Martin's Long Affair with the Square, by Stuart E. Anderson
Martin Gardner began his Scientific American column in 1957. In November 1958 he chose Squared Squares as the subject of his Mathematical Games column and the 33 x 32 perfect squared rectangle of Z. Moroń was featured on the magazine cover. William Tutte wrote an article 'Squaring the Square' for the column on how the square was squared by himself and his Trinity undergraduate colleagues, Brooks, Smith, and Stone.
Martin Gardner revisited the topic of square tilings and squared squares several times in his column and books. As important discoveries were made, Gardner reported the findings, and informed and inspired amateurs and professional mathematicians alike.
The 1958 article 'Squaring the Square' was reprised in the 1961 publication of The Second Book of Mathematical Puzzles and Diversions, with a revised addendum, featuring Bouwkamp and Duijvestijn's tables of squared rectangles, produced in the then unsolved search for the lowest order perfect squared square. In my opinion, this book, and the 'Squaring the Square' article did more than anything to establish squared squares, and squared rectangles, in the imagination of young mathematicians and in the lexicon of recreational mathematics.
One of the most enduring and compelling images of the perfect squared rectangle is the Fibonacci sequence represented as a spiral of whirling squares. In recent times this has become one of the memes of cosmic mystic psycho-babble, with a plethora of bad YouTube videos claiming the circular arcs of the Fibonacci squares whirling spiral are a logarithmic spiral, like the nautilus shell, hurricanes and spiral galaxy arms, all connected in mystic harmony because the spirals look similar. The logarithmic spiral is quite a different beast mathematically to the Fibonacci spiral. Martin Gardner the skeptic and scourge of pseudo-science would have had none of it. On page 93 of The Second Book of Mathematical Puzzles and Diversions he shows a drawing of 'whirling squares' figure 41. This is described as a perfect squared rectangle of order infinity. This is a clear, but beautifully imaginative description, it is also a construction that can be used in proofs, thanks for keeping clear of the mysticism Martin.
Fortunately for readers, Martin updated his publications and postscripts were added to later editions, keeping us informed of all the critical developments, like here:
(POSTSCRIPT, 1987) The most significant new discovery about squared squares was the solution to the task of determining the smallest order for a simple perfect squared square. It is 21. You'll find the details in The Journal of Combinatorial Theory, vol. 35B (1978), pp. 260-63, and in Scientific American of June 1978, pp. 86-87.
One of the impulses and working methods of the mathematician is to generalise problems by changing their assumptions and conditions and abstracting from particular instances to general theories. We see this at work in the way Martin revisits topics in recreational mathematics. In his book Mathematical Carnival, pages 139-149, he has a chapter on, Mrs. Perkins' Quilt and Other Square-Packing Problems. The Mrs. Perkins' Quilt problem is the same as the problem of squaring the square except the squares need not be all different sizes. This changes the situation considerably, increases the number of solutions possible at each order. Progress in quilts since Martin's article has been patchy, although there have been some exciting new developments: arxiv.org, and squaring.net page on Mrs Perkin's Quilts
Squaring the Word, and all manner of things!
Another variation on the squared square theme that Martin entertained was 'Squaring the Word'.
In the Sixth Book of Mathematical Diversions on page 58 we have Squaring the word as creating a kind of crossword square which is symmetrical about the diagonal. Note the little dig at P.J.. Federico, the maestro at finding new methods of making perfect squared squares.
A number of readers tried the more difficult task of squaring the square. All together about 24 different squared squares came in, all with esoteric words such as Square, Quaver, Uakari, Avalon, Rerose, Erinea (hiss. P.J. Federico).
S Q U A R E Q U A V E R U A K A R I A V A L O N R E R O S E E R I N E A
On pages 225-226 he revisits infinite regress and a reiteration of the proof there are no perfect cubed cubes. He returns to this proof in a future article.
In Fractal Music, Hypercards and More pages 289-298, Martin delves into the problem of packing unit squares in a larger square. This can be considered a problem of dissecting (or alternatively packing) squares (all the same size) into a square.
In yet another variation on the theme, on page 296, Martin poses the problem tiling the plane with consecutive squares. This was recently solved by Henle.
There is also a variation on Mrs. Perkins' Quilt — What is the largest number of subsquares (allowing duplicates) into which no square can be cut?
On page 298 he returns to the proof that there are no perfect cubed cubes!
In the Addendum on page 301 he revisits the 70^2 tiling using the squares from 1-24, which is not possible, but asks if it is possible on the torus? The reported remarks by John Horton Conway on the sum of 1 to 24 squared = 70 squared are intriguing!
Another variation on successive square tiling that came from Martin's columns and reader feedback, and appears in this book is the Wainwright partridge tiling. This area has blossomed into a beautiful tiling sub-genre.
Martin also revisits the Fibonacci whirling squares from his 1961 publication, he shows how a Fibonacci tiling of the plane with squares and one perfect squared square is possible.
Squaring the square and the many variations around that activity that Martin Gardner entertained his readers with, have continued to expand and develop in many directions. Thanks Martin, we are reaping the harvest from the garden you have grown.
The Mock Cube Problem
As readers of Lewis Carroll would be aware, the Mock Turtle is a character from Alice in Wonderland, apparently composed of a combination of real creatures.
As aficionados of squared squares know, it is impossible for a perfect squared cube to exist, one cannot 'cube the cube'. Martin Gardner reprised the proof on a number of occasions.
However a construction called the Mock Cube may be possible, but no one has ever found such a thing, though it would be more accurate to say no one has really been looking for it.
What is a Mock Cube? A Mock Cube would resemble a perfect cubed cube on the outside. Each face of the cube would be a squared square, and each squared square would share the elements along each of its 4 edges with 4 other squared squares (for all 6 faces of the cube). Each corner element would be shared with the three squared squares meeting at that cube corner.
How would one construct such a beast? The catalogs of squared squares at squaring.net would be the essential ingredients. (For Mock Cube Soup? --ed.)
If a perfect Mock Cube is not possible, then is it possible to create one using a combination of imperfect and perfect squares?
I should mention that this problem appeared in Jasper Skinner's book from the early 1990s, Squared Squares, Who's Who and What's What. It was probably discussed with Brooks, Smith, Tutte and Stone.
Stuart Anderson 4/10/2014