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Michio Kaku is a professor of physics at the City University of New York, cofounder of string field theory, and the author of several widely acclaimed science. Travel, Immortality, and Our Destiny Beyond Earth [ebook free] by Michio Kaku (epub/mobi) World-renowned physicist and futurist Michio Kaku explores in rich, intimate detail the process by CLICK TO DOWNLOAD. Read {PDF Epub} Download Un día cualquera en by Michio Kaku from the story Contain by noahshier15 with 21 reads. system, range, thousand. Simple.

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Kaku, Michio. Parallel worlds: a journey through creation, higher dimensions, and the future of the cosmos/Michio Kaku.—1st ed. p. cm. Includes bibliographical. Epub Download Physics of the Impossible: A Scientific Exploration into Book Details Author: Michio Kaku Pages: Binding: Paperback. By Michio Kaku. Go to the editions section Cover of: Hyperspace | Michio Kaku Borrow · DAISY for print-disabled Download ebook for print-disabled (DAISY).

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Are you sure you want to Yes No. Be the first to like this. No Downloads. Views Total views. Actions Shares. Embeds 0 No embeds. No notes for slide. Book details Author: Michio Kaku Pages: These experiments, performed on the path of light beams, show that starlight is bent as it moves across the universe.

We do not expect to face the towering pyramids of Egypt. Similarly, when we open the front door, we expect to see the cars on the street, not the craters and dead volcanoes of a bleak, lunar landscape. Without even thinking about it, we assume that we can safely open windows or doors without being scared out of our wits. Our world, fortunately, is not a Steven Spielberg movie. We act on a deeply ingrained prejudice which is invariably correct that our world is simply connected, that our windows and doorways are not entrances to wormholes connecting our home to a far-away universe.

In ordinary space, a lasso of rope can always be shrunk to a point. If this is possible, then the space is called simply connected. However, if the lasso is placed around the entrance of the wormhole, then it cannot be shrunk to a point.

The lasso, in fact, enters the wormhole. Such spaces, where lassos are not contractible, are called multiply connected. Although the bending of our universe in an unseen dimension has been experimen- tally measured, the existence of wormholes and whether our universe is multiply connected or not is still a topic of scientific controversy.

Mathematicians dating back to Georg Bernhard Riemann have stud- ied the properties of multiply connected spaces in which different regions of space and time are spliced together. And physicists, who once thought this was merely an intellectual exercise, are now seriously study- ing multiply connected worlds as a practical model of our universe.

These models are the scientific analogue of Alice's looking glass. When Lewis Carroll's White Rabbit falls down the rabbit hole to enter Won- derland, he actually falls down a wormhole. Wormholes can be visualized with a sheet of paper and a pair of scissors: Take a piece of paper, cut two holes in it, and then reconnect the two holes with a long tube Figure 1.

As long as you avoid walking into the wormhole, our world seems perfectly normal. The usual laws of geometry taught in school are obeyed. However, if you fall into the wormhole, you are instantly transported to a different region of space and time. Only by retracing your steps and falling back into the worm- hole can you return to your familiar world.

Time Travel and Baby Universes Although wormholes provide a fascinating area of research, perhaps the most intriguing concept to emerge from this discussion of hyperspace Worlds Beyond Space and Time 19 Figure 1. Parallel universes may be graphically represented by two parallel planes. Normally, they never interact with each other. However, at times worm- holes or tubes may open up between them, perhaps making communication and travel possible between them.

This is now the subject of intense interest among theoretical physicists. In the film Back to the Future, Michael J. Fox journeys back in time and meets his parents as teenagers before they were married. Unfortunately, his mother falls in love with him and spurns his father, raising the ticklish question of how he will be born if his parents never marry and have children. Traditionally, scientists have held a dim opinion of anyone who raised the question of time travel.

However, in the physics ofwormholes, "acau- sal" effects show up repeatedly. In fact, we have to make strong assump- tions in order to prevent time travel from taking place. The main problem is that wormholes may connect not only two distant points in space, but also the future with the past. In , physicist Kip Thorne of the California Institute of Technol- ogy and his collaborators made the astonishing and risky claim that time travel is indeed not only possible, but probable under certain con- ditions.

