Dihedral group

Two mirrors, placed like an open book and perpen­dic­ular to a base, help to under­stand how kalei­do­scopes work. A segment that is reflected multiple times in this mirror book, which is a dihe­dral mirror angle, can turn into any regular polygon.

The dihe­dral group is a symme­tries (self-mappings) group of a regular polygon that includes both rota­tional and axial symme­tries. All symme­tries of this kind can be produced with reflec­tions. Let’s draw a segment on the model base and look at its reflec­tions in mirrors. To simplify the analysis of the reflec­tions, it’s advised to draw some asym­metric figure or put an object in the corner.

If the mirrors form an angle of $120^\circ$, the segment reflected in mirrors will produce a regular triangle.

If we move the mirrors so that the angle between them is $90^\circ$, the segment will be reflected as a square. The “Math­e­mat­ical Etudes” logo orien­ta­tion in mirrors can be used to deter­mine reflec­tion level.

The main require­ment to a kalei­do­scope is that the image reflected in its mirrors, should be seen by the observer as a real object: if we move rela­tive to the mirrors, the object should not change. For example, a square should always remain a square.

But if we rotate the mirror book as a whole (so that the angle remains right) rela­tive to the base and the segment, the square will turn into a rhombus.

If the angle between mirrors is $72^\circ$, then, as it’s easy to guess, a regular pentagon will be visible in mirrors.

With the angle of $60^\circ$, we’ll get the regular hexagon. If we rotate the mirror book as a whole rela­tive to the base, then at some point we’ll get the regular triangle. Careful observer will notice that the image of the triangle is funda­men­tally different from the case when the right angle was rotated: the triangle, for sure, is regular, but an image of a logo is not correct.

At the angle of $360^\circ/7$, the reflec­tions group will produce a heptagon.

It’s possible to continue: in every case when the angle between mirrors is $360^\circ/n$, the image in the mirror book will always be a regular n-gon. And the image will always be stable!

But if the angle between the mirrors differs from $360^\circ/n$, then only some frag­ments of the primary area between mirrors will be visible near the “spine” of the mirror book. If the observer moves rela­tive to the mirrors, these frag­ments will change — we won’t get a kalei­do­scope.

A basis of kalei­do­scopes is a phys­ical law, which is well demon­strated by a mirror book: mirror reflec­tion in a mirror again works as a mirror.