## Riemann's Revolution

Georg Bernhard Riemann, a shy and timid German mathematician, toppled the world of Euclidean geometry in a single famous lecture to the faculty at Germany's University of Göttingen in 1854. Asked by his mentor, the famous mathematician Carl Gauss, to devise an alternative to the two-thousand-year-old geometry of Euclid, Riemann created the new geometrical framework that was to replace the old Euclidean world forever.

Euclid, in creating his geometrical system, made two fundamental assumptions about space: first, that it is flat, and second, that it has only three dimensions. Ptolemy would later offer the first "proof" that there was no fourth dimension by showing that four mutually perpendicular lines could not be drawn. However, what Ptolemy really proved was that our three-dimensional brains cannot visualize four spatial dimensions.

Riemann's decisive break with Euclid came in two realizations concerning Euclid's assumptions: first, that space was not necessarily flat, and second, that more than three dimensions could exist. In fact, the presence of curved space or higher dimensions renders Euclidean geometry incorrect. This topic will be further discussed later.

## New Concepts of Force

Since Newton's famous laws were published, scientists had believed a force to be some sort of instantaneous interaction between two bodies. This was called action-at-a-distance and postulated that two bodies could affect one another without even touching. Although this principle was criticized, no one before Riemann had ever devised an alternate explanation. Riemann, however, had the insight that perhaps force was not action-at-a-distance but instead a simple consequence of geometry. He conducted a thought experiment by imagining "bookworms" living on a sheet of paper that had been crumpled. He realized that these hypothetical creatures would still believe their world to be flat, but would feel strange forces pushing them to and fro as they moved along the crumpled sheet. Riemann generalized this to our world, imagining that it was crumpled in the fourth dimension. We would not readily notice our misshapen world, but we would feel its effects in the form of forces. He postulated that the natural forces known to him - gravity, electricity, and magnetism - were the effects of higher-dimensional crumpling.

## The Metric Tensor

Riemann's metric tensor consists of ten unique numbers (actually sixteen, but six of these are redundant) that can without fail describe a curved 4-dimensional space. He started with the well-known Pythagorean Theorem, illustrated in the image at right. He then easily extended it to three dimensions, illustrated at left. This can be extended still further into N-dimensional space - if a, b, c . . . are the lengths of the sides of a higher-dimensional cube analog, and z is the diagonal of this analog, the formula for finding z is simply a2 + b2 + c2 + . . . = z2.

Riemann's next step was to think about the curvature of space. If a surface is taken to be flat, Euclid's well-known concepts of geometry apply: the angles of a triangle always add up to 180°, parallel lines never meet, and the shortest distance between two points is a straight line along the surface. Riemann examined surfaces of "positive curvature," like the surface of a sphere, where the angles of a triangle exceed 180°, parallel lines (defined as being great circles of the sphere, not latitudinal lines) always meet, and the shortest distance between two points goes through the surface. He also considered surfaces of "negative curvature," like the surface of a saddle, where the angles of a triangle are less than 180°, an infinite number of lines can be drawn parallel to a given line through a given point, and the shortest distance between two points goes above the surface. These facts are illustrated by the image to the left.

Riemann's main purpose was to create a mathematical way to describe the bends and contours of all surfaces, simple or complex. This logically led him to Faraday fields, or force fields occupying three-dimensional space for which, at any given point, certain numbers can be assigned to describe the force properties of that point. Riemann wanted to create a similar series of numbers that would describe the curvature of any point in space. He found that, for the usual two-dimensional surfaces, three numbers describe its contour. In four-dimensional space, a total of sixteen numbers are needed, as mentioned above, but six of these repeat, so ten unique numbers describe the curvature of any given point in four dimensions. This collection of numbers is the metric tensor mentioned above; the greater its value, the greater the curvature of the surface it describes. The metric tensor can be extended to describe a curved space of N dimensions.

