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| Melencolia I - copper engraving - Albrecht Dürer. Geometry: 2022, DS. |
"Since this knowledge is very useful to workers, and since (it) has been kept hidden and in great secret by the erudite, I propose to bring it in the open and teach it."
- A quote from Albrecht Dürer found in this interesting article by Karel Vereycken: Albrecht Dürer’s fight against “neoplatonic” melancholy.
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"...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."
The quote above is my own - via my first Dürer post. As it happened, the "jury" eventually just walked through the door, and informed me I was wrong on two counts: Dürer's spiral is unlikely passive... and passive spirals are never occulted (deliberately hidden)... they merely occur without conscious intention.
And, what brought me to these conclusions? Well, the story went something like this:
And, so it was. But, did I ever actually communicate with the incorporeal artist? Not that I remember. However, when I awoke, this much was apparent to me: I had missed something in the Dürer "dig" and it was something important; my Dürer work was not yet done.
Now, this could've proved to be a research mine-field but, as it turned out, later that day, I had an epiphany: yes, that ladder in Melencolia is golden, but, it isn't a one-trick-pony; there might be (at least) 2 more triangles tucked inside of it. And, of course, there were. The drawing of the ladder is actually composed of 4 main diagonals (6 if you're a stickler). And, coiling around the diagonal on the inside of the ladder (see above, and inset right) were spirals orientated in the opposite direction of my first (inside left below). Each spiral initially intercepts the comet, then, coils around the polyhedron before passing under the dog's snout, around the edge of the robe, and, finally, over the angel's knee and elbow before terminating around the end of one of the large compasses. (Yes, technically a "compass" is actually composed of a pair of compasses.)
Which is, certainly, quite a different story from the first spiral... (again, inset left) which (ingloriously) terminated somewhere up the dog's... posterior.(!)
Or is it a different story? Maybe its the same one... which begins from the creative point of a compass... but ends up... well, we know where it eventually ends up; hence, melancholy.
Then again, maybe Albrecht was just having his little joke...or, maybe just having a bad day. But, whatever his motives were, I think we can conclude that Dürer consciously knew what he was doing... and that all three spirals are (very likely) active ones... i.e., deliberate.
So, regarding the "Divine" ratio and its GTS, does this mean Dürer was finally informed by the Italian painters and/or Pacioli ? No, actually, I don't think that was the case. The Italian painters and mathematicians were notoriously secretive and unlikely to hand over their knowledge to a foreign rival. Ultimately, he figured it out all by himself...
...because that's how great a geometer he was.
Note the drawing directly above. It is a detail of one of the diagrams from his book on human proportions, although I'm at a loss to explain exactly how and what he was measuring. But, here's a description from the British Museum page (on which it was found):
"There is also a foreshortened section of a circle on the recto, with Dürer's notes on its construction written alongside. The drawing on the verso is accompanied by manuscript notes, in which Dürer gives extensive details of his method of construction with the use of compasses. This is based on the system of human proportion, with the body made up of eight head lengths, outlined by Vitruvius in his treatise 'On Architecture', whereby architects of antiquity were encouraged to follow an ideal human proportion in their planning of temples."
The numerically-sectioned ellipse around the figure's uplifted arm is interesting... an example of Dürer's unusual and innovative techniques. (Below the jump) is the full figure upon which I've positioned a GTS...
