Showing posts with label Albrecht Dürer. Show all posts
Showing posts with label Albrecht Dürer. Show all posts

Monday, 20 September 2021

How Dürer's "Melencolia I" is a painful but liberating metamorphosis!

The title of this post may at first seem rather strange, especially when we know that the main subjects of these pages are "Magic Squares, Spheres and Tori." However, the famous "Melencolia I" engraving by Albrecht Dürer does depict, amongst other symbols, a magic square of order-4 (already examined in "A Magic Square Tribute to Dürer, 500+ Years After Melencolia I," and "Pan-Zigzag Magic Tori Magnify the "Dürer" Magic Square"). 

In the section reserved for correspondence at the end of the post "A Magic Square Tribute to Dürer, 500+ Years After Melencolia I," I have recently received some interesting comments from Rob Sellars. Rob looks at Dürer's engraving from a Judaic point of view and describes the bat-like animal (at the top left) as a flying chimera which has a combination of the Tinshemet features of the "flying waterfowl and the earth mole." Rob's description has made me look harder at this beast, and in doing so, I have noticed some aspects that explain the very essence of "Melencolia I."
 

The Historical Context


The year 1514 CE came during a turbulent historical period, just three years before the Protestant Reformation, and the seemingly endless wars of religion which would follow. When Albrecht Dürer created "Melencolia I," he was expressing the philosophical, scientific, and humanist ideas of fifteenth-century Italy, and thus contributing to the beginning of a new phase of the Renaissance. Dürer was one of the first artists of Northern Europe to understand the importance of the Greek classics, and particularly the ideas of Plato and Socrates. The Renaissance idea was revolutionary, as it suggested that everyone was created "in imago Dei," in the image of God, and was capable of developing himself, or herself, to participate in the creation of the universe. This idea was now being gradually transmitted to all classes of society, thanks to the invention of the printing press; but it required a metaphorical language which could be deciphered by all, especially in a largely illiterate population. Although most of Dürer’s prints were intended for this wide public, his three master engravings (“Meisterstiche”), which include "Melencolia I," were aimed instead at a more discerning circle of fellow humanists and artists. The messages were more intellectual, using subtle symbols that would not be evident for common men, but could be decrypted by those initiated in the art.

The Metamorphosis of the Flying Creature

In the cartouche of Dürer's "Melencolia I", what at first looks like a flying bat is in fact a self-disembowelled flying rat!
Detail of the bat-like beast at the top left-hand side of "Melencolia I" which was engraved by Albrecht Dürer in 1514

At first sight, the cartouche at the top left-hand side of Dürer's "Melencolia I" seems to be a flying bat, bearing the title of the engraving on its open wings. The length and thickness of the tail both look oversized, but we can suppose that Dürer was using his artistic licence to amplify the visual impact of the swooping beast. Nearly all species of bats have tails, even if most (if not all) of these, are shorter and thinner than the one that Dürer has depicted.

But looking again with more attention, we can see that, quite weirdly, the body of the animal is placed above its wings, which is impossible unless the bat is flying upside-down! Closer examination suggests that this is not the case, as the mouth and eyes of the beast are clearly those of an animal with an upright head. All the same, we might well ask where the hind feet are, and how the creature can possibly make a safe landing without these!

Looking once again more closely, we can see another, even more troubling detail, in that the “wings,” which carry the title of the engraving, are in fact two large strips of ragged skin, ripped outwards from the belly, as if the animal has disembowelled itself!

Judging from the thickness of its tail and the form of its head, the airborne creature was initially a rat before it began its painful metamorphosis. It has since carried out an auto-mutilation, and is now showing its inner melancholy to the outer world, but at the same time flying free with its hard-earned wings!

Symbolically, the cartouche is telling us that ""Melencolia I" is a painful metamorphosis which precedes a liberating "Renaissance!""

