Surfing Through Hyperspace

Understanding Higher Universes in Six Easy Lessons

Clifford A. Pickover, Oxford University Press
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"To consider that after the death of the body the spirit perishes is like imagining that a bird in a cage will be destroyed if the cage is broken, though the bird has nothing to fear from the destruction of the cage. Our body is like the cage, and the spirit like the bird. We see that without the cage this bird flies in the world of sleep; therefore, if the cage becomes broken, the bird will continue and exist. Its feelings will be even more powerful, its perceptions greater, and its happiness increased."
- Abdu'l-Baha, Some Answered Questions

   "The bird fights its way out of the egg. The egg is the world. Who would be born must first destroy a world. The bird flies to God. That God's name is Abraxas."
- Hermann Hesse, Demian

Touring Higher Worlds

   I know of no subject in mathematics that has intrigued both children and adults as much as the idea of a fourth dimension -- a spatial direction different from all the directions of our normal three-dimensional space. Philosophers and parapsychologists have meditated upon this dimension that no one can point to but may be all around us. Theologians have speculated that the afterlife, heaven, hell, angels, and our souls could reside in a fourth dimension -- that God and Satan could literally be lumps of hypermatter in a four-dimensional space inches away from our ordinary three-dimensional world. Throughout time, various mystics and prophets have likened our world to a three-dimensional cage:sup.1:esup. and have speculated on how great our perceptions would be if we could break from the confines of our world into higher dimensions. Yet, despite all the philosophical and spiritual implications of the fourth dimension, this extra dimension also has a very practical side. Mathematicians and physicists use the fourth dimension every day in calculations. It's part of important theories that describe the very fabric of our universe.

       I first became excited about the possibility of a fourth dimension as a child watching a TV rerun of the 1959 science-fiction movie The 4D Man. This clever thriller described the adventures of a scientist who develops a method of transposing matter, enabling him to pass through walls, windows, water, and women. Here is a snippet of the movie's dialogue:

Scott Nelson: That's what you've done with your force field. You've compressed the energy of years into a moment.

   Linda Davis: But... that's like... the fourth dimension.

   Captain Rogers: I don't believe it. I'm a cop. I work with facts. Now I have to start looking for something that saps the life out of a man like juice out of an orange.

   Tony Nelson: Nothing can stop him. Can't imprison him or surround him with men or guns or tanks. No walls thick enough or guns strong enough. A man in the fourth dimension is indestructible.

   The movie has a bevy of Hollywood stars -- Patty Duke and Lee Meriwether just to mention two. The plot involved a scientist discovering a dimension in which he can walk through solid matter. I hope I'm not ruining the movie by telling you the bizarre ending where he materializes out of the fourth dimension into our three-dimensional world while passing through a brick wall. Ouch!

   You can't imagine how profoundly affected I was by the blurring of fact and fiction. To a young boy, the strange array of physical and mathematical ideas made the unbelievable seem a frighteningly real possibility. I knew that if an accessible fourth dimension existed, it would actually be possible to escape from a prison by temporarily going into the fourth dimension -- like a bird leaving its nest for the first time, flying upward, and joyfully revelling in its newly found third-dimension.

   My fascination with the fourth dimension was later stimulated by Steven Spielberg's 1982 movie Poltergeist in which a family living in a suburban development is faced with menacing phenomena: a child who disappears, furniture that moves by itself, and weird powers gusting through the house and frightening everyone.

   Do any of you recall the scene in Poltergeist in which balls are thrown into a closet and seem to magically reappear from the ceiling in another location in the house? This could easily be explained if the ball took a route through the fourth dimension -- as you will learn later in this book. Even the early 1960s TV show The Outer Limits touched upon higher dimensions. In one particularly poignant episode, a creature from the Andromeda galaxy lived in a higher dimension than ours, and he is pulled into our universe as a result of terrestrial experiments with a new form of three-dimensional TV. Although the creature is both wise and friendly, its visit to our world causes quite a pandemonium.

   For decades, there have been many popular science books and science-fiction novels on the subject of the fourth dimension. My favorite science book on the subject is Rudy Rucker's The Fourth Dimension which covers an array of topics on space and time. My favorite science-fiction story is Robert Heinlein's --And He Built A Crooked House, first published in 1940. It tells the tale of a California architect who constructs a four-dimensional house. He explains that a four-dimensional house would have certain advantages:

I'm thinking about a fourth spatial dimension, like length, breadth, and thickness. For economy of materials and convenience of arrangement you couldn't beat it. To say nothing of ground space -- you could put an eight-room house on the land now occupied by a one-room house.
Unfortunately, once the builder takes the new owners on a tour of the house, they can't find their way out. Windows and doors that normally face the outside of the house now face inside. Needless to say, some very strange things happen to the terrified people trapped in the house.

   There are many excellent books on the fourth dimension, and these are listed in the "For Further Reading" section at the end of this book. So, why another book on higher-dimensional worlds? I have found that many previous books on the subject were lacking an important element. They don't focus wholeheartedly on the physical appearance of four-dimensional beings, what mischief and good they could do in our world, and the religious implications of a penetration into our world. More importantly, many prior books are also totally descriptive with no formulas with which readers can experiment -- not even simple formulas -- or the books are so full of complicated-looking equations that students, computer hobbyists, and general audiences are totally overwhelmed.

