Albrecht Durer’s polyhedron from “MELENCOLIA” in 1514

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Melencolia I (1514) by Albrecht Dürer

 

Conjectures and theories about Dürer’s solid: an overview

 

 

The copper engraving “Melencolia I” (1514) by the German Renaissance artist Albrecht Dürer (1471 – 1528) remains one of the most enigmatic works in the history of Art. At first view, the composition seems as a jumble of apparently unrelated objects, some of which cannot even be recognized or named. Behind the first layer of perception, that the eye catches immediately, a closer inspection reveals smaller objects, chaotically scattered or even half hidden.

The picture is dominated by a sitting angel, presumably feminine but muscular, whose glance is fixed somewhere in the distance. With her right hand she holds a seemingly idle pair of compasses, while with her left hand she supports her head, gloomy faced, as if dried up of inspiration. At her feet, and half hidden underneath her long garment, lay several carpenter’s tools: a plane, a pair of pliers, a saw and some nails. On her right, a putto sits on a round millstone or grindstone, occupied with scribbling something on a small board fixed on his knees, hiding his work from the viewer with his left hand. Remarkably, the millstone seems to have an axle socket slightly off – centered. Diametrically of the angel stands a large geometric solid, which we will refer to as “the Dürer solid” in this essay. Only four faces of the solid can be seen from the viewer’s point of view. In front of the solid, a dog is calmly rolled up before a sphere. In the background of all these stands what seems to be part of a building, two walls meeting seemingly perpendicularly and fixed on them appear a balance, a hourglass and a bell. A magic square is inscribed on the wall facing the viewer: a square arrangement of the numbers 1 to 16 such as the sum of each row, column and diagonal is equal to 34. A ladder leans on the wall behind the structure while far in the horizon, through the only part of the composition which is not crammed with objects, providing thus an unobstructed view, some sea or lake can be seen and even the houses of some village on its shores. A luminous celestial object on the night sky, possibly a comet, sheds light over the landscape, which is covered by a bright arc, resembling a lunar rainbow. And against the sky, over the whole composition of objects, earthly and heavenly, hovers an imaginary bat – like creature, having only two clawed front legs and the tail of a lizard. The creature stretches its body and its wings to reveal, as if tattooed on its own skin, the inscription “Melencolia§I”.

 

The conundrum presented by the engraving has been studied extensively in the past by many researchers, and several different interpretations have been offered. Most of these are attempts to guess Dürer’s own intentions, while some could even be described as wild guesses or long shots. Sometimes the interpretations given are not very different from the interpretations of a rather abstract poem: it seems that none of them is more or less “correct” than the others. Dürer has left for us a riddle whose answer is highly susceptible to subjectivity, without leaving any instructions on how this enigmatic work should be read after all. Only a few features of the picture are rather clear, such as the year 1514 appearing in the base row of the magic square. Dürer signed his work AD1514, giving himself the solution to this minor puzzle (note that AD stands for “Anno Domini” and “Albrecht Dürer” at the same time). The inverted 5 of the magic square is generally agreed that represents the month of the year 1514 in which Duerer’s mother died, an event undoubtedly linked to melancholy. It has been suggested that the magic square itself is borrowed from a talisman of the German occultist Heinrich Cornelius Agrippa von Netteshime (1486 – 1535). Cornelius Agrippa intended the talisman, or “Jupiter’s square”, as a shield from the bad influence of planet Saturn, which could cause melancholy (in medieval times, scholars associated Saturn with melancholy, one of the four humors of ancient medicine). This explanation is at least possible and reasonable, and is supported by the fact that Agrippa had visited Nurnberg, Dürer’s city, in 1510. Little else, if anything, about the picture has been resolved satisfactorily, let alone undoubtedly.

 

The comet, the bat – like creature, the landscape, the lunar rainbow, the building, the ladder, the scales, the hourglass, the bell, the putto, the millstone, the angel, the tools, the sphere have all been, and still are, subjects of guessing. Even the title of the work, “Melencolia I”, remains largely unresolved, partly due to the misspelled “melancholia” (in latin) and partly due to the mysterious “I”. Of special interest in this essay is the Dürer’s solid, the geometric object dominating the left part of the engraving. The proposed theories or conjectures about the solid can be distinguished in two categories. The first is comprised of theories about the exact geometric nature of the solid. The second collects all theories about the possible message the solid conveys or about what the solid stands for in the general context of “Melencolia I”.

 

Examining the first category, these are some of the proposed solutions of the riddle:

 

1. The truncated rhombohedron hypothesis: Most researchers agree that the solid is probably what in Solid Geometry is called a “truncated rhombohedron”. The name of it may seem somewhat repelling, however it is a rather simple solid with six faces, all of which are rhombi. A rhombus is a quadrilateral with all of its sides having the same length (by school Geometry it can be proved that it is also a parallelogram). The rhombohedron is a solid constructed by repeating a rhombus six times and looks like a dice whose angles are not right. A “truncated rhombohedron” is a rhombohedron with its two facing vertices cut off. It must be stressed that the assertion that the Dürer solid is a truncated rhombohedron is simply a conjecture and is not supported by any of Durer’s writings or by any other data. The only reason leading to a general agreement on the truncated rhombohedron hypothesis is that the Dürer’s solid simply looks like a truncated rhombohedron.

