Albrecht Dürer and the Divine Ratio (Part I)

Melencolia I - copper engraving - 1514, Albrecht Dürer. Geometry: 2022, DS

"Melencolia I is a large 1514 engraving by the German Renaissance artist Albrecht Dürer. The print's central subject is an enigmatic and gloomy winged female figure thought to be a personification of melancholia – melancholy. Holding her head in her hand, she stares past the busy scene in front of her. The area is strewn with symbols and tools associated with craft and carpentry, including an hourglass, weighing scales, a hand plane, a claw hammer, and a saw. Other objects relate to alchemy, geometry or numerology. Behind the figure is a structure with an embedded magic square, and a ladder leading beyond the frame. The sky contains a rainbow, a comet or planet, and a bat-like creature bearing the text that has become the print's title.

The art historian Erwin Panofsky... wrote that 'the influence of Dürer's Melencolia I—the first representation in which the concept of melancholy was transplanted from the plane of scientific and pseudo-scientific folklore to the level of art—extended all over the European continent and lasted for more than three centuries.'"

- Excerpt from Wiki's entry for Dürer's Melencolia 1 (shown above). While many art historians seem to unanimously assume the robed, angelic figure is of the female gender, the figure is most assuredly male, and, judging by its facial expression, Dürer himself. Inset right: an early self-portrait (executed in 1498 at age 26) of the fashionable - but seriously introspective - young dude.

Regarding the central "ladder leading beyond the frame," note that it forms the apex of a large golden triangle.

Note: To give an example of how deeply this image continues to resonate over the years, Wiki mentions Peter-Klaus Schuster's 1991 publication, Melencolia I: Dürers Denkbild, an exhaustive history of the print's interpretation in two volumes.

"It should be noted that even Leonardo was unable to apply his own proportion and anatomical studies to his work given he painted little, or not at all, during the last decade of his life. Hence, the applicability of the study of proportion to practicing artists was still unclear. Dürer would spend nearly three decades working to remedy this ambiguity. He completed two treatises that would be the dominant basis for art theory in Renaissance Germany; their popularity and influence spreading with their subsequent translations. In 1525, Underweysung der Messung, or Four Books on Measurement, was published as a practical guide to geometric perspective for students of the arts; and, in 1528, Vier Bücher von Menschlicher, or Four Books on Human Proportion, appeared a few months after his death. Taken together, the studies illustrated the Renaissance belief that mathematics formed the firm basis and grounding for the arts."

- Excerpt from an commentary regarding Dürer's Vier Bücher von Menschlicher by Giovanni Paolo Gallucci found here. Inset left is one of Dürer's diagrams - a construction of a spiral - found in his Four Books on Measurement.

"There is much speculation as to why Dürer chose this construction rather than Euclid's construction which uses the 'golden ratio' proportion. The speculation stems from the fact that Dürer makes no mention of the golden ratio, although he was no doubt aware of its use in Italian art. It may be that Dürer simply did not feel comfortable with the precepts of the 'divine' ratio. The German architects had their own 'divine' ratio which was the vesica piscis ratio of  √ 1:3. In addition Ptolemy's construction is simpler than Euclid's and these constructions were just a preliminary step in his program."

- Excerpt - along with Dürer's diagram (inset left) - from The Polygons of Albrecht Dürer by G.H. Hughes. (.pdf)

Regarding the diagram, the pentagon is constructed within the mason's "Sacred" tradition utilizing the Vesica Piscis as its generative source.

"Divine truth alone, and no other, contains the secret of what the most beautiful form and measure may be."

- Albrecht Dürer, from his essay Discourse on A
esthetics published as a conclusion to the Third Book of his proportion studies. (See the Giovanni Paolo Gallucci link for the full quote.)

"The greatest miracle that I have seen in all my days, happened in the year 1503, when crosses fell on many people, especially on children more than on other people. Among them all, I once saw one in the shape which I have drawn here; it fell on the linen blouse of Eyer's maid, who was in the Pirckheimer's back-house. And she was so upset about it that she cried and wailed; for she thought she was going to die of it.

Also, I saw a comet in the heavens."

 - Albrecht Dürer from the last page of his 1503 Gedenkbuch regarding an episode of a phenomenon known as Red (or Blood) Rain. Dürer's drawing can be found on this page.

