Tuesday, February 06, 2018

Albrecht Durer's Magic Square

Albrecht Dürer (/ˈdʊərər, ˈdjʊərər/;[1] German: [ˈalbʁɛçt ˈdyːʁɐ]; 21 May 1471 – 6 April 1528)[2] was a painter, printmaker, and theorist of the German Renaissance. Born in Nuremberg, Dürer established his reputation and influence across Europe when he was still in his twenties due to his high-quality woodcut prints. He was in communication with the major Italian artists of his time, including Raphael, Giovanni Bellini and Leonardo da Vinci, and from 1512 he was patronized by emperor Maximilian I. Dürer is commemorated by both the Lutheran and Episcopal Churches. The Expulsion From Paradise by Albrecht Dürer Dürer's vast body of work includes engravings, his preferred technique in his later prints, altarpieces, portraits and self-portraits, watercolours and books. The woodcuts, such as the Apocalypse series (1498), are more Gothic than the rest of his work. His well-known engravings include the Knight, Death, and the Devil (1513), Saint Jerome in his Study (1514) and Melencolia I (1514), which has been the subject of extensive analysis and interpretation. His watercolours also mark him as one of the first European landscape artists, while his ambitious woodcuts revolutionized the potential of that medium. Dürer's introduction of classical motifs into Northern art, through his knowledge of Italian artists and German humanists, has secured his reputation as one of the most important figures of the Northern Renaissance. This is reinforced by his theoretical treatises, which involve principles of mathematics, perspective, and ideal proportions.

Melencolia I 1514

Albrecht Dürer's magic square

Detail of Melencolia I
 
The order-4 magic square Albrecht Dürer immortalized in his 1514 engraving Melencolia I, referred to above, is believed to be the first seen in European art. It is very similar to Yang Hui's square, which was created in China about 250 years before Dürer's time. The sum 34 can be found in the rows, columns, diagonals, each of the quadrants, the center four squares, and the corner squares (of the 4×4 as well as the four contained 3×3 grids). This sum can also be found in the four outer numbers clockwise from the corners (3+8+14+9) and likewise the four counter-clockwise (the locations of four queens in the two solutions of the 4 queens puzzle[36]), 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 (5+9+8+12 and 3+2+15+14), and in four kite or cross shaped quartets (3+5+11+15, 2+10+8+14, 3+9+7+15, and 2+6+12+14). The two numbers in the middle of the bottom row give the date of the engraving: 1514. The numbers 1 and 4 at either side of the date correspond respectively to the letters "A" and "D," which are the initials of the artist.