In the second part of his
eighteenth problem Hilbert formulated: “A fundamental region of each group of
Euclidean motions together with all its congruent copies evidently gives rise to a
covering of the space without gaps. The question arises as to the existence of such
polyhedra which cannot be fundamental regions of any group of motions, but
nevertheless furnish such a covering of the total space by congruent reiteration.”
Following ideas of Heesch this question of Hubert’s will be analysed in detail in this
article, restricting to the case of two dimensions — the Euclidean plane
E2.