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(Note: Originally, the title of this post and the title of the URL were one and the same. That is, till I realized that Melancolia I was one of three designated Master Prints. I am not quite sure who did the designation, but, after reviewing the two other prints involved, it seemed all three might have what I (now) refer to as hidden, occulted, or passive GTS. Unlike the more outrageously active spirals - e.g., those of Caravaggio, which seem as if they were deliberately designed - the passive spirals almost seem to creep into an image with the artist unaware. The thing is, it is logical to assume Albrecht Dürer did know about the golden ratio. Alas, the jury is still out.)

Albrecht Dürer (May 21,1471 - April 6 1528) is, in his own quiet way, possibly one of the most popular artists of the Renaissance period; certainly one of the most prominent. (You know you've arrived when there's a conspiracy blog written about your life!) After all, unlike many other artists, he kept a meticulously written record of his daily affairs. And, then, there were those self portraits... even while he was ill and nearing the end of his life, he sketched his ravaged body for posterity. Perhaps, he just desperately needed to be remembered. But, why is it that all of his self-exposure seems, in the end, superficial? Because, regardless of what we learn, Dürer remains as firmly screwed into his shell as the most resistant of mollusks; he is an enigma even unto himself. In fact, his vital nature seems very much like another cryptic element found in one of his most popular images: Melencolia I (below the jump). Observe...

First off, there is the Magic Square* on the building in the background, and, then, a glaringly weird polyhedron** balanced precariously in some unknown dimension of its own. As for the misfit angels - our resident E.T.s - well, there's no getting around them. Then again, is that a stairway to heaven? Is that a large crystal ball? And, what's with the hound? And, the bat! The rainbow? The comet?

Ah, the comet. Perhaps, we have an answer to that. It seems Dürer saw a comet around the time of his father's death during a year of "Red Rain." As it was, his mother died the same year he completed Melencolia; so, it may be that he identified the comet with death... along with the bat. But, this is conjecture. We do not, cannot, and will not know. Which is why we love Dürer; he has entered our imaginations in a mysterious, almost pentagonal way: obliquely, profoundly, discreetly... but he discloses nothing.

In any case, in an image such as this, one expects to find a golden spiral - especially in light of a prominent triangle - and, as it happens, one does. (See the image introducing this post.) It is a large, loose spiral, but it coils around the angel and all the various tools in the foreground, the sphere, a corner of the polyhedron, until it finally terminates around the hound's hindquarters. The only niggling problem is that the polyhedron's foremost edge does not line up perfectly with the golden triangle... and it should. One would imagine this misalignment is not the sort of thing Dürer would neglect. So, there's that. Also note the rainbow in the upper left-hand corner. It is a GTS as well, although merely a portion. The comet, however, is not in the correct position (i.e., the termination point). As for the termination of the full GTS (which appears to be goosing the dog)... well, if the spiral was deliberate, then maybe Dürer was either showing some humor... or showing his disdain for the golden ratio!

Disdain?  To understand why the golden ratio might disturb Dürer, one would first have to understand his entire approach to geometry. If, as G.H. Hughes surmises (see quote section), Dürer followed the "Rule of the German architects," then "Divine" would apply to the Vesica Piscis exclusively. To introduce another form of divine perfection would be a kind of blasphemy. That Dürer used the Vesica in many of his geometrical diagrams is a given. Inset right is his diagram of (what appears to be) a Bishop's crook. Note his innovative way of measuring the spiral with a series of intercepting circles in a Vesica array. I've highlighted a portion of this progression for you below in red.

Geometry (black): Dürer. Geometry (red) 2021, DS.

"I found no one who has written about a system of human proportions except Jacopo (de Barbari), a native of Venice and a lovely painter. He showed me the figures of a man and woman, which he had drawn based on a canon of proportions... But Jacopo I noticed did not wish to show me his principles clearly; so I set to work on my own and read Vitruvius, who describes the proportions of the human body to some extent... and thereafter, from day to day, I have followed up my search according to my own design."

