Geomagic squares have been defined (and copyrighted) recently by Lee Sallows, a Briton living in the Netherlands, who has described them as “a Copernican revolution” in magic squares. Alex Bellos has written about them on his blog, and Jacob Aron is writing for New Scientist about them.
A magic square is a square array, each cell containing a number, so that the sum of the numbers in any row, column or diagonal (the line sums, I shall say for brevity) is equal to a constant, the magic constant of the square. Traditionally the entries are the integers 1, 2, …, n2. For a geomagic square, rather than a number, we put a geometric shape in two or three dimensions in each cell, and require that the shapes in the cells in each line can be assembled (using Euclidean transformations, that is, combinations of translations, rotations and maybe a reflection) to make a fixed target shape.
What does a mathematician make of this? If the mathematician loves group actions, the answer is to strip away the geometry and pose the problem for an arbitrary group action. The definition almost writes itself:
Let G be a permutation group on a set X. A G-magic square is an n×n square, each cell containing an orbit of G on the power set of X, together with a subset T of X called the target, such that for each line (i.e. row, column or diagonal) of the square, there is a choice of orbit representatives for the orbits in the cells in the line which form a partition of T.
Actually, this definition requires a little tweak, as we will see.
If X is finite, then replacing each cell entry by the cardinality of the sets in the orbit gives an ordinary magic square, called the shadow of the G-magic square.
More generally, if X is a measure space and G consists of measure-preserving transformations, then we create the shadow by replacing each cell entry (which we assume consists of sets of finite measure) by their common measure, again obtaining a magic square. We see that altering the sets by null sets doesn’t alter this, but does alter the fact that the sets in a line partition the target.
In fact, returning to Sallows’ examples, we see that such modification is necessary, since unless we are very clever about which boundary points are included in the sets, we will not strictly obtain a partition. So in this case, we should ask that the union of the sets in a line differs from the target by a null set.
Similarly, if X is a manifold and G a group of homeomorphisms, we shouldn’t worry overmuch about boundary points.
I will concentrate on the finite case, where these problems do not arise, from now on.
At one extreme, we could take G to be the symmetric group on X. Then an orbit of G on the power set consists of all sets of given cardinality. So, if we take any magic square whose entries are natural numbers, we can find a G-magic square (where G is a symmetric group) having the given square as its shadow, by simply replacing the entry k by the collection of all k-element subsets of a sufficiently large domain.
The other extreme is where the group is trivial. We have a purely combinatorial partition problem. There are solutions. For example, take a Latin square of order n whose diagonals are both transversals. Now choose n pairwise disjoint sets, and replace the entry i by the ith set in the collection.
This example has the drawback that each set occurs n times in the square. We can get around this as follows. Choose a pair of orthogonal Latin squares of order n, each of which has both diagonals as transversals. Now choose 2n pairwise disjoint sets, and assign them to the letters of the corresponding Graeco-Latin square; put in each cell the union of the corresponding sets.
It would be interesting to know just what examples are possible.
We see that as the group gets smaller, the examples become more prolific. Is there a point where the examples are few enough to be interesting but many enough to be classifiable?
Jacob Aron made an interesting suggestion. Magic squares used to be a very important mathematical topic. In the eighteenth century, Euler defined Latin squares and used them to construct magic squares. Now Latin squares are serious mathematics, while magic squares have become primarily recreational mathematics.
Is there a notion of “G-Latin square”, or “G-Graeco-Latin square”, which could give rise to G-magic squares by an Euler-type construction? The construction above using Latin squares suggests that this might be so.
One variation which commonly occurs with ordinary magic squares is to replace the lines of a square with another collection of subsets of a set, usually geometrically defined subsets associated with a regular polygon, star, or polyhedron. Of course all these variations can be done for G-magic squares too.
For example, we could have a Sudoku square in which each row, column, or subsquare gives a partition of the target.
Surely there are many more interesting things to investigate here! Any thoughts?
I’m grateful to Jacob Aron for asking a couple of questions which forced me to switch on my brain.