Tiling the plane with congruent tiles.
Dr. H. Heesch in Göttingen
Hilbert's 18th problem ("Mathematische Probleme", Paris 1900. cf this journal 1900) concerns tiling space with congruent polyhedrons among other things. We can ask if Euclidean n-space can be covered simply and completely by congruent copies of the same tile for each n. A certain class of tiles which allows such a tiling presents itself readily, namely the fundamental areas of the cover transformation groups of the space R^n, since a fundamental area together with its images under the group obviously form a cover of the required type. Now Hilbert asks whether polyhedrons exist which cannot occur as a fundamental area of groups of movements, but nevertheless by suitable arrangement of congruent copies can tile the whole space. This question can be affirmed for n = 2 and higher.
The Euclidean plane can be tiled by the following decagon, while it cannot be a fundamental area in a (two-dimensional discontinuous) group of cover transformations (see Figure):
x = 0 for 0 <= y <= 4 y = 4 for 0 <= x <= 2 y - x = 2 for 1 <= x <= 2 y = 3 for 1 <= x <= 2 y + x = 5 for 2 <= x <= 3 y = 2 for 2 <= x <= 3 y - x = 0 for 1 <= x <= 2 y = 1 for 1 <= x <= 2 y + x = 3 for 2 <= x <= 3 y = 0 for 0 <= x <= 3.
An example will clearly set out the connection to the representation of the whole class of questions above. Here the assertion may be confirmed by an elementary constructive consideration: If a complete tiling of the plane is possible with the polygon E, the amounts which are missing from its convex cover must be filled by congruent copies.
At the point #1 in the figure (x=y=1) a concave angle of 45 ° must be covered by neighboring polygons. Since every angle of the polygon is at least 45 °, only one tile can be used to fill this angle. A 45° angle occurs in three places of the polygon. We can ignore the highest such angle - #2 with x=2, y=4, since a second polygon using this angle to fill angle #1 of E, would have to penetrate polygon E in other places at the same time. For both other places there is only one position in which a neighboring polygon covers angle #1. Both situations come from E by glide reflection over the straight line x = 2 with the vertical translations ±1. The second of the two possibilities is drawn in the figure. E' is the chosen neighboring polygon of E.
The fact that a polygon can be a fundamental area of a group means that it is possible in some way to select the transition transformations from an tile to an adjacent tile so that each transformation can appear in a two-dimensional discontinuous group of cover transformations. Particularly there must be transformations whose repetition transfers tile E to a tile in the tiling. Neither of the given glide reflections fulfills this condition however, for their repetition is the translation of ±2 along the y-axis, and this is not a covering transformation. But since one of those two glide reflections must occur in each tiling by the given polygon, then we can see that the area cannot be a fundamental area. On the other hand simple complete tilings can certainly be formed with the polygon: one subjects the rigidly connected drawn polygon pair EE ' to the continued translation of ±4 in the y-direction. Thus a strip 4 wide is first covered, which is bounded by two straight lines, x=0 and x=4. By putting such strips side by side, a tiling of the whole plane arises.
The negative answer to the Hilbert question without using infinitely large polygons should be demonstrated without being dependent on using mirror-image tiles in the tiling, as was not done here.