The 17 plane symmetry groups

Abstract

Several modern mathematics courses contain a description of the 17 distinct ‘wallpaper patterns’. Others contain the definition of “group” and “isomorphism”, together sometimes with a vague statement that these concepts can be used to justify the fact that there are precisely 17 patterns. But neither the passive contemplation of wallpaper patterns, nor the passive contemplation of abstract definitions, is mathematics: the latter is above all an activity in which definitions are used to obtain concrete results. For this reason I have often been asked by teachers what is needed to give a rigorous proof that there are precisely 17 patterns.