They published their claim not in an obscure "fringe" journal, but in the prestigious Physical Review Letters. This marked the first time that reputable physicists, and not crackpots, were scientifically advancing a claim about changing the course of time itself. Their announcement was based on the simple observation that a wormhole connects two regions that exist in different time periods. Thus the wormhole may connect the present to the past.

## The Future of Humanity

Since travel through the wormhole is nearly instantaneous, one could use the wormhole to go backward in time. Unlike the machine portrayed in H.

Wells's The Time Machine, however, which could hurl the protagonist hundreds of thousands of years into England's distant future with the simple twist of a dial, a worm- hole may require vast amounts of energy for its creation, beyond what will be technically possible for centuries to come.

Another bizarre consequence of wormhole physics is the creation of "baby universes" in the laboratory. We are, of course, unable to re-create the Big Bang and witness the birth of our universe. However, Alan Guth of the Massachusetts Institute of Technology, who has made many important contributions in cosmology, shocked many physicists a few years ago when he claimed that the physics of wormholes may make it possible to create a baby universe of our own in the laboratory.

By con- centrating intense heat and energy in a chamber, a wormhole may even- tually open up, serving as an umbilical cord connecting our universe to another, much smaller universe. If possible, it would give a scientist an unprecedented view of a universe as it is created in the laboratory. Mystics and Hyperspace Some of these concepts are not new. For the past several centuries, mys- tics and philosophers have speculated about the existence of other uni- verses and tunnels between them. They have long been fascinated by the possible existence of other worlds, undetectable by sight or sound, yet coexisting with our universe.

They have been intrigued by the pos- Worlds Beyond Space and Time 21 sibility that these unexplored, nether worlds may even be tantalizingly close, in fact surrounding us and permeating us everywhere we move, yet just beyond our physical grasp and eluding our senses.

Such idle talk, however, was ultimately useless because there was no practical way in which to mathematically express and eventually test these ideas. Gateways between our universe and other dimensions are also a favorite literary device. Science-fiction writers find higher dimensions to be an indispensable tool, using them as a medium for interstellar travel.

Because of the astronomical distances separating the stars in the heav- ens, science-fiction writers use higher dimensions as a clever shortcut between the stars. Instead of taking the long, direct route to other gal- axies, rockets merely zip along in hyperspace by warping the space around them.

For instance, in the film Star Wars, hyperspace is a refuge where Luke Skywalker can safely evade the Imperial Starships of the Empire. In the television series "Star Trek: Deep Space Nine," a worm- hole opens up near a remote space station, making it possible to span enormous distances across the galaxy within seconds. The space station suddenly becomes the center of intense intergalactic rivalry over who should control such a vital link to other parts of the galaxy.

Ever since Flight 19, a group of U. Some have conjectured that airplanes and ships disappearing in the Bermuda Triangle actually entered some sort of passageway to another world. The existence of these elusive parallel worlds has also produced end- less religious speculation over the centuries. Spiritualists have wondered whether the souls of departed loved ones drifted into another dimen- sion. The seventeenth-century British philosopher Henry More argued that ghosts and spirits did indeed exist and claimed that they inhabited the fourth dimension.

In Enchiridion Metaphysician , he argued for the existence of a nether realm beyond our tangible senses that served as a home for ghosts and spirits. Nineteenth-century theologians, at a loss to locate heaven and hell, pondered whether they might be found in a higher dimension. Some wrote about a universe consisting of three parallel planes: God himself, according to the theologian Arthur Wil- link, found his home in a world far removed from these three planes; he lived in infinite-dimensional space.

It fascinated such diverse personalities as psychologist William James, literary figure Gertrude Stein, and revolutionary socialist Vladimir Lenin. The fourth dimension also inspired the works of Pablo Picasso and Marcel Duchamp and heavily influenced the development of Cubism and Expressionism, two of the most influential art movements in this century.

Art historian Linda Dalrymple Henderson writes, "Like a Black Hole, 'the fourth dimension' possessed mysterious qualities that could not be completely understood, even by the scientists themselves.

Yet, the impact of 'the fourth dimension' was far more comprehensive than that of Black Holes or any other more recent scientific hypothesis except Relativity Theory after For example, the mathematician Charles L. Dodgson, who taught at Oxford University, delighted generations of schoolchildren by writing books — as Lewis Carroll — that incorporate these strange math- ematical ideas.

