Geomagic squares

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.

You are strongly advised to look at the introduction to Sallows’ book, or Bellos’ description, before continuing.

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.

About Peter Cameron

I count all the things that need to be counted.
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9 Responses to Geomagic squares

  1. Pingback: Android OS news » Ancient Puzzle Gets New Lease on ‘Geomagical’ Life

  2. The New Scientist article is here.

  3. Alex Leibowitz says:

    If one can treat geometrical problems set theoretically, is it possible to treat set theoretical problems geometrically?

  4. Alex Leibowitz says:

    Because to my mind, geometry is always clearer than set theory.

    • In the 19th century, Felix Klein proposed the viewpoint that geometry is the study of properties invariant under prescribed transformation groups. This would mean that potentially anything could be counted as geometry. But I don’t think this is what you mean.

      If geometry is 2- and 3-dimensional Euclidean geometry, then surely the answer is no. For several reasons: first, I could take some set-up whose symmetries were not contained in the Euclidean group; second, I could imagine problems about infinite sets larger than the cardinality of Euclidean space.

  5. Wesley Parish says:

    Any thoughts on the application of this to the (in)famous Rubics Cube?

    • Good question. I don’t think there is a very close connection, since the group of transformations of the Rubik cube is definitely not embeddable into the Euclidean group. However, we could imagine a reformulation, where the 26 small cubes making up the Rubik cube had little bumps and hollows to ensure that they could only fit together in the way which actually solved the cube; you are allowed to move them around but cannot build the cube without actually solving it. Trouble is, I am not quite sure that you can build the constraints of the actual moves in this model.

  6. Your article about the relationship between geometric shapes and numbers in magic squares is very interesting. Although certainly far from a “Copernican revolution” in magic squares, my personal research attempts a new way of understanding the latter. I consider a magic square to be a partial viewpoint of a magic torus and my work explores this concept. A magic torus is a convex or concave 2D surface (depending on whether you look at it from the outside or from within), and visualising magic squares in this way requires serious 3D thinking.
    First published posthumously in 1693, Bernard Frénicle de Bessy’s initial classification “”Des Quarrez Magiques” identified 880 4th-order magic squares. His findings have since been verified by computer and remain generally accepted. Frénicle is also remembered for his “standard form” of magic squares. Ironically, although the procedure is invaluable in eliminating doubles, conversion into Frénicle standard form (by rotation, transposition, and / or reflection) has until today concealed the toroidal continuity of magic squares. My study of 4th-order magic tori reveals that the 880 Frénicle 4th-order magic squares are displayed on 255 essentially different 4th-order magic tori.
    My results are illustrated here:
    The concept of the magic torus is explained here:
    A 3D analysis of a panmagic 4th-order torus is to be found here:
    More recent observations concerning sub-magic squares on 4th-order magic tori are to be found here:
    Magic squares observed this way seem to reveal a very close relationship between geometry and numbers. If you have the time to take a look at these findings I would like to know what you think.

  7. Due to automatic updates of the above-mentioned magic torus links, the main results are now to be found here:
    The concept of the magic torus is explained here:
    A table of 4th-order magic tori is to be found here:
    The interrelationships of 4th-order magic tori can be found here:
    The On-Line Encyclopedia of integer sequences has now published the number of magic tori of order n composed of the numbers from 1 to n^2.
    This sequence can be compared with the number of magic squares of order n composed of the numbers from 1 to n^2, counted up to rotations or reflections.
    Visualising magic squares as partial viewpoints of magic tori reveals interesting relationships between geometry and numbers.

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