Sunday, 7 December 2014

A magic square tribute to Dürer, 500+ years after Melencolia I

Albrecht Dürer's "Melencolia I" contains numerous mathematical and philosophical symbols amongst which we can see a magic square, a polyhedron and a sphere. These 3 symbols are partially interpreted in an earlier article.

1514, the date of the engraving, appears in the lower line of the fourth-order magic square. 500 years later, is it possible to create a fourth-order magic torus tribute to Dürer?

In order to write 2014 we can subtract 1 from all the numbers (1 to 16), and thus obtain the zero that cannot be found in normal fourth-order magic squares. After the subtraction, we can use a flattened viewpoint of the basic magic torus T4.177 (group number T4.05.4.12) that displays not only the Frénicle squares n°99 and 618, but also 14 semi-magic squares. The numbers 0 to 15 are repeated beyond the edges of the square in order to represent the toroidal continuity. As the new figures contain the numbers from 0 to (N²-1) instead of 1 to N², their magic constant is 30 instead of 34.

A magic torus tribute to Dürer - copyright

Detail of the "first state" version of the magic square that appears on Dürer's "Melencolia I"
(National Gallery of Victoria)
Perhaps Dürer hesitated when engraving the left hand side of his square: The number 5 appears to replace a former number 6. It should also be noted that the print detailed in the first version above is, according to The National Gallery of Victoria is "the very rare first state." The reversed number 9 was later corrected in the revised version below. On page 12 of the 1961 Aukland City Art Gallery catalogue "Albrecht DÜRER forty engravings and woodcuts," it is written that "There are two states - the first where the nine on the numerical table is reversed, the second where the nine is corrected." The contemplation of the two versions of Dürer's magic square may well remind the reader of the duality of the “De Umbrarum Regis Novum Portis” engravings in Roman Polanski's film "The Ninth Gate." David Fritz Finkelstein has written that "Dürer's engraving MELENCOLIA I was circulated in two versions not previously distinguished. Besides their conspicuous early Renaissance scientific instruments and tools, they contain numerous apparently unreported concealments whose detection reveals heresies expressed in the work." Frickelstein believes that the second version of the engraving is the one in which the number 9 is reversed, and suggests that the numerals 9, 6, and 0 are curled serpents. To find out more please consult Finkelstein's "Melencolia I.1*"
Detail of the revised version of the magic square that appears on Dürer's "Melencolia I"
(Wikimedia Commons)
Dürer's square, (Frénicle n° 175), is one of 8 semi-pandiagonal squares displayed by the semi-pandiagonal torus T4.077 (torus type n° T4.02.2.03). The seven other semi-pandiagonal squares that are displayed by the same "Dürer torus" are the Frénilce index n° 27, 233, 360, 421, 583, 803, and 850. The semi-pandiagonal "Dürer torus" T4.077 also displays 8 semi-magic squares, and is completely covered by 8 sub-magic squares. More details of the fourth-order magic tori can be found in earlier articles such as:
"255 fourth-order magic tori, and 1 third-order magic torus," "Table of fourth-order magic tori," and "Fourth-order magic torus chart."


A magic torus tribute to Dürer - copyright

Miguel Angel Amela has very kindly authorised me to publish an extraordinary pandiagonal square he created for this anniversary. Again, this is not a normal 4x4 square, as the magic constant of all the columns, rows, and diagonals is not 34, but 1514 - the year that Dürer engraved Melencolia I! Miguel has also managed to write 2014 at the centre of the bottom row, in just the same place that Dürer used to date his square.
Miguel Amela's tribute to Dürer : 2014 - 1514 = 500 years. - copyright

The following figure shows a water retention diagram of Miguel Amela's square that was created by Craig Knecht.

A water retention diagram of Miguel Angel Amela's tribute to Dürer - diagram by Craig Knecht - copyright
Harry White and Craig Knecht have created a giant 196 x 196 water retention magic square that spells the dates 1514 and 2014. Craig Knecht has also devised another 14 x 14 square that honours the 500th anniversary of Durer's famous magic square in Melencolia I. The animation of the latter square can be found at the following link.

Further to the first edition of the present article, on the 21st May 2015, Miguel Angel Amela sent me this very original magic square that he created to celebrate the 544th birthday of Albrecht Dürer (1471-1528):

Miguel Angel Amela's 544th birthday tribute to Albrecht Dürer, created on 21st May 2015 - copyright
This is not a normal 4 x 4 magic square as the magic constant of all the columns, rows, and magic diagonals is not 34, but 544. Miguel Angel Amela's creation (using the reversed nine version of Dürer's writing style) is the product of the Dürer square integers and 16 (which is the number of integers). Miguel Angel Amela points out that the divisors of 544 are: 1, 2, 4, 8, 16, 17, 32, 34, 68, 136, 272, and 544:
1, 2, 4, 8, and 16 all figure on Dürer's fourth-order magic square.
17 is the total of the complementary numbers 1 and 16, 2 and 15, 3 and 14, etc. on Dürer's square.
32 is either the product of 4 and 2 times 4, or it is twice the number of the integers on Dürer's square.
34 is the magic constant of Dürer's square.
68 is twice the magic constant (2 x 34) of Dürer's square.
136 is 4 times the magic constant (4 x 34), and therefore the total of all the numbers on Dürer's square.
272 is 8 times the magic constant (8 x 34) of Dürer's square.
544 is 16 times the magic constant (16 x 34) of Dürer's square.
The square includes the numbers 16 (4 to the power of 2), 64 (4 to the power of 3) and 256 (4 to the power of 4) in its corners, and 4x4 magic square enthusiasts will appreciate this subtle wink! Thank you Miguel for allowing me to publish your very original creation!

Lorenzo D. Sisican, Jr.has kindly authorised me to publish two non-normal, but semi-pandiagonal squares that he has created: The first of these squares has a magic constant of 500 in its rows, columns and alternate diagonals (500 years after Melencolia I).  
Lorenzo D. Sisican Jr.'s first tribute to Dürer, 500 years after Melencolia 1 - copyright
The date of the creation of Melencolia I, the year 1514, is obtained by the numbers 15 & 14. The anniversary year 2014 is obtained by the numbers 20 & 14. Dürer’s date of birth, the 21st May 1471 (21/5/1471), is obtained by the following numbers: (20 + 28 + 167) &14 & 71 = 215 14 71. Dürer died on April 6th, 1528 (4/6/1528), and his date of death is obtained by the following numbers: (161 + 245) & 15 & 28 = 4 06 1528.

The second of Lorenzo D. Sisican Jr.'s squares is also semi-pandiagonal, and has a magic constant of 500 in its rows, columns and alternate diagonals (500 years after Melencolia I). 
Lorenzo D. Sisican Jr.'s second tribute to Dürer, 500 years after Melencolia 1 - copyright
The date of the creation of Melencolia I, the year 1514,  is obtained by the numbers 151 & 4. The anniversary year 2014 is obtained by the numbers 201 & 4. Melencolia I was created 14 years before Dürer’s death, and the number 14 is also prominent on this square.