## Multiply Connected Spaces: Wormholes

Riemann was among the first to discuss what is today one of the most debated ideas in physics: wormholes. Riemann's version of wormholes consisted of two flat surfaces connected by a cut (now called a Riemann cut) that is topologically equivalent to the ordinary picture of a wormhole, but lacks a neck. In the image, the gray surface is the ordinary surface on which the bug lives; the white surface is the space to which the cut provides a link. If the bug walks through the cut, he discovers a new world on the white surface; he can return to his original home by reentering the cut. Although Riemann himself saw these cuts as objects of geometrical interest and not as methods of traveling between universes or areas, the concept of wormholes as shortcuts through time and space is a popular one within the modern physics community and even today's popular culture.

## Higher Dimensions after Riemann

After Riemann's work, research into higher dimensions blossomed, but theorists of the time lacked essential equations necessary for the completion of a geometric theory of gravity, electricity, and magnetism. Despite this, study and especially popularization of Riemann's idea continued even after his death. William Clifford, a British mathematician translating Riemann's work in 1873, furthered Riemann's insight by theorizing that electricity and magnetism as well as gravity are also the result of bending of higher dimensions. Thus, his rudimentary speculation preceded Einstein and Kaluza by 50 years. Hermann von Helmholtz, a renowned German physicist, spent a great deal of time speaking to the public about the import of Riemann's work.

## Higher Dimensions in Popular Culture

As a result of the efforts of Helmholtz and others like him, the public began speculating wildly about beings from the fourth dimension, who would be able to walk through walls, reach through solid barriers, appear or disappear on a whim, and materialize in whatever location they pleased. To understand this idea, imagine a flat sheet of paper with some two-dimensional beings living on it. A three-dimensional human's powers to change shape (his cross-section in two dimensions), see through "impenetrable" walls (lines or circles), and lift the beings off their paper would seem magical to the two-dimensional beings, who cannot conceptualize the idea of "up." The same applies to humans: we cannot conceptualize the fourth spatial dimension, so its inhabitants would seem omnipotent to us. The public did not readily understand the complex physics questions raised by the fourth dimension, but they did recognize the powers of a four-dimensional being. They immediately correlated such beings with the only familiar entities reputed to walk through walls, appear and disappear at will, etc. - ghosts. Similarly, mystics and magicians long claiming supernatural powers latched on to this new, seemingly reputable claim: to be able to access the fourth dimension!

The 1877 London trial of Henry Slade provides an excellent example of this phenomenon. Slade, an American psychic, was visiting London and holding seances when he was arrested for fraud. However, many prominent London physicists suddenly came to his defense, maintaining that his supposed psychic powers derived from ability to access the fourth dimension. Johann Zollner, in particular, defended Slade's abilities and suggested several experiments impossible to complete successfully in three dimensions, but elementary in four: intertwine two separate, unbroken rings; reverse the twisting pattern of a sea shell; tie a knot in a circle of rope without cutting it; tie a right-handed knot in a left-handed position without breaking a wax seal atop it; and remove something from a sealed bottle without unsealing it. Though Slade was eventually convicted of fraud, Zollner and his compatriots did demonstrate that four-dimensional creatures can execute feats obviously impossible in three dimensions.

Religious groups also wholeheartedly embraced the idea of higher dimensions after they saw the powers held by a hypothetical four-dimensional being. They had long been at a loss to answer such seemingly logical questions as, where do angels live? where are heaven and hell? if atmosphere and finally space are above us, where does God reside? With the advent of the fourth dimension, they had an out. A. T. Schofield, a Christian spiritualist, stated that God, heaven, and other religious locales resided in the fourth dimension. Theologian Arthur Willink took the idea one step further, claiming that only an infinity of dimensions was glorious enough for God.

Literature was also infused with the idea of the fourth dimension. Oscar Wilde contributed a clever spoof on the gullible Society for Psychical Research, which had believed Slade, in his 1891 play The Canterville Ghost. H.G. Wells, renowned for his science-fiction stories, added another literary contribution - his 1894 novel The Time Machine, which disseminated the idea that the fourth dimension could be viewed as time; it was not necessarily space. He also published a number of short stories adding other ideas and speculations on the fourth dimension. Perhaps the most lasting of all the contributions of the fourth dimension to popular culture came in the form of a bestselling novel. Written by clergyman Edwin Abbott, the book, called Flatland: A Romance of Many Dimensions by a Square, combined the popular interest in dimensions with a biting social commentary. It was the first mathematically correct description of such matters to reach a wide audience.

