Thursday, July 1, 2010

Madonna and the Hyperbolic Manifold

Visual Mathematics in Paintings by Da Vinci

In the Madonna and the Yarnwinder, the Virgin Mary tries to restrain her child from the yarnwinder, which represents the Passion of Christ; or the Cross. You may have noticed that the trancepts of the yarnwinder are not completely parallel, and were they to be extended they would meet at some point to the left hand-side of the image. For two parallel lines to meet in this manner, mathematicians suggests that space would need to have a constant positive curvature.

 The Madonna and the Yarnwinder earned its creator, Leonardo Da Vinci, the title of ‘the slowest painter in Italy.’ Anyone of his compatriots would have been able to complete a painting of similar size, and content, in a matter of months, and yet it took Leonardo four years to complete Madonna and the Yarnwinder. Why? It has been suggested that Leonardo was constantly thinking about something grander than merely the task at hand, and that his commissions suffered from this lack of focus accordingly. Be that as it may, the bizaare fault in the logic of the perspective of the yarnwinder could not have been overlooked by Da Vinci, in the four years he spent working on this project, and therefore must be considered deliberate.
Rhombicuboctahedron by Leonardo Da Vinci, and Waterfall by MC Escher.

Only one thing could account for the disparity in perspective, and the length of time it took Leonardo to finish the relatively small and simple composition; the impossible triangle or tribar. Unfortunately, it was not discovered until some 400 years after Leonardo’s death. It is well known that Leonardo was able to render perfectly symmetrical 3D politropes onto 2D surfaces with apparent ease. For this reason I think it is possible that he may have come up with the tribar, a projection of a 4D object onto a 2D surface. The tribar, was first created by the Swedish artist Oscar Reutersvard in 1934. It was independantly devised and popularised by Roger Penrose in the 1950’s, who studied it from a maths perspective. MC Escher is considered the first artist to render these mathematical concepts in a graphic medium. His visions of strange worlds, where the laws of gravity seem to no longer apply, are the result of this study. Considering that the tribar was initially discovered by a graphic artist, and later expounded upon by one, it is not so difficult to concieve that Leonardo Da Vinci could have come up with his own version of the object, much earlier.

The tribar is known as an impossible object because it cannot be physically realized in 3-d space. However, it is possible to generate a tribar in some types of 3-manifolds (hyperbolic types). Manifolds are essentially surfaces. Topology describes the surface of a sphere locally, as a 2-d or 2-manifold that encloses a 3-dimensional space. Similarly, a 3-manifold is a 3-dimensional space that is locally 3-dimensional but extends to enclose a 4D space. It is possible to create an approximation of an impossible tribar in 3-d space, either by creating a disjointed or curved representation of it that appears to be coherent or connected from one particular angle. In the instant of the curved model, a number of different varieties can be considered. I prefer my own solution which requires one of the arms of the tribar to be negatively curved, one to be positively curved and the conjoining one to be bent in an S-shape; comprising both positive and negative curvations. Of course, from the point of view of the 3-manifold/4-dimensional object no curvature in any of the arms of the tribar would be necessary.

In order to prove that Leonardo incorporated the mathematics of the tribar into Madonna and the Yarnwinder it will be necessary to distort the appropriate areas of the canvas by these degrees of positive and negative curvature – something that would have been almost impossible to do before the advent of image manipulating software like Photoshop. According to my calculations, the optimum amount of tribar curvature should be plus or minus 32%. After spending many hours on the resulting animation, I got the feeling that I was completing one of Leonardo Da Vinci’s unsung, and unfinished works. It is possible that the visionary who foresaw such marvels, as the submarine, helicopter and animatronic android, could have envisioned the Computer Age, when his images could be stored, recreated and manipulated with comparitive ease. In such an eventuality, I felt I had become almost like the distant understudy of Da Vinci, playing my part in the completion of a masterpiece that took, not just four years, but four hundred to be fully realised.

On the face of it, it does appear as if these tribar based distortions reveal Leonardo’s true intentions where this painting is concerned. Watch, as the head of the Madonna begins to shrink into a more proportioned stance with her body. Watch, as vistas are partially occluded by mountainous terrains, and bodies gesture in life like motions. Could this be the first Renaissance painting to include a code that unlocks a 3-dimensional perspective within its coloured forms? If it is, then, it is the only one that I known of. The entire landscape appears to be alive and breathing with an innate lifeforce. The snowcapped mountains of the background behave like teeth being pressed, from behind, by giant emerging molars of wisdom. This is Renaissance painting on LSD.

It is my belief that the dimensional permutations of this animation focus the mind and permit consciousness expansion. For this reason, I urge you to watch its endless looping permutations for 1 and a half hours (if you can!), for two days, at the end of which time, your brain should be sufficiently conditioned to produce the desired effect without the aid of the animation to help you. The effect is similar to that which you can find in some of those maddening optical illusions, but it is far more dynamic and rewarding, I assure you. Far more interesting, is the impact it has upon your dreams when you sleep!

In the process of contructing this animation, I wound up watching it for hours on end, and noticed some very pleasing results. The piles of clothes littering my bedroom floor began to warp and wiff in the same manner as the painting. When I went out onto the landing of my country home, and tried to walk around in the dark, the angles of the walls began to rotate and distort wildely. They appeared to have more freedom of movement, which presented the possibilty of corridors opening up, where previously there were none. This is the effect of inter-dimensional code embedded in Madonna and the Yarnwinder, a dislocation of the mind from the 3-dimensional temporal sphere.

Considering the minute comparisons between the LSD experience and the effects of watching this video, it should not come as too much of a shock that such temporal distortions should occur. After all, the secret governmental agencies of the world have been experimenting with the combined use of LSD and time travel for decades. See here; and stay tuned for more on that subject later.

1 comment:

  1. Wow what and article!

    It seems to me that this painting is an essay on the perspective and our vision of reality. It clearly looks like that Leonardo was questionning himself about it and that he has imbued this picture with his interrogation and/or answers.

    Look that the imaginary triangle you would create with theses side: the river (or valley) at the right of the madonna, the moutains in her back and the diagonal created by the superposition of her baby's body with her's.
    Don't you see a tribar just as I do?

    P-S I'm french, please forgive any misuse of your language. I hope I expressed myself clearly.