A picture you know from child­hood. Kalei­do­scope. Its name comes from the ancient Greek words καλός — beau­tiful, εἶδος — view, σκοπέω — watch, observe. This optical gadget was invented by a scien­tist in the early XIX century and has quickly become a popular enter­tain­ment in many coun­tries, including Russia.

Those who used to take apart a kalei­do­scope as a child for the “research purposes” will remember that there are three mirrors in the shape of long rectan­gles inside cylin­drical tube. They form a reflec­tive trian­gular prism. Behind the triangle at the base of the prism (we will call it the funda­mental triangle), there is a container in which, as the kalei­do­scope rotates, small coloured objects are shuf­fled around forming a random image. The image formed in the funda­mental triangle is reflected in mirrors and fills the whole image plane in a beau­tiful way.

The word “beau­tiful way” means some­thing different for each person, but let’s try to distin­guish some math­e­mat­ical prop­er­ties in the image formed in the kalei­do­scope.

The image formed in the funda­mental triangle at a partic­ular moment, of course, affects the beauty of the overall image, but it is random and changes with rota­tion, and thus our consid­er­a­tions should not depend on it. Let’s replace it with a simpler one, math­e­mat­i­cally related to the funda­mental triangle itself — three multi-coloured arrows of equal length, laid out from the triangle centre perpen­dic­ular to the mirrors.

The “beauty” of the image in the kalei­do­scope depends on which funda­mental triangle is reflected in the mirrors. The resulting image must fill the whole plane, the different copies-reflec­tions of the funda­mental triangle must not overlap each other, creating a mish­mash, must not be cropped. And the main prop­erty of a “cor­rect” kalei­do­scope is that an observer should see the image formed after the reflec­tion in the mirrors as a real object: the image should not change if one moves rela­tively to the mirrors.

What can be angles of the funda­mental triangle (angles between mirrors) so that the prop­er­ties stated above are fulfilled?

In the most common type of kalei­do­scope, the triangle at the base of the prism is equi­lat­eral, with angles $60^\circ $—$60^\circ $—$60^\circ$. This is also conve­nient from a manu­fac­turing point of view — all mirrors are the same. Are any other sets of angles possible?

Let’s try to make a mirror prism with a base as an arbi­trary triangle. After reflec­tions, the observer will see a lot of debris of the image formed in the funda­mental triangle and the whole image won’t be nice. So a beau­tiful image is a big success.

Besides the equi­lat­eral triangle with angles $60^\circ $—$60^\circ $—$60^\circ$ there are only two other trian­gles that give a beau­tiful image. They are right trian­gles with angles $90^\circ $—$45^\circ $—$45^\circ$ and $90^\circ $—$30^\circ $—$60^\circ$. To verify this, let’s math­e­mat­i­cally build the image appearing in the kalei­do­scope.

Let’s take the stan­dard funda­mental triangle with angles $60^\circ $—$60^\circ $—$60^\circ$. In math­e­mat­ical terms, what does the phys­ical reflec­tion of a triangle in a mirror containing its side and perpen­dic­ular to its plane mean? It is the addi­tion of a triangle symmetric to the orig­inal triangle rela­tive to the side along which the mirror is placed. If we had a single mirror, this would be the end of it; the whole image would consist of the funda­mental triangle and its image in the mirror. But in the case of kalei­do­scope all three sides of the funda­mental triangle are mirrored, so the observer will certainly see the funda­mental triangle itself and its three copies symmetric rela­tively to its sides. In fact, as we know from expe­ri­ence, the image will be much bigger.

The truth is that mirror reflec­tions in a mirror once again “work” like a mirror. So nature continues to reflect copies of trian­gles symmet­ri­cally rela­tively to their “vir­tual” sides.

That is the first condi­tion on the funda­mental triangle: with sequen­tial symme­tries rela­tively to all its sides and then the sides of its copies, the images must sweep (cover without overlap) the whole plane. At the same time, the order in which the reflec­tions are made during sequen­tial construc­tion of the image should not affect the final result — our eye sees all the rays forming the reflec­tions of the first order, the reflec­tions of the second order, etc. at once.

The image observed in a tradi­tional equian­gular kalei­do­scope is indeed the same as that obtained by the math­e­mat­ical method discussed. And it is stable: if you sway the kalei­do­scope, the image will not change. Even where an edge between the mirrors of the kalei­do­scope moves rela­tively to the image, the image remains static regard­less of the posi­tion of the kalei­do­scope and its edges.