When Alice falls down a rabbit hole or steps through the looking glass, she enters Wonderland, a strange place where Cheshire cats disappear leaving only their smile , magic mushrooms turn chil- dren into giants, and Mad Hatters celebrate "unbirthdays.

Some of the inspiration for Lewis Carroll's ideas most likely came from the great nineteenth-century German mathematician Georg Bern- hard Riemann, who was the first to lay the mathematical foundation of geometries in higher-dimensional space. Riemann changed the course of mathematics for the next century by demonstrating that these uni- verses, as strange as they may appear to the layperson, are completely self-consistent and obey their own inner logic. To illustrate some of these ideas, think of stacking many sheets of paper, one on top of another.

Now imagine that each sheet represents an entire world and that each world obeys its own physical laws, different from those of all the other worlds. Our universe, then, would not be alone, but would be one of Worlds Beyond Space and Time 23 many possible parallel worlds. Intelligent beings might inhabit some of these planes, completely unaware of the existence of the others.

On one sheet of paper, we might have Alice's bucolic English countryside. On another sheet might be a strange world populated by mythical creatures in the world of Wonderland.

Normally, life proceeds on each of these parallel planes independent of the others. On rare occasions, however, the planes may intersect and, for a brief moment, tear the fabric of space itself, which opens up a hole — or gateway — between these two universes. Like the wormhole appearing in "Star Trek: Deep Space Nine," these gateways make travel possible between these worlds, like a cosmic bridge linking two different universes or two points in the same universe Figure 1.

## Michio Kaku

Not surpris- ingly, Carroll found children much more open to these possibilities than adults, whose prejudices about space and logic become more rigid over time. In fact, Riemann's theory of higher dimensions, as interpreted by Lewis Carroll, has become a permanent part of children's literature and folklore, giving birth to other children's classics over the decades, such as Dorothy's Land of Oz and Peter Pan's Never Never Land.

Without any experimental confirmation or compelling physical moti- vation, however, these theories of parallel worlds languished as a branch of science.

Over 2 millennia, scientists have occasionally picked up the notion of higher dimensions, only to discard it as an untestable and therefore silly idea. Although Riemann's theory of higher geometries was mathematically intriguing, it was dismissed as clever but useless. Sci- entists willing to risk their reputations on higher dimensions soon found themselves ridiculed by the scientific community.

Higher-dimensional space became the last refuge for mystics, cranks, and charlatans. In this book, we will study the work of these pioneering mystics, mainly because they devised ingenious ways in which a nonspecialist could "visualize" what higher-dimensional objects might look like. These tricks will prove useful to understand how these higher-dimen- sional theories may be grasped by the general public. By studying the work of these early mystics, we also see more clearly what was missing from their research.

We see that their speculations lacked two important concepts: From the perspective of modern physics, we now realize that the missing physical principle is that hyperspace simplifies the laws of nature, provid- ing the possibility of unifying all the forces of nature by purely geometric arguments. The missing mathematical 'principle is called field theory, which is the universal mathematical language of theoretical physics.

Figure 1. Wormholes may connect a universe with itself, perhaps providing a means of interstellar travel. Since wormholes may connect two different time eras, they may also provide a means for time travel. Wormholes may also connect an infinite series of parallel universes. The hope is that the hyperspace theory will be able to determine whether wormholes are physically possible or merely a mathe- matical curiosity.

The Language of Physics Fields were first introduced by the great nineteenth-century British sci- entist Michael Faraday. The son of a poor blacksmith, Faraday was a self- taught genius who conducted elaborate experiments on electricity and magnetism. Fie visualized "lines of force" that, like long vines spreading from a plant, emanated from magnets and electric charges in all direc- tions and filled up all of space. With his instruments, Faraday could measure the strength of these lines of force from a magnetic or an elec- tric charge at any point in his laboratory.

Thus he could assign a series of numbers the strength and direction of the force to that point and any point in space. He christened the totality of these numbers at any point in space, treated as a single entity, a field. There is a famous story concerning Michael Faraday. Because his fame had spread far and wide, he was often visited by curious bystanders.