On the other hand, Dürer was always hungry for new knowledge, especially in regards to human proportions and perspective. He was especially drawn to the Italian artists, and made two trips to Italy during his lifetime. The Plague was raging in Nuremburg and he had just embarked on an arranged marriage when he took his first journey (alone) to Venice. Few documents remain of this trip, but he was already making plans for a second journey before he returned. As it happened, in 1500, before he set out on his second trip to Italy, Venice more or less came to him in the form of the painter he had met during his first trip, Jacopo de Barbari. Unfortunately, as we find in the quote (above) while Barbari may have fanned the flames of Dürer's technological, theoretical passions, he did not satiate them. And, one gets the impression that, after Dürer's father's death in 1503 - when he saw the comet - his "own design" sent him back to Italy to do a little more digging.

Interestingly, Jacopo de Barbari is the artist who (allegedly) produced the painting inset right. Created between 1495 and 1500, Portrait of Luca Pacioli features Renaissance mathematician Luca Pacioli - author of Divina Proportione (see here) - and a figure which might have possibly been one of his students. It's also been hypothesized that the younger man was actually Dürer... but, according to Jane Campbell Hutchinson's 1990 biography of Dürer (one of my sources for this post), Dürer left Italy (the first time) in 1495... and there is no record of him having met Pacioli.  However, through his friend, the humanist writer and aristocrat Willibald Pirckheimer, he may have met Galleazo de Sanseverino, who was considered "a great friend and protector of Leonardo (da Vinci) and fra' Luca Pacioli." In fact, Luca Pacioli dedicated the manuscript of Divina Proportione to him, and it is he who may be the younger figure in the painting. In any case, Hutchinson hypothesizes that Galeazzo "may have helped to stimulate his (Dürer's) growing interest in mathematics and the theory of perspective - an interest which would grow stronger with the return to Nuremburg of the great mathematician and astronomer, Johannes Werner..." Interestingly, in 1509, Dürer bought the house and observatory of one of Werner's friends, astronomer, Bernhard Walther, which is now a museum (see here).

As for the two remaining Master Prints and the GTS, well, above is Knight, Death and the Devil, executed the previous year before Melencolia. What sold me on this spiral was the presence of the the "landscape" golden triangle, found in the position of the knight's sword and staff. In fact, the triangles formed are not only equal in size and orientation to the central GTS but share a diagonal (side); this second spiral terminates around the knight's elbow. See figure on the right, below.

Note, too, that with this additional triangle, 2 pentagrams (stars) are in the process of forming directly over the knight! The larger star will incorporate the Devil; the smaller will incorporate Death's head.

The sad fact about this etching, is that it seemed to be a favorite of Adolf Hitler, who (possibly recognizing himself) (he's the one with the uni-horn) considered Dürer "the most German of all German artists" and this didn't improve Dürer's reputation in the 20th century.

What Hitler and the Nazis failed to recall was that Dürer, while nationalistic in regards to German art and artists - he wanted to bring the Renaissance to Germany - was foremost a humanist: "Generally, the term refers to a focus on human well-being and advocates for human freedom, autonomy, and progress. It views humanity as responsible for the promotion and development of individuals, espouses the equal and inherent dignity of all human beings, and emphasizes a concern for humans in relation to the world." The reality is, Albrecht would've rolled over in his tomb had he known of his future fan-base.

The problem I have with Saint Jerome in His Study (above) considered to be the third Master Print, is that while the first two images in the triad seem to be of a sort, this one seems like the odd one out. Is it just me?

In any case, I'm not sure about the GTS. I found three possibilities; below are the others. Although the spiral terminates very neatly in the first example, it is most fully expressed in the third example (below, right). But, no matter which spiral you choose, the proportion seems to be present throughout the image.

Also, if it means anything, both Leonardo Da Vinci and Caravaggio had their tributes to Jerome.

In any case, after all is said and done, we are left with a did-he-or-didn't-he? situation. Was Dürer designing with the golden spiral or was the golden spiral designing with him?  As it was, he used many forms of spirals in his work - in fact, he seemed to be a little in love with both triangles and spirals, so the pentagonal golden ratio would've been a perfect fit for him. But, was he bothered by it in some way? (I keep thinking about that rectally-directed spiral in Melancolia...)  If so, perhaps, he was reluctant to see it as a measure of beauty... or, just dismayed that he learned of it too late in his career. Keep in mind, a mere fourteen years after the prints were executed, Dürer died after contracting some strange, unidentified disease during one of his trips to the Netherlands.

