Tilings by Regular Polygons

A tiling of the plane is a family of sets called tiles that cover the plane without gaps or overlaps. ("Without overlaps" means that the intersection of any two of the sets has measure (area) zero.) Tilings are also known as tessellations, pavings, or mosaics; they have appeared in human activities since prehistoric times. Their mathematical theory is mostly elementary, but nevertheless it contains a rich supply of interesting and sometimes surprising facts as well as many challenging problems at various levels. The same is true for the special class of tilings that will be discussed here more or less regular tilings by regular polygons. These types were chosen because they are accessible without any need for lengthy introductions, and also because they were the first to be the subject of mathematical research. The pioneering investigation was done by Johannes Kepler, more than three and a half centuries ago. Additional historical data will be given later (in Section 6) but as an introduction we reproduce in FIGURE 1 certain drawings from Kepler [1619]. We shall see that these drawings contain (at least in embryonic form) many aspects of tilings by regular polygons which even at present are not completely developed. As is the case with many other notions, the concept of "more or less regular" tilings by regular polygons developed through the centuries in response to the interests of various investigators; it is still changing, and no single point of view can claim absolute superiority over all others. Our presentation reflects our preferences, although many other definitions and directions are possible; some of these will be briefly indicated in Sections 4, 5 and 7. For most of our assertions we provide only hints which we hope will be sufficient for interested readers to construct complete proofs. Initially we shall use only regular convex polygons as tiles: if such a polygon has n edges (or sides) we shall call it an ngon, and use for it the symbol {n}. Thus {3} denotes an equilateral triangle, while {4}, {5}, {6} denote a square, a (regular) pentagon, and a (regular) hexagon, respectively. All the polygons are understood to be closed sets, that is, to include their edges and vertices.