Dürer's Magic Square

Engraving "Melancholia I"

Albrecht Dürer (1471-1528) is generally considered to be Germany's most famous Renaissance artist. He was about 20 years younger than Leonardo da Vinci and around 1500 became greatly interested in the relationship between mathematics and art. Leonardo and his contemporary, mathematician Pacioli, almost certainly influenced Dürer in these studies. In 1514 he created the copperplate engraving named "Melancholia I" which contained his magic square - the first magic square published in Europe.


Engraving Melancholia I / 24 x 19 cm

This copper engraving by Albrecht Dürer has been viewed as a self portrait of the artist and the essence of German Humanism. The winged figure illustrates simultaneously the dangers and satisfactions of the intellectual activity. It is the image of the creative spirit of the man alone with his inner thoughts. The ladder leaning against the building indicating that this is unfinished, the unsolved geometry problem, the cupid above a wheel, the hydrophobic dog, the falling sand of the hourglass, and the empty oscillating balance ... all this sums up despair to Melancholia.

Dürer's magic square


Dürer's square from engraving

Albrecht Dürer (self portrait at 28 years)

Dürer's magic square, a best-known and most enigmatic, is a magic square with magic constant 34 used in an engraving entitled Melancholia I by Albrecht Dürer (The British Museum, Burton 1989, Gellert et al. 1989). 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.

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!

In the applet below, these numbers are reproduced in more modern style. 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. If you click on one of the cells, then the numbers in its group will be enlarged and highlighted in red. These groupings of cells are not arbitrary. 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.


Applet by Christopher J. Henrich

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 (Hunter and Madachy 1975, p. 24). 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, illustrated above, 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.

Interchanging columns does not change the column sums or the row sums. It usually changes the diagonal sums, but in this case both diagonal sums are still 34. So now our magic square matches the one in Dürer's etching. Dürer probably chose this particular 4-by-4 square because the date he did the work, 1514, occurs in the middle of the bottom row.

The 8 forms of Dürer's square

The eight figures of the magic square of Dürer are represented above. Remember that any magic square has eight different "forms" or eight different "figures", obtained by rotation (4 figures, including the original), by symmetry with respect to median (2 figures) and by symmetry in relation to the main diagonal (2 figures).

163213
510118
96712
415141
49516
156103
147112
112813
114154
12769
811105
132316
138121
211714
310615
16594
Original

Rotation ¼

Rotation ½

Rotation ¾

132316
811105
12769
114154
415141
96712
510118
163213
112813
147112
156103
49516
16594
310615
211714
138121
Symmetry
1st median

Symmetry
2nd median

Symmetry
2nd diagonal

Symmetry
1st diagonal


Dürer's magic square with lines
From Suzanne Alejandre

Numerology of Melancholia I

This section presents some funny results pointed by David Bowman.

The year of the engraving is written as 15-14, at bottom line. Even more, the number on the right is 1 which is the same as A in simple Masonic code of values [A=1, B=2, C=3, ... Z=26], and the number on the left of the date is 4 equaling D according to gematric value. Both numbers are therefore inter-changeable with letters A, and D, which were almost religiously used as Albrecht Durer's monogram.

The sum of all the numbers in the Durer's magical square is 136, the number that conceals both the master and the title of the masterwork. Dürer used another, Latin, variant of the spelling of his name - ALBRECHT DVRER:

ALBRECHT DVRER = (1+12+2+18+5+3+8+20)+(4+22+18+5+18) = 136

MELENCOLIA I is the engraved title, I could be read as Eins [in German, I = One = Eins]:
MELENCOLIA EINS = (13+5+12+5+14+3+15+12+9+1)+(5+9+14+19) = 136

There is also another mathematical joke concealed in the numbers. If two fields are taken for a number (16 and 3 = 163, 2 and 13 = 213, 5 and 10 = 510, and so on) and added together than the value of 2368 is obtained. This number is significant since it equals the value of Jesus Christ in Greek:

ΙΗΣΟΥΣ ΧΡΙΣΤΟΣ = 888+1480 = 2368

163+213+510+118+96+712+415+141 = 2368

This could be Dürer's homage to the Demiurge, the Grand Architect, which is also one of the aspects of the central figure of the Melencolia I, holding an open compass, a frequent symbol of many orthodox icons representing Christ as Pantocrator.

The dimensions of Durer's engraving are approximately 18.9 x 24.1 cm, which comes very close to the ratio of 11:14. This specific ratio was used in classical compositions very often, mostly because it is derived from the ratio 22:7 that closely approximates Pi = 3.14159... and was known since ancient times. The importance of Pi for the composition of Melencolia I is emphasized by the sphere in the lower left corner of the engraving, which is simultaneously also a module of the composition.

Mathematics of Dürer's Magic Square

The magic constant of a normal magic square depends only on n and has the value M = (n3 + n)/2. Here is the proof. Given an  normal magic square, suppose M is the number that each row, column and diagonal must add up to. Then since there are n rows the sum of all the numbers in the magic square must be . But the numbers being added are 1, 2, 3, ... n2, and so 1 + 2 + 3 + ... + n2. In summation notation, . Using the formula for this sum, we have , and then solving for M gives . Thus, a Lo Shu's normal magic square must have its rows, columns and diagonals adding to , a Albrecht Dürer's to M = 34, a Benjamin Franklin's  to M = 260, and so on.

Symbolic Elements in Melencolia I and interpretations

Elements of the picture

Among the most conspicuous are:

  1. The tools of geometry and architecture surround him, unused
  2. The 4 × 4 magic square, with the two middle cells of the bottom row giving the date of the engraving: 1514.
  3. The truncated rhombohedron[4] with a faint human skull on it (over the years, there have been numerous articles disputing the precise shape of this polyhedron[5])
  4. The hourglass showing time running out
  5. The empty scale (balance)
  6. The despondent winged figure of genius
  7. The purse and keys
  8. The comet and rainbow in the sky
  9. Mathematical knowledge is referenced by the use of the symbols: compass, geometrical solid, magic square, scale, hourglass
  10. The lamb is often associated with the myth of Argonauts and the Golden Fleece.

Interpretations of Melencolia I

The square in the Melencholia is a special type of magic square; the sum in one of its four quadrants, and also the sum of the central square, are 34, the magical value of the square. This is a gnomon magic square. This square should be a positive influence against melancholy. In the middle of the last line, the numbers 15 and 14, which correspond to the date of the engraving, 1514, it is also the date of the death of Barbara, mother of Albrecht Dürer.

The engraving Melancholia I, features a frustrated thinker sitting by an uncommon polyhedron. Suggestions that a series of engravings on the subject was planned are not generally accepted. Instead it seems more likely that the "I" refers to the first of the three types of melancholia defined by the German humanist writer Cornelius Agrippa. In this type, Melencholia Imaginativa, which he held artists to be subject to, 'imagination' predominates over 'mind' or 'reason'. Erwin Panofsky (a German Jewish art historian) proposed the most authoritative interpretation of Melencolia I as Dürer's "spiritual self portrait". Patrick Doorly has shown that the engraving is much indebted to Plato's Hippias Major and even more to Luca Pacioli's book De Divina Proportia. John Read has commented on the alchemic symbolism of the engraving.

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