Kalei­do­scopes built on funda­mental trian­gles with angle sets $90^\circ $—$45^\circ $—$45^\circ$ and $90^\circ $—$30^\circ $—$60^\circ$ also satisfy all the described prop­er­ties. Are there any other cases?

Consider a triangle with angles $120^\circ $—$30^\circ $—$30^\circ$. Geomet­ri­cally this triangle seems to be suit­able, the reflec­tions rela­tive to its sides produce a plane tiling. But let us start reflecting... An atten­tive observer may notice that even in the reflec­tions of the first order — rela­tive to the sides of the funda­mental triangle itself — there is an incon­sis­tency. The images obtained by the reflec­tions with respect to the sides adja­cent to the $120^\circ$ angle are not symmetric. Thus the image obtained by math­e­mat­ical construc­tion depends on the order in which the reflec­tions are made. Specif­i­cally, if all possible reflec­tions are consid­ered, the image will be the “sum” of the orig­inal image and its mirror copy.

The surprises of the $120^\circ $—$30^\circ $—$30^\circ$ triangle don’t end there. If you make a kalei­do­scope with these angles, at first sight it would seem that, unlike the math­e­mat­ical construc­tion which says that the image will, albeit well, overlap, the resulting optical system, gives a beau­tiful image. However, that is not true. We notice that even the simplest image (colored arrows perpen­dic­ular to the mirrors) is reflected unequally even in small orders of reflec­tion. Some­where the arrows of one color are seen near the centers of the formed hexa­gons, and some­where the arrows of another color. Moving farther away from the funda­mental triangle, there are other irreg­u­lar­i­ties. So the real image does not coin­cide with the math­e­mat­i­cally predicted one. The fact is that the image is formed in each of the mirrors sepa­rately according to the already mentioned prin­ciple “the image of the mirror in the mirror, works like a mirror again”. But the image formed in one of the mirrors is not re-reflected in the other mirror.

If we sway a kalei­do­scope built on a funda­mental triangle with angles $120^\circ $—$30^\circ $—$30^\circ$, we see that the image depends on the mutual posi­tion of the observer and the kalei­do­scope axis — the image changes near the mirror prism edge as we sway.

In the case of an arbi­trary triangle, if you start making all kinds of reflec­tions of it on the plane, they will overlap each other and you can’t speak about any beau­tiful image. If one builds an optical system as a mirror prism over such a triangle, the whole image will be composed of somehow shuf­fled frag­ments of the orig­inal image and will not be regular.

So, a kalei­do­scope can be constructed using a triangle with angles $60^\circ $—$60^\circ $—$60^\circ$, $90^\circ $—$45^\circ $—$45^\circ$ or $90^\circ $—$30^\circ $—$60^\circ$ as the base of the prism. How can we math­e­mat­i­cally under­stand that a triangle with angles $120^\circ $—$30^\circ $—$30^\circ$, suit­able geomet­ri­cally for plane tiling using symme­tries, is not suit­able for constructing a kalei­do­scope? Are all possible trian­gles already listed?

The require­ments on the resulting image in the kalei­do­scope can be stated explic­itly: The triangle at the base must have angles $\frac{180^\circ }{k}$, $\frac{180^\circ}{m}$, $\frac{180^\circ}{n}$, where $k$, $m$, $n$ are natural numbers, and $\frac{180^\circ}{k}+\frac{180^\circ}{m}+\frac{180^\circ}{n}=180^\circ$. If we ignore the order, the only solu­tions $\{k, m, n\}$ of this equa­tion are the triples $\{3, 3, 3\}$, $\{2, 4, 4\}$ and $\{2, 6, 3\}$, giving the already familiar sets of angles $60^\circ $—$60^\circ $—$60^\circ$, $90^\circ $—$45^\circ $—$45^\circ$ and $90^\circ $—$30^\circ $—$60^\circ$. There are no other “kalei­do­scopic” trian­gles.

If the base of the mirrored prism is not a triangle but an arbi­trary polygon, the right kalei­do­scope is only possible by using four mirrors placed at the sides of the rectangle.

This reasoning about the prin­ciple of kalei­do­scope is the starting point of a very inter­esting area of math­e­matics, the theory of reflec­tion-gener­ated groups.