   The fourth dimension need not remain confined to Hollywood and the realm of science fiction, beyond the range of exciting experiment and careful thought. Many of the ideas, thought exercises, and numerical experiments in this book are accessible to both students and seasoned scientists. A few pieces of computational recipes are included so that computer hobbyists can explore higher-dimensional worlds. But those of you with no interest in computing can easily skip these sections and explore the mental realms, unaided by computation. In this book, I'll discuss concepts such as "degrees of freedom" and then gradually work my way up to more sophisticated concepts such as the possibility of stuffing huge whales into tiny four-dimensional spheres. The Appendices discuss a number of stimulating problems -- from a four-dimensional version of Rubik's cube and the evolution of four-dimensional biologies, to four-dimensional fractal quaternions with infinitely complex structures. However, the emphasis will be on the powers and appearances of four-dimensional beings. I want to know if humankind's Gods could exist in the fourth dimension.:sup.2:esup.

   If a fourth dimension did exist, God could be so close we could hear His breath, only inches away, but impossible to see because our perception is seemingly confined to three dimensions.

   What if you could visit a four-dimensional world filled with intelligent lifeforms? Would the aliens have heads, arms, and legs, or even be vaguely humanoid? What capabilities would they possess if they were to visit our world? The challenging task of imagining beings from other dimensions is useful for any species that dreams of understanding its place in a vast universe with infinite possibilities.

   Is there really a fourth dimension we can explore and understand? This question is an old one posed by philosophers and scientists, and it has profound implications for our world view. There seems to be no reason why a four-dimensional world of material, four-dimensional objects could not exist. The simple mathematical methods in this book reveal properties of shapes in these higher spaces, and with special training, our dimensionally-impoverished minds may be able to grasp the "look and feel" of these shapes. As we speculate, we touch on the realm of mysticism and religion -- because in the fourth dimension the line between science and mysticism grows thin.

   Hyperbeings living in a four-dimensional space can demonstrate the kinds of phenomena that occur in hyperspace. For example, a hyperbeing can effortlessly remove things before our very eyes, giving us the impression that the objects simply disappeared. This is analogous to a three-dimensional creature's ability to remove a piece of dirt inside a circle drawn on a page without cutting the circle. We simply lift the dirt into the third dimension. To two-dimensional beings confined to the piece of paper, this action would appear miraculous as the dirt disappeared in front of their eyes. The hyperbeing can also see inside any three-dimensional object or life form, and if necessary remove anything from inside. As will be explained in this book, the being can look inside our intestines, examine our nervous system, or remove a tumor from our brain without ever cutting through the skin. A pair of gloves can be easily transformed into two left or two right gloves, and three-dimensional knots fall apart in the hands of a hyperbeing who can lift a piece of the knot up into the fourth dimension. In his correspondence and verse, eminent physicist James Clerk Maxwell referred to the fourth dimension as the place where knots could be untied:

My soul is an entangled knot
Upon a liquid vortex wrought
The secret of its untying
In four-dimensional space is lying.

   I call such four-dimensional beings "Gods". If we ever encounter beings that can move in a fourth spatial dimension, we would find they can perform levitation, bloodless surgery, disappear in front of our eyes, walk through walls.... It would be very difficult to hide from them no matter where we went. Objects locked in safes would be easy for them to retrieve. If such a being were observed in biblical times, it would be considered a God with many characteristics of omniscience, omnipresence, and omnipotence.

A Note on Terminology Used in This Book

Most Earthly cultures have a vocabulary with words like up, down, right, left, north, south, and so forth. While the terms "up" and "down" have meaning for us in our three-dimensional universe, they are less useful when talking about movements from the three-dimensional universe into the fourth dimension. To facilitate our discussions, I use the words "upsilon" and "delta." The worlds "upsilon" and "delta" can be used more or less like the words up and down, as you shall see when first introduced to the terms in Chapter 3.

   The term "hyperspace" is used popularly when referring to higher dimensions, and hyper- is the correct scientific prefix for higher-dimensional geometries. Like other authors, I have adhered to the popular custom of using "hyperspace" when referring to higher dimensions. The word "hyperspace" was coined by John W. Campbell in his short story "The Mightiest Machine" (1934), and the term has been used both by science-fiction writers and physicists ever since. Moreover, physicists sometimes use "hyperspace" in discussions of the structure of our universe. For example, if we cannot move faster than light in this universe, perhaps we can take a short cut. Astrophysicists sometimes speculate that there may be a way to slip out of ordinary space and return to our own universe at some other location via a crumpling of space. This severe folding takes place in hyperspace so that two seemingly far-away points are brought closer together. Some physicists also view hyperspace as a higher dimension in which our entire universe may be curved -- in the same way that a flat piece of paper can be flexed or rolled so that it curves in the third-dimension.