How Melancholy leads to Renaissance

During 1514 CE the artist's mother, Barbara Dürer (née Holper), passed away, or “died hard” as he described it, and we can therefore suppose that Dürer’s grief would have been a strong catalyst of the very melancholic atmosphere depicted in his “Melencolia I.” The melancholy, referred to in the title of the engraving, is illustrated by an extraordinary collection of symbols that fill the scene. Some of these are tools associated with craft and carpentry. Others are objects and instruments that refer to alchemy, geometry or mathematics. In addition to the bat-like beast, the sky also contains what might be a moonbow and a comet. Further symbols include a putto seated on a millstone, and a robust winged person, also seated, which could well be a metaphorical self-portrait. These, and many other symbols, are the object of multiple interpretations by various authors. Some scholars consider the engraving to be an allegory, which can be interpreted through the correct comprehension of the symbols, while others think that the ambiguity is intentional, and designed to resist complete interpretation. I tend to agree with the latter point of view, and think that the confusion symbolises the unfinished studies and works of the main melancholic figure; an apprentice angel, who believes that despite his worldly efforts, he lacks inspiration, and is not making sufficient progress.

Notwithstanding the melancholy that reigns, there is still hope: The 4 x 4 magic square, for example, has the same dimension as Agrippa's Jupiter square, a talisman that supposedly counters melancholy. The intent expression of the main winged person suggests a determination to overcome his doubts, and transcend the obstacles that continue to block his progression. Positive symbols of a resurrection or "Renaissance" are also plainly visible, not only in the hard-earned wings of the flying creature, but also in the growing wings that Dürer gives himself in his portrayal as the apprentice angel.


"Melencolia I," engraved by Albrecht Dürer in 1514, is an illustration of the artist's melancholy, and is filled with symbols.
"Melencolia I" engraved by Albrecht Dürer in 1514

On page 171 of his book entitled "The Life and Art of Albrecht Dürer," Erwin Panofsky considers that "Melencolia I" is the spiritual self-portrait of the artist. There is indeed much resemblance between the features of the apprentice angel, and those of the engraver in previous self-portraits.

Dürer had already adopted a striking religious pose in his last declared self-portrait of 1500 CE, giving himself a strong resemblance to Christ by respecting the iconic pictorial conventions of the time. In other presumed self-portraits, (but not declared as such), Dürer had also presented himself in a Christic manner; in his c.1493 "Christ as a Man of Sorrows;" and in his 1503 "Head of the Dead Christ." What is more, Dürer inserted his self-portraits in altarpieces; in 1506 for the San Bartolomeo church in Venice ("Feast of Rose Garlands"); in 1509 for the Dominican Church in Frankfurt ("Heller Altarpiece"); and in 1511 for a Chapel in Nuremberg's "House of Twelve Brothers" ("Landauer Altarpiece" or "The Adoration of the Trinity"). Thus Dürer was already a master of religious self-portraiture when he engraved "Melencolia I" in 1514, and he might well have continued in the same manner. But this time, probably because the theological, philosophical and humanistic ideas of the Renaissance were not only spiritually, but also intellectually inspiring, he went even further, and gave himself wings!

Acknowledgement

Passages of "The Historical Context" are inspired by the writings of Bonnie James, in her excellent article "Albrecht Dürer: The Search for the Beautiful In a Time of Trials" (Fidelio Volume 14, Number 3, Fall 2005), a publication of the Schiller Institute.  

Latest Development

After reading this article, Miguel Angel Amela (who like me, is not only interested in magic squares, but also in "Melencolia I") sent me his thanks by email, and enclosed "a paper of 2020 about a painful love triangle..." His paper is entitled "A Hidden Love Story" and interprets the "portrait of a young woman with her hair done up," which was first painted by Albrecht Dürer in 1497, and then reproduced in an engraving by Wenceslaus Hollar, almost 150 years later in 1646. Miguel's story is captivating, and I wish to thank him for kindly authorising me to publish it here.


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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 written in French.

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.
Magic torus of order-4 tribute to Dürer
A magic torus tribute to Dürer
Magic Square of Dürer's Melencolia I first state
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 magic square of Dürer's Melencolia I revised state
Detail of the revised version of the magic square that appears on "Melencolia I"
Albrecht Dürer [Public domain], via 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 Frénicle 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." Please note that since the 21st January 2018 "Fourth-Order Magic Torus Chart" has now been superseded by "Multiplicative Magic Tori."