When one asked what his work was good for, he answered, "What is the use of a child? It grows to be a man. Knowing nothing about science, Gladstone sarcastically asked Faraday what use the huge electrical con- traptions in his laboratory could possibly have for England.

Faraday replied, "Sir, I know not what these machines will be used for, but I am sure that one day you will tax them. Simply put, a field is a collection of numbers defined at every point in space that completely describes a force at that point. For example, three numbers at each point in space can describe the intensity and direction of the magnetic lines of force. Another three numbers every- where in space can describe the electric field.

Faraday got this concept when he thought of a "field" plowed by a farmer. A farmer's field occu- pies a two-dimensional region of space. At each point in the farmer's field, one can assign a series of numbers which describe, for example, how many seeds there are at that point.

Faraday's field, however, occu- pies a three-dimensional region of space. At each point, there is a series of six numbers that describes both the magnetic and electric lines of force. What makes Faraday's field concept so powerful is that all forces of nature can be expressed as a field. However, we need one more ingre- dient before we can understand the nature of any force: We must be able to write down the equations that these fields obey.

The progress of the past hundred years in theoretical physics can be succinctly summa- rized as the search for the field equations of the forces of nature. In , Einstein discovered the field equations for gravity.

After innumerable false starts, the field equations for the subatomic forces were finally writ- ten down in the s, utilizing the earlier work of C. Yang and his student R.

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These fields, which govern the interaction of all subatomic particles, are now called Yang-Mills fields. However, the puzzle that has stumped physicists within this century is why the subatomic field equations look so vastly different from the field equations of Einstein — that is, why the nuclear force seems so different from gravity. Some of the greatest minds in physics have tackled this problem, only to fail. Perhaps the reason for their failure is that they were trapped by com- mon sense.

Confined to three or four dimensions, the field equations of the subatomic world and gravitation are difficult to unify. The advan- tage of the hyperspace theory is that the Yang-Mills field, Maxwell's field, and Einstein's field can all be placed comfortably within the hyperspace field.

We see that these fields fit together precisely within the hyperspace field like pieces in ajigsaw puzzle. The other advantage of field theory is that it allows us to calculate the precise energies at which we can expect space and time to form wormholes. Unlike the ancients, therefore, we have the mathematical tools to guide us in building the machines that may one day bend space and time to our whims.

The Secret of Creation Does this mean that big-game hunters can now start organizing safaris to the Mesozoic era to bag large dinosaurs? Thorne, Guth, and Freund will all tell you that the energy scale necessary to investigate these anomalies in space is far beyond anything available on earth. Freund reminds us that the energy necessary to probe the tenth dimension is a quadrillion times larger than the energy that can be produced by our largest atom smasher. Twisting space-time into knots requires energy on a scale that will not be available within the next several centuries or even millennia — if ever.

Even if all the nations of the world were to band together to build a machine that could probe hyperspace, they would ultimately fail. And, as Guth points out, the temperatures necessary to create a baby universe in the laboratory is 1, trillion trillion degrees, far in excess of any- thing available to us. In fact, that temperature is much greater than anything found in the interior of a star. So, although it is possible that Worlds Beyond Space and Time 21 Einstein's laws and the laws of quantum theory might allow for time travel, this is not within the capabilities of earthlings like us, who can barely escape the feeble gravitational field of our own planet.

While we can marvel at the implications ofwormhole research, realizing its poten- tial is strictly reserved for advanced extraterrestrial civilizations. There was only one period of time when energy on this enormous scale was readily available, and that was at the instant of Creation.

In fact, the hyperspace theory cannot be tested by our largest atom smash- ers because the theory is really a theory of Creation. Only at the instant of the Big Bang do we see the full power of the hyperspace theory com- ing into play. This raises the exciting possibility that the hyperspace the- ory may unlock the secret of the origin of the universe.

Introducing higher dimensions may be essential for prying loose the secrets of Creation. According to this theory, before the Big Bang, our cosmos was actually a perfect ten-dimensional universe, a world where interdimensional travel was possible. However, this ten-dimensional world was unstable, and eventually it "cracked" in two, creating two separate universes: The universe in which we live was born in that cosmic cataclysm.

