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.