They begin by describing Hecke's main correspondence theorem and establish a couple of useful lemmas, then work through a fundamental region
These are two four-sided polygons that tile the plane only in a nonperiodic way, that is, without a fundamental region
that repeats periodically like the hexagonal tiling of a bathroom floor, or the amazing tesselations of the Dutch artist M.
Finally, students repeatedly traced around their fundamental regions five times on a sheet of paper.
To consider transitional, reflective, and rotational symmetry, each student examined a sample of Escher's artwork and followed the activity of one of the fundamental regions as it is repeated in his artwork.