Equilateral Convex Pentagons Which Tile the Plane

It is shown that an equilateral convex pentagon tiles the plane if and only if it has two angles adding to 180 o or it is the unique equilateral convex pentagon with Although the area of mathematical tilings has been of interest for a long time there is still much to be discovered. We do not even know which convex polygons tile the plane. Furthermore, for those polygons which do tile, new tilings are being found. It is known that all triangles and quadrilaterals tile the plane and those convex hexagons which do tile the plane have been classified. It is also known that no convex n-gon with n ≥ 7 tiles. In this paper we consider the problem of finding all equilateral convex pentagons which tile the plane. The upshot of our study is the following: THEOREM. An equilateral convex pentagon tiles the plane if and only if it has two angles adding to 180 o , or it is the unique equilateral convex pentagon X with Thus the list of equilateral convex pentagons which tile, to be found in Schattschnei-der's paper [2], is complete. We also note that, with this theorem, the only convex polygons whose ability to tile is still in question are the nonequilateral convex pentagons. It should be remarked that in obtaining this result we make no assumptions regarding periodicity of any tiling. (Yet it is a fact that every equilateral convex pentagon which tiles does so in a periodic manner.) Our method of proof is interesting if only for the fact that it works only for the problem at hand–it could not, for instance, handle the problems of finding all convex pentagons or all equilateral convex hexagons which tile. In various places in the proof computer calculations are used.