# Dust circles

The subject of the exhibit “Dust Circles” is an interesting mathematical theorem about plane motions. First of all you can look at the exhibit: It consists of two transparent plastic plates, each of which has black dust embedded in a congruent manner. The upper of the two is clamped in a wooden frame, which can be moved against the lower fixed plate. What can you observe now?

It seems that circles are involuntarily formed by this displacement — as the name of the exhibit suggests.

#### And now the mathematics of it:

But what is the reason for this phenomenon? For this purpose we want to introduce some mathematical terms which allow a precise description of the observed. For this purpose we first introduce the so-called Euclidean distance on the plane : Let and be two points in the plane. Then their Euclidean distance is defined as . A mapping of the plane in itself preserves the Euclidean distance, i.e., holds for all . Why do we consider such mappings? Well, the reason is that moving the upper plastic disk against the lower one corresponds exactly to such a distance-preserving self-mapping of the plane, because in this case the distance between two points remains the same, no matter whether we move the disk.

The question the exhibit indirectly asks the visitor is this: What distance-preserving self-images of the Euclidean plane are there anyway?

We will fully address this question in the course of this short in-depth text. To do this, we first observe that any translation around a given vector is distance-preserving, because it holds But the translations are now in bijection with the vectors themselves. So if is a distance-preserving mapping, we can form the mapping which still preserves the Euclidean distance but now fixes the zero point . Therefore, we may assume that . Now let us consider the images of the two standard unit vectors and under . The right triangle is determined by the lengths of its sides (namely , and ) except for congruence. However, these are preserved under , so and must again be unit vectors perpendicular to each other. Consider further that each vector is uniquely determined by its distances to the three points , , . Namely, these are exactly , , , so that and uniquely fix the parameters and . But this must give , because this vector has the same distances to , , as to . Thus the mapping must now even be linear (because this corresponds exactly to the observation just made). But since also maps the orthonormal basis to such a basis again, corresponds to an orthogonal matrix. However, these matrices can now be described very simply: To do this, suppose that maps to the unit vector . Because , we find an angle such that and (since the equation exactly describes the unit circle). The vector must now be mapped to a unit vector which is perpendicular to . But it already follows that , since there are only two such vectors. Thus holds. So for we get the matrix Here is to be chosen arbitrarily. If now , this simply corresponds to a rotation by the angle . This is the fact brought out by the exhibit. If, on the other hand, , we obtain a reflection on the straight line intersecting the axis at an angle of .

Thus we have fully understood the distance-preserving self-mappings of the Euclidean plane: Any such mapping is of the form for a rotation or reflection matrix as above and a vector corresponding to the subtracted translation. However, if we start at the identical self-mapping (where holds) and take it to a certain other distance-preserving mapping, the parameter cannot make a “jump” (since otherwise the orientation would suddenly reverse and thus the plate would be flipped, which is not allowed in the exhibit). So we can limit ourselves to the case (if , we can show that then always represents the reflection at a straight line). Now two cases can occur:

Case 1: has a fixed point . Then holds. From this follows . So is then given by , which is simply the rotation around the point by the angle .

Case 2: has no fixed point. Since the equation is always solvable for (because is then not singular), consequently must hold. is then simply a translation.

The two cases 1 and 2 now completely clarify the phenomenon observed at the exhibit: Either one sees large dust circles (case 1) or at least dust straight lines (case 2).

Finally, it should be noted that observed interesting fact for the group of all orientation-preserving ( ) self-mappings of the Euclidean plane lies group-theoretically in the fact that it is a so-called Frobenius group. This is expressed precisely in the fact that the elements which do not fix a point together with the identity form a subgroup (the translations).