## The Fourth Dimension and Politics

Interestingly, the fourth dimension also has a unique history in the Bolshevik party of Russia. After the failed 1905 revolution, a sect within the Bolsheviks called the Otzovists (God-builders) suggested that before the peasants could be introduced to socialism, they had to be won over through religion and spirituality. Drawing on the musings of German physicist Ernst Mach and the new discoveries of radioactivity by Henri Becquerel and Marie Curie, they suggested that religion, especially the fourth dimension, be integrated with socialism. Vladimir Lenin, the leader of the Bolshevik party, was appalled at the idea and wrote Materialism and Empirio-Criticism, an enormous philosophical work defending materialism, while in exile in Geneva. He wrote that a new dialectic was emerging in which matter and energy are one and the same (he did not know of Einstein's discovery only 3 years before). Lenin's interest and insight into the fourth dimension would shape Russian physical inquiry for the next 70 years, and indeed, Russian physicists have played a major part in developing higher-dimensional theories in modern times.

## Seeing The Fourth Dimension

Charles Howard Hinton, an English mathematician, became the man who "saw" the fourth dimension, as well as the messenger of the fourth dimension to America. An Oxford graduate working for the Uppingham School in England, he was arrested on charges of bigamy in 1885. His first wife, who apparently knew about her husband's other interest, decided not to press charges and the couple (without the second wife) departed for America. Hinton then took a job at Princeton University, where he was sidetracked from physics by baseball. Curiously enough, he invented the baseball machine, now considered a staple of the sport. After being fired from Princeton, he took a job with the United States Naval Observatory, then switched to the Washington Patent Office in 1902. (Apparently patent offices provide plenty of opportunity to think; Einstein worked at a Swiss patent office when devising his famous relativity!) While there, he worked on developing methods of visualizing the fourth dimension, eventually devising a type of cube that allowed people with enough concentration to visualize a cube in four dimensions, or a hypercube. These cubes were advertised as allowing their users to view the world of the dead, as well as giving admittance to heaven to those capable of visualizing hypercubes in their true form. He also coined the term tesseract, the name for an unraveled hypercube represented in three dimensions and possibly familiar to modern readers from Madeleine L'Engle's famous Wrinkle in Time series.

## Four-Dimensional Shapes

Without intense thought and concentration, three-dimensional beings cannot visualize a four-dimensional hypercube. Similarly, two-dimensional beings cannot visualize a three-dimensional cube; however, they can understand the idea by visualizing the two-dimensional analog of a cube, or an "unraveled" cube, as shown in the first step of the image at right. This diagram shows the steps involved in folding a cube from its two-dimensional representation into its true three-dimensional form. As this folding is performed, a two-dimensional being would see the squares disappear one by one, eventually leaving only one in the original plane. Similar steps would have to be performed on the tesseract shown at left in order to transform it into a true hypercube; three-dimensional beings would see the cubes disappear, finally leaving only one.

Another method of "seeing" the fourth dimension was also devised by Hinton. He theorized that, although a two-dimensional being cannot see a cube, he can see the shadow of a cube as a square within a square. Imagine holding a stick-figure cube directly below a lamp. The resulting shadow, similar to the figure at right, is what would appear in two dimensions. If the cube was rotated, the squares' motions would seem impossible to a strictly two-dimensional being. Similarly, the shadow of a hypercube in three dimensions is a cube within a cube. If the original hypercube is rotated, the cubes would execute seemingly impossible motions.

In addition to developing ways of visualizing higher-dimensional objects still used today, Hinton also made other contributions to the understanding of the fourth dimension. Just as our familiar three dimensions can be described as the "back-forth," "up-down," and "left-right" dimensions, he gave the terms ana and kata to describe motion in the fourth dimension. (It should be noted that these terms do not refer to time, the "back-future" dimension, but instead to the fourth spatial dimension. Throughout this page, the convention of numbering spatial dimensions consecutively and time separately has been observed.) Hinton also pioneered the idea that dimensions beyond the familiar extended ones must be curled up in very tiny form in order to escape detection. Additionally, Hinton followed Riemann's famous idea that light is the vibration of the fourth dimension - an idea that would later contribute to Theodor Kaluza's brilliant insight.

