Chasles’ theorem

Any motion of a plane, that main­tains its orien­ta­tion is either a rota­tion or trans­la­tion.

Any motion of a plane, that changes its orien­ta­tion is a glide reflec­tion (or trans­flec­tion).

It is possible to make a model, illus­trating the first part of this impor­tant and inter­esting geomet­rical theorem, which has appli­ca­tions, for example, in mechanics. To do this, one needs a sheet of A4 paper, a sheet of trans­parency film of the same A4 size and a black-and-white printer.

The same drawing — grid with randomly filled cells — is printed on both sheets. Initially, the trans­parency film is over­laid on the sheet of paper in such a way that the draw­ings match. Then the film is slightly moved in an arbi­trary manner. As a result, the filled cells will almost always be placed on concen­tric circles and there will be a feeling of seeing these circles (unless one were so unlucky to do a trans­la­tion). This clearly shows that the motion was a rota­tion.

The circles are espe­cially promi­nent when the centre of a rota­tion falls within the sheet of paper and the rota­tion angle is not too large. Of course, the theorem is true for any orien­ta­tion-main­taining film motions rela­tive to a sheet but this example only works for minor trans­la­tions.

To make such a model, it is just enough to print the same drawing of large number of squares on the sheet of paper and on trans­parency film. One can create a drawing himself or use the given one. If you are going to do it your­self, be ready to make several attempts: too many or too few squares won’t work well.

You may have seen patterns like this in real life, they are called moirés.

Other models in “Geometric transformations”