The Mystery of Dürer’s Magic Square

Zuzanna Stańska 23 July 2022 min Read

Many strange symbols appear in Albrecht Dürer’s famous engraving Melencolia. Among them is a magic constant 34, a well-known and enigmatic example. The engraving shows a disorganized clutter of scientific equipment lying around while a scientist sits absorbed in thoughts. “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.

Dürer Magic Square: Albrecht Dürer, Melencolia I, 1514, National Gallery of Art, Washington, DC, USA.
Albrecht Dürer, Melencolia I, 1514, National Gallery of Art, Washington, DC, USA.

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), 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 Dürer’s square that add up to the sum of the magic number, 34!

Dürer Magic Square, Albrecht Dürer, Melencolia I, 1514, National Gallery of Art, Washington, DC, USA. Detail.
Albrecht Dürer, Melencolia I, 1514, National Gallery of Art, Washington, DC, USA. Detail.

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 13 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 additional properties: the sum 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. In addition, any pair of numbers symmetrically placed about the center of the square sum to 17, making the square even more magical.



Weisstein, Eric W. “Dürer’s Magic Square.” MathWorldA Wolfram Web Resource. Accessed Jan 31, 2022.


Michelangelo the Creation of Adam Bizarre

Left-Handness: Artists Who Proved It Was a Gift, Not a Curse

Are you left-handed? Don’t believe ancient stories about the devil’s hand! It would be a blessing if you’d like to become an artist, like many...

Camilla de Laurentis 12 December 2022

Art for Halloween: Roberto Comandante del Caos Diz, Story Twist, 2022. Artist's profile. Bizarre

Scary Monsters and Super Creeps for a Frightening Halloween!

Throughout the history of mankind, art has always been a window to the soul. We have witnessed ethereal beauty, magical nature, bizarre sceneries,...

Errika Gerakiti 31 October 2022


Trompe L’oeil: How Artists Deceived Our Eyes Throughout the Ages

Without knowing, you have more than likely seen an example or two of trompe l’oeil art. It is a term that encompasses many art forms in its...

Rachel Witte 18 August 2022


Vile Bodies? Anatomy in Art

Are you ready to journey into the fascinating world of anatomical art? Read on, as we reveal flesh and blood art through the...

Candy Bedworth 7 March 2022