The Great Beyond

John Wiley & Sons

Chapter One

March on, symbolic host! with step sublime, Up to the flaming bounds of Space and Time! There pause, until by Dickenson depicted, In two dimensions, we the form may trace Of him whose soul, too large for vulgar space, In n dimensions flourished unrestricted. -James Clerk Maxwell to the Committee of the Cayley Portrait Fund, 1887

Shadow Play

Is the cosmos just a shadow play? Such is its portrayal in the sacred Indonesian tradition of Wayang Kulit. Part religious ritual, part entertainment, Wayang Kulit is a type of puppet theater acted behind a backlit screen. One of the oldest storytelling traditions in the world, its nightlong dramas depict the endless struggles of gods and demons as they set the course of cosmic history.

A typical show begins with the audience seated in front of a stretched white sheet. An oil lamp bathes the screen in an other-worldly glow. The Dalang, or puppeteer, takes his place behind the screen and chooses fromamong two sets of colorful handcrafted leather puppets. One set represents the heroic characters, the other the villains. Behind the Dalang are musicians, whose surreal cadences lend aural texture to the tales. During the course of the performance, the Dalang never addresses the musicians; rather, they shape their sounds around the ever-changing moods of the stories. As they play on, the Dalang conjures up the memories of generations of storytellers and delivers his one-man epic. From nightfall until the first stretches of the Sun's awakening rays, the consummate puppeteer never takes a break. With his well-practiced repertoire of voices and movements, he evokes the bravery of legendary warriors as they grapple with horrific ten-headed monsters, relays the blood feuds of times untold, and sketches the tales of impassioned lovers as they woo and betray each other.

With all eyes gazing intently at the screen, the audience sees only projected images of the backstage drama. Passing through each other, blinking out and then suddenly reappearing, these shadows are able to act in a manner impossible for more solid figures. The specter of a gorgon might easily and instantly devour the projection of a sword-wielding lad, with nary a bulge or burp. Two other creatures might merge their shadows and form a ghastly behemoth. Well aware of the varied laws of the two kingdoms-the colorful one behind the sheet and the murky one on its surface-the Dalang extracts whatever magic he can from the difference.

Strange as it would seem, this exotic fiction could represent the truth-in artifice rather than content. A new movement in physics imagines the universe itself as a shadow theater. The world we see around us, according to this novel vision, is but a mere projection of a more fundamental reality. The true drama takes place beyond the curtain on a higher-dimensional stage. Possessing at least one extra dimension beyond space and and time, this backstage area, called the bulk, can never be seen (with visual means at least) because it admits no light. We can only witness the shadow play on the curtain itself, a three-dimensional volume called the brane, and surmise what lies beyond.

Nevertheless, researchers are trying to test this new model of physics, known as M-theory. Their strategies make use of experiments that rely on gravity rather than light. Like the Dalang and his musicians, gravity is thought to have special access to backstage. According to M-theory, it can penetrate the bulk, and emerge in other places along the brane. If this indeed is the case, then gravity could conceivably jump from one region to another at a rate faster than the speed of light. Physicists are currently using accelerator data and other means in attempts to substantiate such a hypothesis. They are also examining alternative higher-dimensional models of the universe, in various versions of Kaluza-Klein theory.

Plato's Cave

Although M-theory has been around for only about a decade, and more basic Kaluza-Klein theory for less than a century, the notion that the visible world represents mere shadows of the truth is quite ancient. Philosophers have long been intrigued by Plato's idea of "forms." All we see around us, according to the Greek sage, is just an illusion-an incomplete projection of the perfect domain of forms. These forms constitute the ideal versions of everything we know: the perfect Sun, the perfect Moon, the flawless human being, refreshingly pure air and water, and so on.

