Alex's Adventures in Numberland - Alex Bellos [90]
Taoist woodblock print with the yubu.
China was not the only culture to see the mystical side of the lo shu. Magic squares have been objects of spiritual importance for Hindus, Muslims, Jews and Christians. Islamic culture found the most creative uses. In Turkey and India vgins were required to embroider magic squares on the shirts of warriors. And if a magic square were placed over the womb of a woman in labour, it was believed that the birth would be an easier one. Hindus wore amulets with magic squares as protective charms, and Renaissance astrologers associated them with the planets in our solar system. It is easy to mock a predisposition for the occult in our ancestors, yet modern man can understand their fascination with magic squares. Both simple and yet subtly complex, a magic square is like a numerical mantra, an object of endless contemplation and a self-contained expression of order in a disordered world.
Melencolia I: Dürer’s famous woodcut shows an angel lost in thought surrounded by mathematical and scientific objects, such as a compass, a sphere, a set of scales, an hourglass and a magic square. Art historians, especially those with a mystical bent, have long pondered the symbolism of the geometrical object in the middle left of the image, which is known as ‘Dürer’s solid’ mathematicians have long pondered the mystery of how on earth to construct it.
One of the pleasures of magic squares is that they are not restricted to 3 × 3 grids. A famous example of a 4 × 4 square comes from the work of Albrecht Dürer. In Melencolia I (shown opposite), Dürer included a 4 ×4 square that is best known for containing the year in which he engraved it: 1514.
Dürer’s square, in fact, is übermagic. Not only do the rows, columns and diagonals add up to 34, but so do the combinations of four numbers marked by dots and linked in the squares below.
The patterns Dürer’s square produces are amazing, and the more you look, the more you find. For example, the sum of the squares of the numbers on the first and second rows adds up to 748. You get the same total by adding up the squares of the numbers in rows 3 and 4, or the squares of the numbers in rows 1 and 3, or the squares of the numbers in rows 2 and 4, or the squares of the numbers in both diagonals. Wow!
For further amazement, rotate Dürer’s square by 180 degrees, then subtract 1 from the squares containing 11, 12, 15 and 16. The result is the following:
The image is from the side of the Sagrada Família cathedral in Barcelona, designed by Antoni Gaudí. Gaudí’s square is not magic, as two numbers are repeated, but it is still pretty special. The columns, rows and diagonals now all add up to 33: the age of Christ at his death.
Hours of fun can be had by playing around with magic squares, and marvelling at the patterns and harmonies. In fact, no other area of non-practical maths has attracted as much attention from amateur mathematicians over such a long period. In the eighteenth and nineteenth centuries, literature on magic squares flourished. One of the most notable enthusiasts was Benjamin Franklin, one of the Founding Fathers of the United States who, as a young clerk of the Pennsylvania Assembly, got so bored during debates that he would construct his own squares. His best-known square is the 8 × 8 variation shown opposite, which he is said to have invented as a boy. In this square Franklin included one of his own enhancements to the theory of magic squares: the ‘broken diagonal’, which are the number in the black squares and grey squares, shown in A and B below. While his square isn’t a proper magic square because the full diagonals don’t add up to the number 260, his newly invented broken diagonals do. The sums of the black squares in C and D and E, and the sum of the grey squares in E and, of course, the sum of every row and column, also add