n_1,\mathbf n_2\rangle)$ . In these new more suitable coordinates we can now easily illustrate what happens to a point $\mathbf x=(x,y,z)$ . Under $s_1$ it is mapped to the point $\mathbf x_1=s_1(\mathbf x)=(x,-y,z)$ , which then goes under $s_2$ to the point

$\mathbf x_2=s_2(\mathbf x_1)=\begin{pmatrix}\cos(2\alpha)x-\sin(2\alpha)y\ \sin(2\alpha)x+\cos(2\alpha)y\ z\end{pmatrix}$

goes. See also figure 1 below:

So it is simply a rotation by the angle $2\alpha$ around the $z$ axis. In the same way it can be determined that thus the point $\mathbf x_2'=s_1(s_2(\mathbf x))$ is simply the point $\mathbf x$ rotated by $-2\alpha$ around the $z$ -axis. This again also confirms our above observation that the mappings $s_1\circ s_2$ and $s_2\circ s_1$ are exactly equal for $\alpha=90^\circ=\pi/2$ , because then both are simply equal to a reflection at the straight line $g$ !

This now even explains the observation that the image you see in the two rotating mirrors with two mirror planes rotates when you put the construction into rotation. Because the intersection line $g$ turns then before you and thus also the mirror images with.

Three and more mirrors

If we now add another mirror, it becomes even more curious: Let’s assume that three mirrors with the mirror planes $E_1$ , $E_2$ , $E_3$ are perpendicular to each other (thus form the coordinate planes $xy$ , $xz$ and $yz$ except for rotation). Similar considerations as above, now show that the threefold mirrored point $\mathbf x$ is then transformed into the point $-\mathbf x$ (independent of the order of the mirrorings; see figure 2). So if you look into such a coordinate cross, you will again see no break at the intersection lines. It gets even better: No matter from which direction you look into this construction, you always see your face, because the point $\mathbf x$ is always exactly opposite to the triple mirror image $-\mathbf x$ . This technique is also used in navigation, for example for bridges.

Mirror groups

If you look into two mirrors like in the exhibit rotating mirror, you may have noticed that from some point of view it seems as if not only two but several mirrors are standing on top of each other at the same angle. How many mirrors you see depends on the intersection angle $\alpha$ . For example, if the two mirrors meet at an angle of $90^\circ$ , you will see four mirrors evenly spaced around the intersection line. At a smaller angle of intersection the number becomes larger. Where does this strange phenomenon come from?

Well, above we have seen that the sequence in which the reflections take place is important. So running $s_1\circ s_2$ and $s_2\circ s_1$ one after the other results in a different image (at least, if the intersection angle $\alpha$ is not exactly $90^\circ$ ).

What happens now is that you see not only the mirror image of the mirror image, but the mirror image of the mirror image of the mirror image and so on. This means on the mathematical side that you determine all possible mappings $s_1\circ s_2\circ s_1\circ\cdots$ that can somehow be composed of the reflections $s_1$ and $s_2$ . This is called the group generated by the reflections $s_1$ and $s_2$ . A reflection always has the property that applied twice it gives the identity function again: if you change the sign of a basis vector of an orthonormal basis twice, you get back the original basis. So that means $s_1\circ s_1=s_2\circ s_2=\mathrm{id}$ . We have also already considered that $s_1\circ s_2$ represents a rotation by angle $2\alpha$ (and $s_2\circ s_1$ represents a rotation in the opposite direction). This makes it appear to you that many mirrors are arranged around the straight line $g$ so that two intersect each other at the angle of $\alpha$ (because the plane $E_1$ then intersects with the mirror image $s_1(E_2)$ exactly at the angle $\alpha$ ).

The corresponding group is called a dihedral group. These are groups generated by exactly two reflections (these are also called involutions, i.e. $s^2=\mathrm{id}$ ). How many elements the Dieder group has now depends on the number of different mappings, which can be written as concatenations of the two basic mirrorings. This in turn depends on the angle $\alpha$ : For example, if $\alpha=90^\circ$ , then $s_1$ and $s_2$ interchange so that $s_1\circ s_2=s_2\circ s_1$ . It is then easy to consider that there are only four fundamentally different mappings: $,\mathrm{id},\s_1,s_2,s_1s_2$ . This corresponds to the dieder group $D_2$ of order $4$ . This is the symmetry group of a distance in the plane. Now, if $\alpha=2\pi\frac{p}{q}$ is a rational multiple of the total angle $2\pi$ with $p,q\in\mathbb N_+$ divisible, then the group generated by $s_1$ and $s_2$ is the dieder group $D_q$ of order $2q$ , that is, the transformation group of a regular $q$ -corner, because $(s_1\circ s_2)^q=\mathrm{id}$ . If, on the other hand, $\alpha$ is not such a rational multiple of $\pi$ , then the rotation $s_1\circ s_2$ never returns to its initial state, i.e., it does not satisfy any equation of the form $(s_1\circ s_2)^q=\id$ . This gives us the infinite Dieder group $D_\infty$ .

For two mirrors, this is all that can happen. If, on the other hand, we take three mirrors or more, things get more complicated: with three mirrors perpendicular to each other, we get a group with eight elements, each of which transforms the unit cube into itself.

Such groups generated by finitely many reflections can be studied and classified in detail, see [1].

The exhibits “kaleidoscope mirror”, “kaleidoscope”, “mirror funnel” and “polyhedron crown” are also relevant for this purpose. The first three again show a group generated by a certain arrangement of mirrors. Especially the “mirror funnel” is interesting here, because it seems as if one sees here the side faces of a dodecahedron. This connection is no coincidence, because a platonic body is transformed into itself by each reflection at a plane which runs through its center and contains one of its edges.

Literature

[1] https://de.wikipedia.org/wiki/Wurzelsystem#Spiegelungsgruppen

[2] https://de.wikipedia.org/wiki/Diedergruppe

[3] https://de.wikipedia.org/wiki/Regelmäßiges_Polygon

[4] https://de.wikipedia.org/wiki/Platonischer_Körper

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