Our four-dimensional universe expanded explosively, while our twin six-dimensional universe contracted violently, until it shrank to almost infinitesimal size. This would explain the origin of the Big Bang. If correct, this theory dem- onstrates that the rapid expansion of the universe was just a rather minor aftershock of a much greater cataclysmic event, the cracking of space and time itself. The energy that drives the observed expansion of the universe is then found in the collapse of ten-dimensional space and time.

According to the theory, the distant stars and galaxies are receding from us at astronomical speeds because of the original collapse of ten-dimen- sional space and time. This theory predicts that our universe still has a dwarf twin, a com- panion universe that has curled up into a small six-dimensional ball that is too small to be observed.

This six-dimensional universe, far from being a useless appendage to our world, may ultimately be our salvation. Evading the Death of the Universe It is often said that the only constants of human society are death and taxes. For the cosmologist, the only certainty is that the universe will one day die. Some believe that the ultimate death of the universe will come in the form of the Big Crunch. As the stars contract, temperatures will rise dramatically until all matter and energy in the universe are con- centrated into a colossal fireball that will destroy the universe as we know it.

All life forms will be crushed beyond recognition. There will be no escape. Scientists and philosophers, like Charles Darwin and Bertrand Russell, have written mournfully about the futility of our pitiful exis- tence, knowing that our civilization will inexorably die when our world ends. The laws of physics, apparently, have issued the final, irrevocable death warrant for all intelligent life in the universe.

According to the late Columbia University physicist Gerald Feinberg, there is one, and perhaps only one, hope of avoiding the final calamity.

He speculated that intelligent life, eventually mastering the mysteries of higher-dimensional space over billions ofyears, will use the other dimen- sions as an escape hatch from the Big Crunch. In the final moments of the collapse of our universe, our sister universe will open up once again, and interdimensional travel will become possible.

As all matter is crushed in the final moments before doomsday, intelligent life forms may be able to tunnel into higher-dimensional space or an alternative universe, avoiding the seemingly inevitable death of our universe. Then, from their sanctuary in higher-dimensional space, these intelligent life forms may be able to witness the death of the collapsing universe in a fiery cataclysm.

As our home universe is crushed beyond recognition, temperatures will rise violently, creating yet another Big Bang. From their vantage point in hyperspace, these intelligent life forms will have front-row seats to the rarest of all scientific phenomena, the creation of another universe and of their new home.

Masters of Hyperspace Although field theory shows that the energy necessary to create these marvelous distortions of space and time is far beyond anything that mod- ern civilization can muster, this raises two important questions: How long will it take for our civilization, which is growing exponentially in knowl- edge and power, to reach the point of harnessing the hyperspace theory?

And what about other intelligent life forms in the universe, who may already have reached that point? What makes this discussion interesting is that serious scientists have tried to quantify the progress of civilizations far into the future, when space travel will have become commonplace and neighboring star sys- Worlds Beyond Space and Time 29 terns or even galaxies will have been colonized.

Although the energy scale necessary to manipulate hyperspace is astronomically large, these scientists point out that scientific growth will probably continue to rise exponentially over the next centuries, exceeding the capabilities of human minds to grasp it.

Since World War II, the sum total of scientific knowledge has doubled every 10 to 20 or so years, so the progress of science and technology into the twenty-first century may surpass our wildest expectations.

Technologies that can only be dreamed of today may become commonplace in the next century. Perhaps then one can discuss the question of when we might become masters of hyperspace. Time travel. Parallel universes. Dimensional windows. By themselves, these concepts stand at the edge of our understanding of the physical universe.

However, because the hyperspace theory is a genuine field theory, we eventually expect it to produce numerical answers determining whether these intriguing concepts are possible. If the theory produces nonsensical answers that disagree with physical data, then it must be discarded, no matter how elegant its mathematics. In the final analysis, we are physicists, not philosophers.

But if it proves to be correct and explains the symmetries of modern physics, then it will usher in a revolution perhaps equal to the Copernican or Newtonian revolutions. To have an intuitive understanding of these concepts, however, it is important to start at the beginning.

