Albrecht Dürer’s in his famous engraving Melencolia I showed many weird things. Among them is a magic square, a best-known and most enigmatic, with magic constant 34. The engraving shows a disorganized jumble of scientific equipment lying unused while an intellectual sits absorbed in thought. Dürer’s magic square is located in the upper right-hand corner of the engraving. The numbers 15 and 14 appear in the middle of the bottom row, indicating the date of the engraving, 1514.

Albrecht Dürer, Melencolia I, 1514, Staatsgalerie Stuttgart

The sum 34 can be found in the rows, columns, diagonals, each of the quadrants, the center four squares, the corner squares, the four outer numbers clockwise from the corners (3+8+14+9) and likewise the four counter-clockwise, the two sets of four symmetrical numbers (2+8+9+15 and 3+5+12+14) and the sum of the middle two entries of the two outer columns and rows (e.g. 5+9+8+12), as well as several kite-shaped quartets, e.g. 3+5+11+15. Actually, there are 86 different combinations of four numbers from the Dürer’s square that sum to it’s magic number, 34!

Albrecht Dürer, Melencolia I, detail, 1514, Staatsgalerie Stuttgart

Each row, each column, and each diagonal adds up to 34; these are the traditional magic properties. But there is more magic to be found here. There are actually thirteen different ways of dividing this square into four groups of four cells, with each group of four cells adding to 34. The menu in the applet can be used to select among these. The positions in the Dürer square can be seen as a finite vector space, in which each set of four groups of four cells is a set of parallel affine planes.

Dürer’s magic square has the additional property that the sums in any of the four quadrants, as well as the sum of the middle four numbers, are all 34. It is thus a gnomon magic square. A gnomon magic square is a 4×4 magic square in which the elements in each 2×2 corner have the same sum. Dürer’s magic square,is an example of a gnomon magic square since the sums in any of the four quadrants (as well as the sum of the middle four numbers) are all 34. In addition, any pair of numbers symmetrically placed about the center of the square sums to 17, a property making the square even more magical.