Monday, December 30, 2013

Dürer's Melencolia I: The Math

We keep seeing the motifs of Dürer as craftsman return again and again in our examination of Melencolia I. Indeed, the image itself is replete with these symbols: the compass in the celestial figure's hand, the woodworking tools at its feet, and the scales, hourglass, and bell above. But look more closely at the image and you see another symbolic thread, parallel and complementary to craftsmanship, and that is the presence of mathematics.

Of course, without math one cannot correctly construct. But the math in Melencolia I is obscure, occult, surreal, magical. It defies the earthy concerns of a saw, nails and a plane. It begs us to leave the real world behind and enter, yes, an imaginary, higher realm.

We've spoken of the magic square in detail before; all that remains to be said is that Dürer's magic square in Melencolia I has many eyebrow-raising references and properties. Everyone knows the inclusion of "15/14" in the bottom row of the square to reflect the year it was created. But there is also another monogram of Albrecht Dürer here, hidden in the corners of the square: A and D are the 1st and 4th letters of the alphabet. In addition, it is a gnomon magic square, where the four quadrants of the square add to the same sum, as well as the center four numbers. Lots of satisfying mathematical symmetry here (not the least of which that opposite values on the edge of the square total to 17). But what of the mystical significance of this particular type of square?

The 4x4 square was dedicated, in Agrippa, to Jupiter. Now, one would naturally wonder or expect a magic square in a piece titled after melancholy to include the Saturnine, 3x3 square. But instead of the leaden, ponderous square of Saturn, we are giving the shining, beneficent, literally-jovial square of Jupiter. Is the magic square in Melencolia I meant to something to aspire to? It hangs above the angelic figure's head, conspicuous in its placement among the muddle and mess of the rest of the scene, the bell waiting to be rung above it. Perhaps it is even a square of warding, a way of keeping the confusion and overwhelming stimuli of melancholia imaginativa at bay? An exercise to settle the mind, a meditation on order? Think of the time Dürer likely spent in lining up all these numbers to work just as he wished. Perhaps he saw the square as a way to quiet and focus his mind.

In Agrippa, the square of Saturn can be used as both "white" or "black" magic, to either confer stability or discord upon the subject of its spell, but Jupiter's square in unequivocally beneficent magic:
They say that if [a four-by-four magic square] be impressed upon a Silver plate with Jupiter being powerfull, and ruling, it conduceth to gain, and riches, favor, and love, peace, and concord, and to appease enemies, to confirm honors, dignities, and counsels, and dissolve enchantments if it be engraven on a corall. (French, 1651)
Moving on to the other mathematical enigma in the image, we come to Dürer's solid, on the left hand side of Melencolia I. And it's quite an uncanny piece of business. Let's start with the fact that it contains the primary optical illusions in Melencolia I. First, there is the face, or according to Finkelstein, faces plural. These "ghosts" that appear in the solid are put there via the method of foreshortening; an optical illusion to us today, but in the early decades of the birth of perspective in Western art, a new and bizarre way of communicating graphical information.

The solid itself is a rhombohedron, with truncated shavings taken off two of the vertices, turning the 6-sided figure into an 8-sided one. We, of course, cannot see the back four surfaces of the solid, which means we are making assumptions of symmetry. It's often said that this solid has "fascinated scholars for centuries," but what really is going on here I think is the inherent uncanniness of taking of a regular, nay Platonic, solid – the cube – turning its sides into rhombuses and then chopping off two of the vertices. It feels wrong. Especially when you compare its "irregular" shape to the conspicuous sphere on the ground. Why a sphere, right there of all places? Again, I believe the placement is intentional and meant to unnerve us. The sphere is the most perfect shape; the octahedron, odd and seemingly out-of-place.

But what if the solid is exactly what it should be? A man like Dürer would not arbitrarily consign an irregular solid to his image for no reason. What is it? Perhaps it is the very liminality, mutability, and process of transition of the solid from cube to rhombohedron to octahedron that makes it meaningful? The cube in Platonic metaphysics represents the element of earth; the octahedron, air. Perhaps Dürer's unsubtle shaving of the already-warped rhombohedron was a way of describing the feeling in Agrippa of being gripped as an earthly being by a spiritual, mystical melancholia imaginativa? The earthbound transforms into a realm of pure idea? There is something to be said about the fact that would have necessitated a transformation of cube into quintessential dodecahedron and not airy octahedron, but who's to say what exists on the other side of the solid, facing the distant landscape and eerie rainbow arcing its way across the sky?

1 comment:

  1. was wondering if this was of any interest to square centered individuals?
    Gnome magic square in the trillions where the numbers 1 thru 9 have triple digits and the 6's, 7's and 8's have repeating triple digits and one cell has quadruple 5's. each cell contains never more than on triple set The grouping of like numbers happens in the same position in the cell, always followed by two final numbers.

    magic constant 83562883710972

    37383395344383 8796093022207 6597069766655 30786325577727
    13194139533311 24189255811071 26388279066623 19791209299967
    21990232555519 15393162788863 17592186044415 28587302322175
    10995116277759 35184372088831 32985348833279 4398046511103

    Just playing around with squares.