Plato encapsulated his thoughts on this subject in his famous "Allegory of the Cave." He imagined prisoners constrained to spend their entire lives inside a cave, close to the entrance. Shackles restricted their motions so they could gaze only at a stony wall. From their vantage point, however, they could observe the interplay of shadows from the outside world, cast by a fire blazing outside in the distance. As people carrying all sorts of goods and vessels walked between the fire and the mouth of the cave, the prisoners saw only their silhouettes. Because the captives were unfamiliar with external reality, they presumed that the flat shadows, not the solid bodies, were all that there was.

Viewed in the modern context, Plato's cave allegory seems to imply that our three-dimensional world is but a projection of an even higher-dimensional reality as well. Clearly, though, that wasn't Plato's intention. The ancient Greeks had no known interest in higher dimensions. Plato set his perfect realm in a metaphysical domain, not in a multidimensional extension of our own space. Thus his tale was explicitedly a metaphor for unseen perfection, not for unseen dimensions. On the contrary, the Greeks saw reason to believe that nature was limited to only three dimensions: length, width, and height. Plato's student Aristotle, born in 384 B.C., emphasized this fact in his work "On the Heavens." Recognizing the natural progression from a line to a plane and then to a solid, Aristotle stressed that nothing of higher dimension lies beyond. He considered the solid to be the most complete type of mathematical object, unable to be augmented or improved. Therefore, no other body could surpass it in number of dimensions. To further bolster his case, he pointed out the Pythagorean idea that three was a special number, because everything has a beginning, a middle, and an end. The Pythagoreans were a learned society in ancient Greece that had a great interest in the power of mathematics and the mystical properties of various numbers. Hundreds of years later, a treatise by Ptolemy entitled On Dimensionality amplified Aristotle's thesis. In it Ptolemy showed that one couldn't construct a set of more than three mutually perpendicular lines passing through a single point.

Due perhaps to the hallowed Pythagorean tradition, as further developed by Plato, Aristotle, and others, Greek society maintained a keen fascination with three-dimensional geometry. This extended to its art and architecture, from precisely proportioned sculpture to the grand symmetrical structures of the Parthenon. In mathematics, the Greeks were the first to discover that there are only five regular (equal-sided) three-dimensional polyhedra, known as the Platonic solids. These are the tetrahedron (four-sided pyramid), the cube, the octahedron (eight sides), the dodecahedron (twelve sides), and the icosahedron (twenty sides). This limitation seemed quite mysterious, given that there are an infinite number of regular two-dimensional polygons (triangles, squares, and so forth). Such striking differences between two and three dimensions made the latter seem even holier.

From such rudimentary seeds, Euclid made beautiful structures bloom. He used his postulates to prove virtually all of the basic geometric properties known at the time. Many of those results are familiar to every high school student. For example, if two triangles have exactly the same shape, the sides of one must be proportional to the sides of the other. With an argument based on angles, Euclid also explained why there are only five Platonic solids.

The postulates upon which Euclid based his proofs were so compelling they were considered sacrosanct until the nineteenth century. With the first four, it's clear why. Every set of two points defines a straight line, he noted. A line segment can be extended forever. A circle can be drawn with any given center and radius. All right angles are equal to each other. Who, familiar with simple plane geometry, could argue with these?

Euclid's fifth postulate is markedly more complex than the other four. Consider two straight lines, and a third line crossing them. This creates two intersections. Suppose the angles on one side of both intersections are each less than right angles. Then that is the side where the first two lines eventually meet.

Because of its bearing on the subject of parallel lines, the fifth postulate has come to be known as the parallel postulate. Mathematically it is equivalent to the following alternative statement, called Playfair's axiom: given a line and a single point not on it, there is precisely one line parallel to the first through that point. When expressed in this manner, one can readily see how the parallel postulate serves as a "duplicating machine" for producing parallel lines throughout all of space. If one wants to construct a set of parallel lines, just take one line and choose a point that happens to be somewhere else. The point automatically acts as the basis for a parallel line.