Before we can feel comfortable with ten dimensions, we must learn how to manipulate four spatial dimen- sions. Using historical examples, we will explore the ingenious attempts made by scientists over the decades to give a tangible, visual represen- tation of higher-dimensional space.

The first part of the book, therefore, will stress the history behind the discovery of higher-dimensional space, beginning with the mathematician who started it all, Georg Bernhard Riemann. Anticipating the next century of scientific progress, Riemann was the first to state that nature finds its natural home in the geometry of higher-dimensional space.

Mathematicians and Mystics Magic is any sufficiently advanced technology. Arthur C.

## Epub Download Physics of the Impossible A Scientific Exploration into…

Clarke O N June 10, , a new geometry was born. The theory of higher dimensions was introduced when Georg Bernhard Riemann gave his celebrated lecture before the faculty of the University of Gottingen in Germany.

In one masterful stroke, like open- ing up a musty, darkened room to the brilliance of a warm summer's sun, Riemann's lecture exposed the world to the dazzling properties of higher-dimensional space. His profoundly important and exceptionally elegant essay, "On the Hypotheses Which Lie at the Foundation of Geometry," toppled the pillars of classical Greek geometry, which had successfully weathered all assaults by skeptics for 2 millennia.

The old geometry of Euclid, in which all geometric figures are two or three dimensional, came tumbling down as a new Riemannian geometry emerged from its ruins. The Riemannian revolution would have vast implications for the future of the arts and sciences. Within 3 decades of his talk, the "mysterious fourth dimen- sion" would influence the evolution of art, philosophy, and literature in Europe.

Within 6 decades of Riemann's lecture, Einstein would use four-dimensional Riemannian geometry to explain the creation of the universe and its evolution. And years after his lecture, physicists 30 Mathematicians and Mystics 31 would use ten-dimensional geometry to attempt to unite all the laws of the physical universe. The core of Riemann's work was the realization that physical laws simplify in higher-dimensional space, the very theme of this book.

Brilliance Amid Poverty Ironically, Riemann was the least likely person to usher in such a deep and thorough-going revolution in mathematical and physical thought. He was excruciatingly, almost pathologically, shy and suffered repeated nervous breakdowns.

He also suffered from the twin ailments that have ruined the lives of so many of the world's great scientists throughout history: His personality and temperament showed nothing of the breath-taking boldness, sweep, and supreme confidence typical of his work.

Riemann was born in in Hanover, Germany, the son of a poor Lutheran pastor, the second of six children. His father, who fought in the Napoleonic Wars, struggled as a country pastor to feed and clothe his large family.

As biographer E. Bell notes, "the frail health and early deaths of most of the Riemann children were the result of under- nourishment in their youth and were not due to poor stamina. The mother also died before her children were grown. Riemann exhibited his famous traits: Painfully shy, he was the butt of cruel jokes by other boys, causing him to retreat further into the intensely private world of mathematics.

He also was fiercely loyal to his family, straining his poor health and constitution to buy presents for his parents and especially for his beloved sisters. To please his father, Riemann set out to become a student of theology.

His goal was to get a paying position as a pastor as quickly as possible to help with his family's abysmal finances. It is difficult to imag- ine a more improbable scenario than that of a tongue-tied, timid young boy imagining that he could deliver fiery, passionate sermons railing against sin and driving out the devil.

In high school, he studied the Bible intensely, but his thoughts always drifted back to mathematics; he even tried to provide a mathematical proof of the correctness of Genesis. He also learned so quickly that he kept outstripping the knowledge of his instructors, who found it impos- sible to keep up with the boy. The book was Adrien-Marie Legendre's Theory of Numbers, a huge page master- piece, the world's most advanced treatise on the difficult subject of num- ber theory.

Riemann devoured the book in 6 days. When his principal asked, "How far did you read? I have mastered it. Beset by the daily struggle to put food on the table, Riemann's father might have sent the boy to do menial labor. Instead, he scraped together enough funds to send his year-old son to the renowned University of Gottingen, where he first met Carl Friedrich Gauss, the acclaimed "Prince of Mathematicians," one of the greatest mathematicians of all time. Even today, if you ask any mathematician to rank the three most famous mathematicians in history, the names of Archimedes, Isaac New- ton, and Carl Gauss will invariably appear.

