A “magic square” is a grid of numbers that add up to the same total across every row, every column and both long diagonals: The main picture for this post shows a detail from Albrecht Durer’s engaving Melencolia I, which includes (among many other symbols of sixteenth century mathematical learning) a 4×4 magic square whose magic number is 34. In fact there are many other ways to make 34 from the numbers in this square: adding the four corners is one. How many can you find?

Latin squares, on the other hand, are made from a collection of symbols — traditionally letters from the Latin alphabet — so that each occurs exactly once in each row and column (the two long diagonals don’t matter for Latin squares). Here are two Latin squares, one made from chess pieces and another from colours (source — well worth a look):

Here’s a brief excerpt from a lecture by Robin Wilson describing some of the history of Latin squares:

Such things are often found in the domain of “recreational mathematics” — maths as a source of puzzles to wile away a Sunday afternoon — but Latin squares also have important applications to cryptography and coding theory, agriculture and the design of medical trials; and here’s a surprising application to a problem in pure geometry that might interest a sculptor (see the image on p.6). If you happen to have a partially filled-in Latin square, the task of finishing it off is NP complete, meaning there’s probably no efficient algorithm for doing it.

In literature, French novelist Georges Perec employed a Graeco-Latin square (a similar principle) to construct the novel Life: A User’s Manual. Much earlier, T. S. Eliot used a Latin Square composed purely of rotations of a triangle in the composition of his epic late poem The Four Quartets, a variation discussed in detail in this book (a bit more maths here), and perhaps it gives us a whiff of the repetitive, combinatory work of the Latin square in the way the same words and themes keep returning in different permutations. Here’s Eliot reading the whole thing (this is a manner of poetry performance that’s almost entirely vanished, incidentally):

Composer Sir Peter Maxwell Davies, who sadly died this week at the age of 81, was one of many of the post-war years who employed combinatorial techniques in his music. In A Mirror of Whitening Light, a magic (not Latin!) square is used as a kind of territory that the musical process “walks around”, as briefly explained in this PDF and these slides from the same presentation. Note, too, that the composer chose to make some alterations to the square — mathematical methods can inform artistic practice without turning into a straitjacket.

These objects aren’t very interesting visually, perhaps, although some have tried making paintings or drawings of them. Their real potential is behind the scenes, as all these (primarily non-visual) examples suggest. And in fact there’s a much deeper connection between Latin squares and “symmetry” in the most abstract sense; but that belongs to group theory, a topic (we hope) for a future FAMC course.