Wednesday, July 7, 2010

Madonna and the Hyperbolic Manifold 2

Seeing life in 4-dimensions

If the landscape depicted in Madonna and the Yarnwinder describes a 3-manifold ( a 3-dimensional object which encloses a 4-dimensional space) then it also describes a method of space-time travel. If you were to travel along the circuitous routes that lead around the edge of the painting, you would arrive back in time to meet yourself before you left. Here walking in space is equivalent to walking through time.

In his seminal work Valis (1980) Phillip K.Dick discusses the idea of space-time reversal from the perspective of Wagner's opera Parsifal

PARSIFAL: I move only a little and already I seem to have gone far.
GURNEMANZ: You see, my son, here space turns into time.

If space can turn into time, PKD speculates, can time turn into space?

According to Einstein's Theory of Relativity, space and time are one thing. So it should be possible to reverse their symmetries without breaking any of the laws of physics. 

A) Consequence of time becoming space; choosing a particular direction and walking in it means that we would either be walking into the past or the future. 
B) Consequence of space becoming time, our surroundings would noticeably move in a four-dimensional direction.

If space is time, and time space, then all the matter that exists within the universe is a type of time condensate. The mountains and valleys could represent the rise and fall of civilizations, and the tufts of grass and clumps of earth, the trials and tribulations of the common man. This phenomenon is not limited to landscape and trees, but would extend to all life forms; including our own. Thus, using a piezoelectric needle, it could be possible decode the voice of King Henry VIII from the grooves in people's fingerprints and skin.

Our minds have the ability to perceive space (the objective world outside) and time (the subjective world of our internal memories) with seamless continuity; indicating that space and time are not two sides of the same coin, but a mobius loop with only one side and one edge (and a half twist). This half twist is the consciousness of man. Since space is beheld outside the body and time beheld inside the mind, all that is needed for time travel is to look within and explore the past and future already contained there.

In order to see the depth of the third dimensions, we rely on the tiny differences in position between our right and left eyes. This difference is known as the paralax shift. To see into the fourth dimensional plane our bodies would need the addition of a third eye. Conincidently, in PKD's book, the time traveling aliens, which might well be us from the future, have 3 eyes. What would the world look like if you viewed it through three eyes?

To answer this question I would like to recourse to a short anecdote;

One morning, while walking along a beach, I came across the plastic cap of a motorcycle's filter light that had washed up on the shore. The filter cap had four holes or apertures in it. I placed the cap over the lens of my Nikkon camera I had with me at the time; it was a snug fit. Then I aimed the central aperture of the cap at the horizon, where the ocean meets the sky, and took a picture. 

When I got the roll back from the developers, I was shocked to see, not one, but two horizons captured at different levels and in different apertures. I could tell that what I had captured in the top-most aperture was the sea, and not the sky, because of the ripples/waves clearly visible there. Also the hues and ratio of both spy-holes are remarkably similar.

I deduced that light from the horizon had entered through both apertures; by dint of their different angular alignments to the horizon, and created two slightly different perspectives of the same thing. To validate my theory I constructed a pinhole camera with multiple apertures and began taking and developing my own photographs. Here are a few examples;

This image of a wash-hand basin in a college lavatory,
 was taken using an 11-aperture pinhole camera.

This is a partial image of a widow with railings and stacks of chairs in front of it - taken by the same camera. It reminds me of a theory I heard once about the action of light on a mirror. The theory goes that light bounces off of mirrors in all directions, so if you were to view a mirror 'properly' (i.e. from all angles), it would just look like a bright shining light. As we know––and as these pictures demonstrate–– light is bouncing off everything at every angle all the time. So,  if we were able to view the world 'properly,' or from every conceivable angle possible, the whole world would look aglow. It is the aim of magi and holy men to transform the world into a world of light. In order to do this, I now believe, we must look at the world from every aspect and from every  angle possible; only then will we be able to see our true worth, and the worth of others, and realise that, in reality, we are all shining beings of light.

These images show what it would be like to have multiple eyes. But our question was; what would the world look like viewed through just 3 eyes? Well, according to mystical interpretation, the human body does possess a third eye; the spiritual eye of the pineal gland. This eye would mostly likely not give you the different spatial aspects, as seen in these images, but rather add a different temporal aspect to your reality.

If you have ever had the good fortune to have your third eye activated by a gnat while meditating by a stream when a train went by, you will know something of what it is like to see into 4-dimensional space. As I concentrated on the gnats convolutions, my slightly crossed eyes observed the different sections of the train collide harmoniously with each other. The back of the train appeared to arrive into the station after the train had left.

The Crab Story

Coincidentally, the aliens in Valis, who might also be evolved humans from the future, have three eyes and clawed hands; like a crab. This reminds me of the three-eyed aliens in Toy Story, who worship "The Claw"––their God. Don't forget the U2 stage of the same name, which Bono compared to a House of God. The implication here being that an celestial being, possibly extra-terrestrial in origin, will reach down from the sky and pluck the U2 boys to safety (or perhaps, so they think)...

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.