Given the relative elaborateness of the parallel postulate, for generations mathematicians wondered if it could be derived from the other four postulates. In that case it would be a secondary proposition instead of a basic assumption. Euclid himself considered it inferior to his other postulates and, in his proofs, avoided using it as much as he could. Various mathematicians' attempts to dethrone the fifth postulate all failed, however. It wasn't until the early nineteenth century, and the discovery of non-Euclidean geometries, that Gauss, Bolyai, and Lobachevsky demonstrated that the parallel postulate was wholly independent of the others, and could in fact be replaced with other assumptions.

Until then, Euclid's Elements reigned supreme in the field of geometry. It is a record thus far unsurpassed by any other scientific work, and a tribute to the magnificence of Greek thinking on the subject.

Leonardo's Perspective

When Rome conquered Greece, it acquired a cargo of natural and philosophical knowledge, which it unpacked and wore with great enthusiasm. Though the Romans had many erudite thinkers, much of their scholarship came secondhand. They had little interest in developing their own theories. Still the older ideas were none the worse for wear, and helped them construct magnificent temples, statuary, and other public works, with designs derived from Greek mathematical principles.

The rise of Christianity and the fall of Rome led to a radical change of attitude in Europe toward science and culture. The extravagance of Greco-Roman art and architecture became replaced by austerity and uniformity. Thoughts turned to preparations for the world to come, rather than ways to understand the world that is. Throughout the Middle Ages, a period dating roughly from the fifth until the fourteenth centuries, an emphasis on unadorned design resulted in a two-dimensional approach to painting. Portraits from that era appear flat and unrealistic, like paper dolls. The notion of depth was almost forgotten, as painters reproduced staid likenesses of Jesus, Mary, the Apostles, and other New Testament figures.

Then in the Renaissance era, the sleeping giant of creative art arose from its slumber. Rubbing its eyes, it gazed at the world anew. It began to scrutinize the precise details of the way nature appears, capturing those impressions in increasingly realistic depictions.

One of the harbingers of the new movement was the early fourteenth-century Florentine artist Giotto di Bondone. When Giotto painted scenes, he imagined them from the point of view of someone standing a certain distance away. Then he sketched the images with those lines of sight in mind. The result was sharply different from the flat paintings of his predecessors, far more vivid and true to form.

With his discovery of perspective, Giotto brought the third dimension back into art. Onlookers stood entranced when looking at Giotto's paintings, like children watching television for the first time. They marveled at his ability to make them feel as if they were actually at the scenes he rendered. Soon other artists began to imitate his style, hoping to create some of their own striking images.

To improve upon the illusion of three-dimensionality, artists began to study the long-ignored field of Euclidean geometry. They also began to realize that the proper placement of light and shadows would enhance the realism of their portrayals. Thus, the best artists also became naturalists and mathematicians, calculating the best color and placement for every aspect of their works.

Perhaps the quintessential Renaissance artist was Leonardo da Vinci. Leonardo, who worked in the late fifteenth and early sixteenth centuries, was determined to render his portraits as lifelike as possible. To turn his canvas into a mirror of nature, he studied mathematics, mechanics, optics, anatomy, and other scientific subjects, exploring them in groundbreaking ways. His notebooks contain some of the most detailed studies of human and animal forms ever rendered, indicating the precise exertions of various muscles in a variety of movements. These sketches helped him create realistic portrayals of his subjects that almost seem to be gazing, or even smiling (in his best-known masterpiece), back at the beholder.

Leonardo was very interested in the subtle dance of light and shadow that the Sun's rays produce on a subject. He noticed that lighter and darker areas can be used to convey a sense of either proximity or remoteness. By mixing his colors with appropriately sunny or dusky shades, he found he could enhance the three-dimensionality of his works.

Corresponding to the development of depth in painting came a revived interest in sculpture. Even more so than its Greek and Roman antecedents, Renaissance sculpture captured the flesh and blood humanity of its models.

Continues...

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Excerpted from The Great Beyond by Paul Halpern Excerpted by permission.
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