The magic of magic squares – 7/4/2023 – Marcelo Viana

“Melancholia I”, from 1514, is one of the most famous engravings by the German master Albrecht Dürer (1471–1528). In the center, a winged female figure, believed to be the representation of melancholy, supports her enigmatic and somber face in one of her hands. Around it, objects from the world of technology — compass, planer, hourglass, scales, saw, hammer — and others that refer to mathematics and numerology. One of them, located above the woman’s head, always catches my eye: a magic square.

A magic square of order N consists of an N x N square filled with the numbers 1, 2, 3, …, N^two in such a way that the sum of the numbers in each row, in each column and in each diagonal always has the same value. This value, called the magic constant, is given by the formula N(N^two+1)/2.

We know that there is essentially a single magic square of order 3: all others can be obtained from it through simple operations like symmetries and rotations. In this case, the magic constant is 15. Dürer’s magic square is of order 4 and, therefore, his magic constant is 34.

There are exactly 880 different magic squares of order 4. The number grows rapidly: for N=5 there are 275,305,224, and for N=6 they go from 10¹⁹ (1 followed by 19 zeros). In fact, for orders greater than 5 we only have rough estimates of the number of distinct magic squares.

The study of magic squares has a long history. The first known mention —a square of order 3— is from 190 BC in China. The first record of a magic square of order 4 dates from 587 in India. The “Encyclopedia of the Brethren of Purity”, published in Baghdad in 983, contains examples of all orders up to 9.

Interest in magic squares spread to many other cultures: Japan, the Middle East, Africa and the Iberian Peninsula, from where it reached Europe as a whole. By the end of the 12th century, the main methods for building these squares had been discovered, but there was still much progress in the Renaissance and beyond.

As such advances took place, the mystical character that surrounded these mathematical objects in their early days also diminished. But they retained their fascination, particularly among artists who, like Dürer, incorporated magic squares into their creations. Another fine example, also of order 4, can be seen on the facade of the spectacular Sagrada Familia church in Barcelona, ​​by the Spanish architect Antoni Gaudí (1852–1926).

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