Squaring the Plane I

The question was first posed by Solomon Golomb in a 1975 article in the Journal of Recreational Mathematics [1]. He asked if the infinite plane can be tiled by different squares with every positive integer side-length represented. He called it the “heterogeneous tiling conjecture” and asked readers if they could find a solution.

Martin Gardner wrote a number of times about the problems concerned with tiling squares in his 'Mathematical Games' / Mathematical Recreations' column in Scientific American.

Martin Gardner also published collections of his columns, and the solutions sent in by readers in a series of books beginning in 1961[2]. In 'Fractal Music, Hypercards and More ...' (1992) [3] Gardner mentioned

"I gave a way to tile the plane with distinct integer-sided squares based on the dissection of a square into 21 unequal squares (using Duijvestijn's 1978 discovery of 21:112A). Another way to do it is to start with a unit square so dissected, then whirl around it squares of 1,2,3,5,8,13,... in the Fibonacci sequence. However, Gardner noted "Golomb's problem of tiling the plane with consecutive squares, starting with 1, remains unsolved. However, Brian Astle, of Princeton, New Jersey, wrote to explain how the plane could be tiled with consecutive squares in such a way as to leave an arbitrary small fraction of the plane untiled. Every tile in the sequence is at a finite distance from the origin point. Astle's method, too complicated to explain here, remains unpublished. Golomb tells me that he may have found an algorithm for solving his problem, but there are still gaps in the proof that he has yet to fill."

Grunbaum and Shepard wrote about the problem in their 1987 book ¨ Tilings and Patterns [4]. They described there a second way in which a squared square S can generate a tiling of the plane (in addition to the method indicated in the present paper): Take a second copy of S and expand it to a square S1 such that the smallest square in S1 is the size of the original square S and fit S into that square. Take another copy of S and expand it to S2 so that its smallest square is the size of S1, and so on.

In 2008 Fred Henle and James Henle published a solution [5]. Their approach focuses on rectangles and ells. By “ell,” they mean any six-sided figure whose sides are parallel to the coordinate axes.

A figure is perfect if it is composed entirely of squares of different sides. The key to their result is a Lemma, which states that given any perfect ell, it is possible to add squares to it to form a perfect rectangle. When they add squares to a perfect figure, keeping it perfect, they say they are “puffing it up.” When they puff an ell up to form a perfect rectangle, they say they are “squaring up the ell.” They can then “square the plane” as follows:

  1. Start with any perfect ell and square it up.
  2. Create a new ell by attaching to the rectangle the smallest square not yet used.
  3. Square this ell up, making sure that new squares are added in all four directions.
  4. Repeat steps 2 and 3 ad infinitum.

Henle and Henle noted ;

The algorithm presented in this paper is extravagant in that the ratio of the largest square used so far to the smallest square not yet used rapidly diverges. The procedure for squaring up, for example, when applied to the smallest possible ell, a 2 × 2 square next to a 1 × 1 square, ends in a rectangle with dimensions 1106481365205154721693 × 2659648557852203795117. The smallest square not used at this point is 4 × 4. The squaring-up procedure can certainly be improved. By way of illustration, take the ell in Figure 6. .... Our procedure, however, doesn’t square the ell up until a rectangle is reached with dimensions approximately (5.0 × 1014272) × (5.8 × 1014272). Can our squaring-up procedure be improved in some well-defined way? Does an algorithm exist for tiling the plane that methodically expands a connected island of squares in such a way that the ratio of the largest square used to the smallest not yet used is bounded by a polynomial?

Henle and Henle also posed a number of other questions and problems related to squaring of the plane, some of which they subsequently solved;


  1. S. W. Golomb, The Heterogeneous Tiling Conjecture, The J. of Rec. Math. 8 1975.
  2. M. Gardner, The Second Scientific American Book of Mathematical Puzzles & Diversions: A New Selection, Simon and Schuster, New York, 1961
  3. M. Gardner, Fractal Music, Hypercards and More ... , W. H. Freeman, New York, 1992
  4. B. Grunbaum and G. C. Shephard, ¨ Tilings and Patterns, W. H. Freeman, New York, 1987
  5. Frederick V. Henle and James M. Henle, “Squaring the Plane,” The Am. Math. Monthly, 115(1): 3-12, 2008.
  6. A. Berkoff, J. Henle, A. McDonough, A. Wesolowski, "Possibilities and Impossibilities in Square-Tiling", Int. J. of Comput. Geom. & Apps., Vol. 21, No. 5 (2011)
  7. F.V. Henle and James Henle, "Squaring and Not Squaring One or More Planes", Online Journal of Analytic Combinatorics, 01: 1-18, 2015
  8. Karl Scherer, New Mosiacs, privately printed, 1997
  9. James Henle's website