Vol. 112, No. 2, 1984

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Contributions to Hilbert’s eighteenth problem

Wolfgang Wollny

Vol. 112 (1984), No. 2, 451–495

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.

Mathematical Subject Classification 2000
Primary: 52A45, 52A45
Secondary: 51M20
Received: 6 March 1981
Revised: 25 May 1982
Published: 1 June 1984
Wolfgang Wollny