Squaring and Not Squaring One or More Planes

A set of natural numbers tiles the plane if a square-tiling of the plane exists using exactly one square of sidelength n for every n in the set. In [9] it is shown that N, the set of all natural numbers, tiles the plane. We answer here a number of questions from that paper. We show that there is a simple tiling of the plane (no nontrivial subset of squares forms a rectangle). We show that neither the odd numbers nor the prime numbers tile the plane. We show that N can tile many, even infinitely many planes. A square-tiling is perfect if no two squares used are the same size. In 1903 Max Dehn [2] asked: Is there a perfect square-tiling of a square? In 1925 Zbigniew Moroń found perfect square-tilings of several rectangles [10]. Dehn’s question was ultimately answered affirmatively in 1938 by Roland Sprague [12]. The problem and its solution were the subject of a memorable paper, “Squaring the Square” by Tutte [13], reprinted in Martin Gardner’s column in Scientific American (see [6]). Papers have continued to appear on the subject ever since (see for example, [5], [3], [4]). In 1975 Solomon Golomb [7] asked if there was a perfect square-tiling of the infinite plane with every side-length represented. In 1997, Karl Scherer [11] found an imperfect square-tiling of the plane—squares of all integral sides are used, but each size is used multiple times. The number of squares of side n used, s(n), is finite but the function s is not bounded. Golomb’s question was ultimately answered affirmatively in “Squaring the Plane.” (2008, [9]). The solution opened a host of questions, for example, Which sets tile the plane? Is there a three-colorable tiling? Can the half-plane be tiled? There are connections between squaring planes and squaring squares (see for example the proof of 5.1). There are also curious disconnects. There is a clever proof that a cube cannot be cubed ([13]). But the technique has not yet shown us that space cannot be cubed. In section 1, we find a large class of sets, including the set of odd numbers, that do not tile the plane. In section 2, we show that N can tile many planes at once. In section 3, we show that the prime numbers do not tile the plane. In trying to square squares, Tutte and his fellow researchers especially prized “simple” tilings, tilings in which no nontrivial subset of the squares forms a rectangle. One question in [9] asked if there is a simple N-tiling of the plane. In section 4, we construct a simple, perfect, square-tiling of the plane. In section 5, we report on the questions posed in [9], give a Z-tiling of the half-plane, and pose a number of new questions. Since the results in this paper were obtained, additional research has been done, [1], which deepens the mystery of tiling sets. It is now known, for example, that a set with Date: November 19, 2014. 2010 Mathematics Subject Classification. 05B45, 52C20.