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BRANKO GRUNBAUM AND G. C. SHEPHARD

To enumerate monohedral tilings that satisfy S.l it is necessary to find all polygons T with the property that, using tiles congruent to T, it is possible to form a "patch" of tiles of which translates can tile the plane. Although the tion of all such tiles T has not been carried out, several systematic (if laborious!) approaches are conceivable.

The present paper is mainly concerned with proper isohedral tilings by arbitrary polygons, that is, with polygonal tilings satisfying conditions 1.2 and S.2. The history of the problem of determining such tilings is of interest and we shall review it briefly.

The eighteenth of Hilbert's famous problems [26] asks whether (in our terminology) there is a tile that admits a monohedral tiling of the 3-dimensional space, but admits no isohedral tiling. From the context (see also [38]) it appears that Hilbert assumed that the corresponding planar problem has a negative solution. A similar (although rather vaguely expressed) assumption was made earlier by Fedorov [5, § 64]. The same opinion was shared by K. Reinhardt, who was Hilbert's assistant during part of the years of World War I, and whose dissertation [36] investigated planar tilings by polygons. (We shall mention some of the results of this dissertation below.) Later, Reinhardt [37] found examples of 3-dimensional tiles that admit only non-isohedral tilings, and in the same paper he asserted that no such tiles exist in the plane. He even announced that a paper proving this assertion was in preparation - but Heesch [23] found a ple, a tile that admits a periodic tiling of the plane but no isohedral tiling. Heesch's tile (see Figure 2) is non-convex, and leads only to improper tilings, but it was the first example to demonstrate that conditions S.l and S.2 are genuinely distinct. Variants of Heesch's tile were given in [22, 24, 35]. Another example, which is not related to these and uses a convex tile, will be described below. Milnor's recent account [35] of developments related to Hilbert's teenth problem devotes a section to monohedral tilings but goes no further than Heesch's contribution [23] from 1935.

One of the aims of Reinhardt's thesis [36] was the determination of all convex tiles that admit monohedral tilings of the plane; he expected to establish Hilbert's conjecture that each such tile admits isohedral tilings. Reinhardt observed that all

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