Computational Geometry Column 47

A remarkable theorem is described: “It is possible to tile the plane with nonoverlapping squares using exactly one square of each integral dimension. ” Thus, one can “square the plane.” More than thirty years ago, Solomon Golomb posed [Gol75] the question of whether or not the infinite plane could be tiled using one copy of each square of integer side length: one copy of the squares (1×1), (2×2), (3×3), and so on. The problem was subsequently discussed in a Martin Gardner column, and in Grünbaum and Shephard’s Tiling and Patterns [GS87, p. 79], but remained unsolved. The challenge is to avoid repeating any square, at the same time as using all of them. Golomb’s “heterogeneous tiling conjecture ” has now been established by Henle and Henle [HH06]. Here we sketch the main idea of the proof, which is both elementary and subtle. Define a figure to be perfect if it is composed entirely of non-overlapping squares of different sizes. Define an L to be any six-sided orthogonal polygon, i.e., one whose edges meet at right angles. (An example is in Fig. 3a.