Magic torus of order 4 - a tribute to Dürer
A magic torus tribute to Dürer

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 cells that Dürer used to date his square.
Order 4 Magic square tribute to Dürer
Miguel Amela's tribute to Dürer : 2014 - 1514 = 500 years.

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

order 4 water retention magic square tribute to Dürer
A water retention diagram of Miguel Angel Amela's tribute to Dürer - diagram by Craig Knecht
Craig Knecht had already devised a 14 x 14 magic square in 2013 to honour the 500th anniversary of Durer's famous "Melencolia I" magic square. The animation of this water retention 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):

order 4 magic square tribute to Dürer 500 years
Miguel Angel Amela's 544th birthday tribute to Albrecht Dürer, created on 21st May 2015
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).  
order 4 magic square tribute to Dürer 500 years after
Lorenzo D. Sisican Jr.'s first tribute to Dürer, 500 years after Melencolia I
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). 
Order 4 magic square tribute to Dürer 500+ years
Lorenzo D. Sisican Jr.'s second tribute to Dürer, 500 years after Melencolia I
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.

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Tuesday, 10 August 2010

Les Carrés Magiques et les Sphères - 3

Le carré magique et la sphère sont associés dans la gravure sur cuivre d’Albrecht Dürer « Melencolia I » datée de 1514. « Melencolia I » est un œuvre symbolique majeur qui fait l’objet d’interprétations multiples. Les symboles illustrés dans cette allégorie semblent décrire des connaissances qui sont perdues, ou des défis qui dépassent le commun des mortels – tels que le problématique impossible de la quadrature du cercle. On peut y trouver un carré magique, un polyèdre, et une sphère.

"Melencolia I" by Albrecht Dürer, engraved in 1514, and showing a magic square.
Albrecht Dürer's Melencolia I [Public domain], via Wikimedia Commons

Le carré de Dürer n’est pas panmagique mais il a la particularité d’être gnomon : En plus des caractéristiques d’un carré magique classique ayant un constat magique de 34, les sommes de chacun de ces quatre quadrants ainsi que la somme des nombres du carré du milieu sont toujours de 34. 3 et 4 sont significatifs car ils apparaissent en symétrie dans la Tetrakys Pythagoricienne : 1+2+3+4 = 10 = 3+4+3.

Le_carré_magique_Dürer

Le polyèdre est d’une construction originale qu’on appelle « polyèdre de Dürer ». Ce polyèdre de 8 faces est sphérique, ayant tous ses sommets situés à égale distance par rapport à un centre. Dans « A new hypothesis on Dürer’s enigmatic polyhedron in his copper engraving ‘Melencholia I’ » publié en 1999, P. Schreiber fait remarquer que ce solide peut être obtenu à partir d’un cube. Ce polyèdre sphérique contient donc le secret du passage du cube à la sphère. Le rapport entre le polyèdre et le carré magique est frappant car il y a une correspondance des faces projetées du solide et le carré arithmétique – voir l’illustration et la description qui proviennent de Wikimedia Commons:
Dürer's solid, and it's relationship with the magic square in Melencolia I.
Dürer's solid by Ael 2 [Public domain], via Wikimedia Commons
La sphère et le carré magique se trouvent dans deux angles diamétralement opposés sur un axe de composition de la gravure.

Est-ce que Dürer voulait mettre en exergue une relation particulière entre le carré magique, le polyèdre, et la sphère ? Le carré magique semble être la clé d’un énigme pythagoricienne – visible au spectateur mais négligé par le penseur, ou la penseuse, Melencolia.

En 2014, soit 500 ans après 1514, la date de création de Melencolia I, une nouvelle page est consacrée à cet anniversaire. Il s'agit de l'édition de carrés magiques commémoratives.
 
Aussi en 2014, Günter M. Ziegler a écrit un article très intéressant intitulé "Dürer's polyhedron: 5 theories that explain Melencolia's crazy cube." Cet article, publié dans la rubrique d'Alex Bellos, "Adventures in Numberland," est paru dans The Guardian le 3 décembre 2014.

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