Arts and Sciences

The arts and sciences were far more closely intertwined in the past than they are today. Among the artists who were fascinated by mathematics was the Northern Renaissance master Albrecht Dürer (1471-1528). Like his somewhat older counterpart, the Italian Renaissance master Leonardo da Vinci (1452-1519), Dürer was interested in the theory of human proportions, perspective (an emerging “science”), and technology of all sorts, particularly machines of war. He undoubtedly was familiar with Leonardo’s studies, having traveled in Italy as early as 1494.

About 1508 Dürer started to collect information on mathematics for a book, which in its complete form would never be published. But portions of this work were used in other books, such as his Course in the Art of Measurement, an introduction to geometry and perspective, and the postumously published Four Books on Human Proportion (1528).

One of Dürer’s most famous engravings is Melancholia, in which the winged figure of Melancholy symbolizes the depressed state of mind that robs artists of enthusiasm for work. Aristotle observed that such depression often afflicts talented people. Dürer surrounds his Melancholy with other symbols, many having to do with geometry, such as a compass, a polyhedron, and a sphere. This engraving is justly famous for a unique element: a magic square.

A “magic” square is an arrangement of integers in the form of a square, such that each column and row and each of the two major diagonals adds to the same sum. The figure shows Dürer’s magic square from Melancholia. It is interesting to note that the center cells in the bottom row give the date of the engraving, 1514.

The magic square dates back to ancient China. According to legend, the first magic square appeared on the back of a sacred tortoise that crawled from the Yellow River about 2200 B.C. Knowledge of magic squares spread to India about the eleventh century, and they were written about by the Japanese during the sixteenth century. Emanuel Moschopulus, a fifteenth-century Byzantine writer, apparently introduced magic squares to Europe. But it was Albrecht Dürer who is believed to have created the first original European magic square.

In the past, magic squares were supposed to possess mystical powers. Some were constructed to represent aspects of the zodiac. Carrying a magic square engraved on silver protected a person from the plague. Interest in magic squares revived during the past two centuries, and algebra and calculus have been applied by more recent square makers. Indeed, there are several different types of magic squares, each with its own rules of construction.

This bit of information is adapted from Visual Knowing: Connecting Art and Ideas Across the Curriculum, which I wrote to help teachers make connections that can enhance students’ acquisition of content knowledge and understanding in all areas of learning. (More about this book at http://www.corwinpress.com/booksProdDesc.nav?prodId=Book226697)