One point I would like to make here, however, is that pentagonal Phi, when fully investigated, incorporates the Vesica Piscis in a few of its manifestations. For instance, the Rose Pentagram, introduced here, utilizes 5 perfect Vesicas in its construction. So, pentagonal Phi is not outside the Sacred Geometrical realm, and my feeling is that, at least, some medieval masons and architects were as aware of this as some artists and artisans of that time (and previously).

On a final note, Albrecht himself was to expand his interests into architecture, which is not unusual for an artist for whom proportion, perspective and geometry were the basis of all art. In the last years of his life he wrote one other book: Etliche Unterricht, zur Befestigung der Stadte, Schlosser und Flecken (Instruction on the Fortification of Cities, Castles, and Towns). Whereas he admitted to feeling like a "parasite" in Nuremburg (but a "gentlemen" in Italy), the city he envisioned and drafted plans for in Etliche Unterrichtan was designed primarily to provide the optimum living and working conditions for artists and artisans! (See Tess Morrison's Albrecht Dürer and the Ideal City).

Lastly, but not leastly - in keeping with our theme - we have this decadent little snail by Dürer (inset right). It is one tiny element of possibly one of the largest woodcuts ever created: Triumphal Arch, executed for Holy Roman Emperor Maximilian I between 1512 and 1515. While Albrecht didn't design the actual arch, he and his crew did do the numerous drawings of its heavily illustrated facade. I'm not sure where the snail was tucked in, but I just had to include it here! (Another hat-tip to Hunting For Snails!)

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*   In ways my statement is inaccurate. The angel (while most certainly Dürer) is androgynous. Some see it as female, and some see it as male because it is, in reality, both. As an angel it has transcended gender.

Unfortunately, it has not yet transcended the dead weight of the material world with its antagonistic physical forces and its existential threat of death.

But, then again, Melencolia is primarily a mirror. The angst is our own.

On an optimistic note: Dürer has given himself wings. 

**  Note that the spiral goes (primarily) through numbers 14, 6 and 5... which, as far as I know, is a combination which has nothing to do with Dürer's life nor the mechanics of his magic square... (all rows of numbers add up to the number 34). See here.

Ah, but guess where this magic square first appears? Apparently somewhere between 1501-1503 (predating Melencolia by a decade) Luca Pacioli used it as the example of a magic square of "the order of 4" in his De viribus quantitanis being "a compilation of material gathered" in preparation for De divina proportione! See here.

Now this could be an important link between Albrecht and the divine ratio... but, while I've seen some of  Leonardo's illustrations, I've yet to see an English translation of Pacioli's text. Does he even mention the spiral?

In any case, it appears Dürer was tipping his hat to Pacioli. 

This just in: and there's, yet, one more bit of info tying Dürer and Pacioli together (found in Tess Morrison's article):

"At the end of this stay in Venice in 1506, Durer wrote to his friend Pirckheimer... and stated 'I should like to travel to Bologna to learn the secrets of the art of perspective, which a man there is willing to teach me. I should stay there about eight or ten days and then come back to Venice'...Durer did accomplish his intention to visit Bologna. There remains speculation as to who instructed him in the art of perspective but it is thought to have been the mathematician Luca Pacioli."

But, alas, "thought to have been" is not quite the same as definitely was. In other words, in much the way of all things pentagonal, this rabbit hole is unlikely to ever end!

*** Technically, if the information I've gathered is correct, Albrecht's infamous polyhedron is a truncated triangular trapezohedron, which forms Durer's graph. But, adding to the list of theories that surrounds this figure, we have another understanding from artist Yvo Jacquier:

"The polyhedron of Albrecht Dürer has been a long time a great enigma. Mathematicians have recently solved its proportions (especially the circum and the inscribed spheres), thanks to the discovery of sketches from the hand of  the artist. Two important points emerge: - the form is organized by the Pentagram, that explains the presence of φ in the results.  - This solid is the missing element to the set of Plato to achieve the model of Kepler."

And, as you can see below, Dürer's solid emerges from my pentagonal "fold" (template) quite nicely, although not completely accurately... in that the triangular plane is not in it's correct perspective.

Speaking of my pentagonal template, however - which can also be found utilized here - if one changes it's orientation by 18 degrees so that the horizontal lines become vertical ones, a new lattice is formed. It seems one can create a 3rd dimension out of two, simply by shading selected areas with color.

Welcome to the pentagonal world! :-)

Albrecht Dürer and the Divine Ratio (Part II) - Dürer Reconsidered