 

2. The truncated rhombohedron – 72° hypothesis: Some researchers have even made a step further, to assume that the angles of each face of the rhombohedron are 72 and 108 degrees (i.e. 1/5 and 3/10 of the full turn). Mathematically, these values are of special importance and link the Dürer’s solid to the notorious number φ, the ancient “golden ratio” or “golden mean”. Indeed, the above angles refer to a special kind of polygon, called a regular convex pentagon, which is a shape with five equal sides and five equal angles (all of them equal to 108 degrees). It can be proved that the chords of such a shape are in golden ratio to its sides.

 

3. The scaling hypothesis: It has also been argued that a vertical compression of the engraving by a factor sqrt(φ) turns the picture into a square and the solid into a truncated cube, providing thus another link to the golden ratio.

 

4. The truncated rhombohedron – 80° hypothesis: According to another point of view, resulting from measurements based on quantitative assumptions, the angle of the rhombohedron is approximately 80 degrees, which by mathematical standards is a rather humble number with no miraculous properties as the ones mentioned above.

 

5. The heptagon hypothesis: Despite the apparent humbleness of the number 80 mentioned above, it has been observed that it is suspiciously close to the number 77,2, which would be the value of an angle of the irregular pentagon produced if five of the seven vertices of a regular convex heptagon were joined.

 

6. The truncated rhombohedron – circumscribed sphere hypothesis: According to this theory, Dürer observed that while six of his rhombohedron vertices lie on the same sphere, the other two, those at the bottom and the top, stick out of it. In order to make his solid beautiful, the artist cut of the two protruding vertices in a way that the resulting truncated solid is circumscribed in a sphere. The theory is supported by the familiarity of the artist to the so called “plan and elevation” method, i.e. the using of projections of solids on planes, which would have been of great assistance in the process of the truncation.

 

7. The rectangular slab hypothesis: Apart from the rhombohedron conjecture, the Dürer’s solid has also been viewed as a nearly rectangular slab, or parallelepiped (a matchbox), with two opposite corners cut off.

 

8. The ambivalence hypothesis: This can be summarized in physicist David Finkelstein’s phrase “I propose that Dürer designed the Octahedron [meaning the solid] to be ambivalent, irresistibly construed as a truncated rhomboid in one orientation, as a truncated slab in another, and as something else from yet another”.

 

Number 1 of the above hypotheses may be traced to Erwin Panofsky (“The Life and Art of Albrecht Dürer”, 1943), a standard reference on Dürer. Numbers 2 and 6 have been proposed by Peter Schreiber (Historia Mathematica 26, 1999, 369 – 377). Numbers 5 and 7 are found in works by David Finkelstein, who attributes number 7 to personal communication with Dr. Basimah Khulusi, apparently a medical doctor. Number 4 has been proposed by C. MacGillavry (Mac Gillavry C., Nederl. Akad. Wetenschap. Proc. Series B 84, 1981, 287).

 

A number of explanations have been proposed about some meaning conveyed by the solid, or why Dürer included it in “Melencolia I”, putting it in such a prominent position. Some of these are:

 

1. The philosopher’s stone hypothesis: According to a point of view, the Dürer’s solid is a symbol, or image, of the medieval philosopher’s stone or the “stone of Saturn” (this is the same stone that, in ancient Greek mythology, Saturn swallowed instead of Jupiter). Such alchemic positions can be found in www.alchemylab.com.

 

2. The star of David hypothesis: David Finkelstein sees a Jewish star (a hexagram) of David on the projection of the solid on the ground. In his opinion the “circumcised rhomboid” is a sign of Hebraicism from Dürer’s part.

 

3. The human skull hypothesis: Many see a rather deformed human skull in the strange stain on the front face of the Dürer’s solid, with unclear meaning.

 

4. The four ghosts (or hidden faces) hypothesis: David Finkelstein again noticed in 2004 that “anyone who steps back several paces from a good print and focuses on the shading of the front face of the polyhedron patiently for a minute or so, will soon find or construct a face” which, by some sort of Necker’s cube illusion, could be at the same time either a man’s or a woman’s face. After mentioning that these faces may represent Dürer’s mother and father, Finkelstein denotes that “I am less certain of a third hidden face”. By 2008 Finkelstein had discovered two more faces on the solid which he claims first to have inferred and then found. His overall theory about “Melencolia I” is a medley of subliminal images, gematria, rebuses and anagrams, to arrive, as far as the solid is concerned, to the conclusion that it is a puzzle “unsolvable in principle but appears to solve itself in some views. It declares that the Intellectual World may have a mathematical design but, if so, that design is inaccessible to us”.