## The Fourth Dimension and Relativity

Albert Einstein, arguably the greatest physicist of all time, created two of the most famous theories of physics: special and general relativity. They showed that time can be viewed as a dimension and combined our three spatial dimensions with one temporal dimension to form the idea of spacetime, the fundamental fabric of our universe. Using time as the fourth dimension, four-dimensional beings would see humans as an infinite series of static forms that represent all motions of life moving through time as seen all at once; a section of such a worldsheet is seen in the image at right. Relativity, a description of gravity, was one of the first theories to simplify the laws of nature in higher dimensions and was based on Riemannian geometry. Maxwell's famous theory of electromagnetism consists of a total of eight equations when space and time are treated separately; these simplify to one equation when written relativistically. Relativity has one consequence important to the current topic: it implies that space itself is curved.

## Life in Curved Space

If a man lived on a hypersphere (four-dimensional sphere) of small radius, light would easily circle the hypersphere and return to his eyes, giving him a vision of a person standing facing away from him. Touching this person, he would discover that the person is real and solid - not just a fake created by a trick of the light. But looking far off in front of him or behind him, the man would discover an infinite series of people, each touching the next person in precisely the same manner that he was touching the man in front of him. In reality, he is the only person on this hypersphere, and he is not touching another person, but reaching around the hypersphere and touching himself. The series of people is a consequence of the topology of the hypersphere. This example is especially relevant since it is not widely believed by physicists that our universe takes the form of a giant hypersphere so large that light cannot possibly traverse it in any observable amount of time.

This example and others like it show that, contrary to the pre-Einsteinian view of space as a passive arena in which the game of life is played, space and geometry are active participants in our universe. Moreover, space's curvature in higher dimensions is describable with Riemannian geometry and is responsible for the forces felt by three-dimensional beings. However, problems remained; light and gravity, like oil and water, could not be mixed into a unified theory.

## Kaluza-Klein Theory

Theodor Kaluza's brilliant idea, possibly spawned by Hinton and other popularizers of spatial dimensions, was to add another dimension - this time a spatial one. He used the Riemann metric tensor, expanded to fit five dimensions in a 5 × 5 array, to unite Einstein's and Maxwell's theories - that is, the theories of gravity and electromagnetism. Kaluza's theories were later refined by Oskar Klein. This was a great step forward in the search for a geometrical description of the universe, but fundamental problems in the theory arose, and it was abandoned in favor of point-particle quantum mechanics.

## Unification

Physicists spent long decades researching the remaining two fundamental forces - the strong and weak nuclear forces. In the 1970s, C.N. Yang and his student R.L. Mills successfully wrote down the equations that govern these forces, collectively called Yang-Mills fields. However, the success ended there: these equations could not be combined with those of gravity, although unification with the electromagnetic force was eventually achieved. It was later discovered that adding the Yang-Mills fields to the metric tensor gives an expanded version of kaluza-Klein theory that successfully unites the four forces.

Next remaining was the challenge of unifying matter into this framework. This could be done with supergravity by using the super Riemann tensor, a vastly expanded version of the original tensor including 11 dimensions. Supergravity was the closest anyone ever came to uniting the forces and matter in one framework, but it had certain deficiencies. It has recently been shown that it is the low-energy approximation of a new and mysterious form of string theory, called M-theory (see the String Theory series).

## An Interesting Consequence

Although supergravity eventually succumbed to the mainstream introduction of string theory, it did spawn an unusual reaction in the public. The idea of unification, so important in both supergravity and its successor, string theory, is vital to new-age meditation. In fact, the followers of one group published a poster containing the complete 11-dimensional metric tensor and stated that each of its terms represented a different favorable attribute, such as harmony, love, and brotherhood. This is one of the few, if not the only, times an equation has sparked a religious movement.

Created by Dan Corbett, Kate Stafford, and Patrick Wright for ThinkQuest.