Life for Riemann, however, was an endless series of setbacks and hardships, overcome only with the greatest difficulty and by straining his frail health. Each triumph was followed by tragedy and defeat. For exam- ple, just as his fortunes began to improve and he undertook his formal studies under Gauss, a full-scale revolution swept Germany. The working class, long suffering under inhuman living conditions, rose up against the government, with workers in scores of cities throughout Germany taking up arms.

The demonstrations and uprisings in early inspired the writings of another German, Karl Marx, and deeply affected the course of revolutionary movements throughout Europe for the next 50 years. With all of Germany swept up in turmoil, Riemann's studies were interrupted. He was inducted into the student corps, where he had the dubious honor of spending 16 weary hours protecting someone even more terrified than he: Beyond Euclidean Geometry Not only in Germany, but in mathematics, too, fierce revolutionary winds were blowing.

The problem that riveted Riemann's interest was the impending collapse ofyet another bastion of authority, Euclidean geom- Mathematicians and Mystics 33 etry, which holds that space is three dimensional. Furthermore, this three-dimensional space is "flat" in flat space, the shortest distance between two points is a straight line; this omits the possibility that space can be curved, as on a sphere. In fact, after the Bible, Euclid's Elements was probably the most influ- ential book of all time.

For 2 millennia, the keenest minds of Western civilization have marveled at its elegance and the beauty of its geometry. Thousands of the finest cathedrals in Europe were erected according to its principles. In retrospect, perhaps it was too successful. Over the cen- turies, it became something of a religion; anyone who dared to propose curved space or higher dimensions was relegated to the ranks of crack- pots or heretics. For untold generations, schoolchildren have wrestled with the theorems of Euclid's geometry: However, try as they might, the finest mathematical minds for several centuries could not prove these deceptively simple propositions.

In fact, the mathematicians ofEurope began to realize that even Euclid's Elements, which had been revered for 2, years, was incomplete. Euclid's geometry was still viable if one stayed within the confines of flat surfaces, but if one strayed into the world of curved surfaces, it was actually incorrect.

To Riemann, Euclid's geometry was particularly sterile when com- pared with the rich diversity of the world. Nowhere in the natural world do we see the flat, idealized geometric figures of Euclid. Mountain ranges, ocean waves, clouds, and whirlpools are not perfect circles, tri- angles, and squares, but are curved objects that bend and twist in infinite diversity.

The time was ripe for a revolution, but who would lead it and what would replace the old geometry? The Rise of Riemannian Geometry Riemann rebelled against the apparent mathematical precision of Greek geometry, whose foundation, he discovered, ultimately was based on the shifting sand of common sense and intuition, not the firm ground of logic.

It is obvious, said Euclid, that a point has no dimension at all. A line has one dimension: A plane has two dimensions: A solid has three dimensions: And there it stops. Nothing has four dimensions.

In On Heaven, he wrote, "The line has magnitude in one way, the plane in two ways, and the solid in three ways, and beyond these there is no other magnitude because the three are all. First, he said, draw three mutually perpendicular lines. For example, the corner of a cube consists of three mutually perpendicular lines. Then, he argued, try to draw a fourth line that is perpendicular to the other three lines.

No matter how one tries, he reasoned, four mutually perpendicular lines are impossible to draw. Ptolemy claimed that a fourth perpendicular line is "entirely without measure and without def- inition. What Ptolemy actually proved was that it is impossible to visualize the fourth dimension with our three-dimensional brains.

In fact, today we know that many objects in mathematics cannot be visualized but can be shown to exist. Ptolemy may go down in history as the man who opposed two great ideas in science: Over the centuries, in fact, some mathematicians went out of their way to denounce the fourth dimension. In , the mathematician John Wallis polemicized against the concept, calling it a "Monster in Nature, less possible than a Chimera or Centaure.

Length, Breadth, and Thickness, take up the whole of Space. The Unity of Ail Physical Law The decisive break with Euclidean geometry came when Gauss asked his student Riemann to prepare an oral presentation on the "foundation of geometry.

Decades before. Gauss had privately expressed deep and extensive reservations about Euclidean geometry. He even spoke to his colleagues of hypothetical "bookworms" that might live entirely on a two-dimensional surface.

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