 

I must say that some of the above theories and explanations seem to be, to me at least, too farfetched and perhaps many of them did not cross Dürer’s mind, not even as a distant possibility. It must be taken into account that Dürer was an artist of the Renaissance era and, a genius as he might be and a master of the burin, his capabilities were certainly within the boundaries of human ability. It is rather doubtful whether, apart from some basic principles of perspective, Dürer could have rendered his solid with such precision and mastery to deliberately create such a subtle illusion as the ambivalence hypothesis requires. As pointed out in a previous essay (Albrecht Dürer’s eyer lini), the bell of the upper right corner of the picture is after all inaccurately drawn. The 72 degrees hypothesis is plausible and rather tempting to adopt, however there is nothing specific in the engraving to support it. I have more respect to the 80 degree hypothesis, as it is the result of a calculation with data taken directly from the picture, be it so under certain assumptions. The scaling hypothesis is simply wrong: it takes some imagination to link the dimensions of the picture to number φ, and the solid does not look like a cube under the scaling by a factor sqrt(φ). The circumscribed sphere hypothesis is the only one providing a reasonable explanation of why and how Dürer arrived to his solid. The heptagon hypothesis is supported by a newly found sketch of Dürer, showing an irregular pentagon inscribed in a circle with an apex angle approximately equal to 80 degrees (Weitzel, Hans. A further hypothesis on the polyhedron of A. Dürer, Historia Mathematica 31 ,2004, 11). If this is accepted as a rough sketch of the solid, then MacGivallry’s result is confirmed. The rectangular slab hypothesis is plausible but unsupported.

 

I have several times ventured to discover for myself the ghosts that allegedly haunt Dürer’s solid and I have indeed found several figures that might be taken for human faces. I have also found other figures that might be taken for the head of a rat, the head of an alien and the head of a lamb. It seems to me that there might be still other such figures that remain undetected, which however are of not very different nature than the face on Mars.

 

Instead of trying to explore Dürer’s mind, by sometimes wild guesswork, in order to understand what he could possibly have intended to depict with this enigmatic solid of his, I find much more interesting the problem of understanding what he actually did depict. And the only way to achieve this, in my opinion, is to build up solid arguments based on direct measurements and calculations. – Article by Tim Razo – Video by Zak Zaurus

 

6 Comments

  1. Avatar
    تحميل اغانى January 31, 2017

    Hello,I log on to your new stuff named “Albrecht Durer’s polyhedron from “MELENCOLIA” in 1514 | Max Resistance” regularly.Your story-telling style is awesome, keep doing what you’re doing!

  2. Avatar
    Lorenzo December 23, 2014

    The “true” Durer’s polyhedron has already been found. The earlier hypothesis can now be set aside. How Durer did it is described in the methodology.

  3. Avatar
    Lorenzo Sisican. Jr. September 24, 2014

    I am a Filipino who is a true-lover of science and mathematics. I studied the Durer’s solid in Melencholia I for almost two weeks. The individual subject has its own scale or proportion according to a limited space available, i called this spacial proportioning. It is apparent in the size of objects that are supposed to be far but are drawn closer, objects that are smaller are drawn to “oversize” and object that are closer are prominently exposed/emphasized.. Take the size of the man sitting who was believed to Durer and the size of other things relative to him. For the solid, it is super-emphasized that a normal form in a normal (drawing) perspective is altered but the illusion or a view of a solid a painter wanted to show is still there. In my initial investigation and geometrical analysis the solid is not made out of a rhombohedron and the angles are not of those mentioned. It can be constructed using two types of solid.

  4. Avatar
    Terence F. Dick August 07, 2014

    Hello M R

    I am a professional engineer looking at the truncated tetrahedron in plain sight. What I have found intriguing about the shape was that the faceted panels had been paired in reverse opposition to each other. Much has been said about all its angular construction so I have considered if there was perhaps another possibility. Having made a model like most researchers it seemed to me that maybe a hidden internal construction had not been considered. Because of my experience in hidden detail it occurred to me that if all the panels including the top and bottom triangles in there geometric forms were joined together something quite remarkable might take place at the centre. What did become apparent was a geometric collision taking place at the core creating a spiky ball absorbing all of the projected panels, if you take just two opposing panels separately and join them together a six pointed star presents itself – if you combine all the rest of them at the centre a stylised object not unlike the one shining in the sky above is possible.

    (An interesting quotation by Albrecht Durer: Whoever then proves his point and demonstrates the prime truth geometrically should be believed by all the world, for there we are captured)

    • Avatar
      Lorenzo Sisican. Jr. September 24, 2014

      How the person of present generation be believed, by today’s and of those many decades past since the first analysis of the Durer’s solid came out where the acceptance had gone 5 centuries and counting. Though the new proof is simple and easy, the change to accept it is complex and difficult.

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