   Not only may hyperspace play a role on the galactic and universal size scale, but it may help characterize the ultrasmall. Physicist John A. Wheeler has suggested that empty space may be filled with countless tiny wormholes connecting different parts of space, like little tubes that run outside of space and back in again at some distant point. Wheeler describes the wormholes as running through "superspace," which seems similar to what science fiction has called "hyperspace" for over a half-century.

The FBI's Four-Dimensional Smorgasbord

This book will allow you to travel through dimensions, and you needn't be an expert in physics, mathematics, or theology. Some information is repeated so that each chapter contains sufficient background information, but I suggest you read the chapters in order as you gradually build your knowledge. I start most chapters with a dialog between two quirky FBI agents who experiment with the fourth dimension from within the (usually) safe confines of their FBI Headquarters office in Washington D.C. You are Chief Investigator of four-dimensional phenomena. Your able student is a novice FBI agent initially assigned to work with you to debunk your outlandish theories. But she gradually begins to doubt her own skepticism.... This simple science fiction is not only good fun but it also serves a serious purpose, that of expanding your imagination. We might not yet be able to easily travel into the fourth dimension like the characters in the story, but at least the fourth dimension is not forbidden by the current laws of physics. I also use science fiction to explain science because, over the last century, science fiction has done more to communicate the adventure of science than any physics book. As you read the story, think about how humans might respond to future developments in science that could lead to travel in a fourth dimension.

   When writing this book, I did not set out to write a systematic and comprehensive study of the fourth dimension. Instead, I have chosen topics that interested me personally and which I think will enlighten a wide range of readers. Although the concept of the fourth dimension is more than a century old, its strange consequences are still not widely known. People often learn of them with a sense of awe, mystery and bewilderment. Even armed with the mathematical theories in this book, you'll still have only a vague understanding of the fourth dimension, and various problems, paradoxes, and questions will plague you. What are the chances that we could learn to communicate with a 4-D extraterrestrial? Would they have internal organs like our own? We'll encounter all these and other questions as we open doors....

   I've attempted to make Surfing Through Hyperspace a strange journey that unlocks the doors of your imagination with thought-provoking mysteries, puzzles, and problems on topics ranging from hyperspheres to religion. A resource for science-fiction aficionados, a playground for philosophers, an adventure and education for mathematics students, each chapter is a world of paradox and mystery.

   Hopefully, my army of illustrators will also stimulate your imagination in ways that mere words cannot. Imagery is at the heart of much of the work described in this book. To better understand and contemplate the fourth dimension, we need our eyes. To help visualize higher-dimensional geometrical structures like hypercubes, I use computer graphics. To help visualize higher-dimensional beings, I recruit artists from different backgrounds to produce visual representations from myriad perspectives. For many of you, seeing hypothetical 4-D beings, and their intersections with our ordinary three-dimensional world, will clarify concepts. I often use the technique of explaining phenomena in lower dimensions to help understand higher dimensions.

   Why contemplate the appearance of 4-D beings and their powers? Mathematicians and artists feel the excitement of the creative process when they leave the bounds of the known to venture far into unexplored territory lying beyond the prison of the obvious. When we imagine the powers of hyperbeings, we are at the same time holding a mirror to ourselves, revealing our own prejudices and preconceived notions. The fourth dimension appeals to young minds, and I know of no better way to stimulate students than to muse about higher-dimensional worlds. Creative minds love roaming freely through the spiritual implications of the simple mathematics.

   Could creatures be hiding out in the fourth dimension at this very moment observing us? If you had the opportunity of stepping off into the fourth dimension, even for a few minutes, and looking down on our world, would you do it? (Before answering, remember that you would be peering into the steaming guts of your best friends.... You'll learn more about the unavoidable "X-ray" vision effect later.) None of these questions can be answered to scientists', theologians' or psychologists' satisfaction. Yet the mere asking of these questions stretches our minds, and the continual search for answers provides useful insights along the way.

   As in all my previous books, you are encouraged to pick and choose from the smorgasbord of topics. Many of the chapters are brief to give you just the tasty flavor of a topic. Those of you interested in pursuing specific topics can find additional information in the referenced publications. In order to encourage your involvement, the book is loaded with numerous what-if questions for further thought. Spread the spirit of this book by posing these questions to your students; to your priest, rabbi, mullah, or congregation; to your buddies at the bowling alley and local shopping mall, to your family the next time you plunk down on the couch to watch The X-Files, or the next time you can't seem to find your keys and wonder if they have escaped your notice by temporarily retreating into the fourth dimension...

   Whatever you believe about the possibility of a fourth dimension, the dimensional analogies in this book raise questions about the way you see the world and will therefore shape the way you think about the universe. For example, you will become more conscious about what it means to visualize an abstract object in your mind.

   By the time you've finished this book, you will be able to

You might even want to go out and buy a CD of the music from the X-Files TV show.

   In closing this Preface, let me remind you -- as I do in many of my books -- that humans are a moment in astronomic time, a transient guest of the Earth. Our minds have not sufficiently evolved to comprehend all the mysteries of higher dimensions. Our brains, which evolved to make us run from predators on the African grasslands, may not permit us to understand four-dimensional beings or to understand their thought processes. Given this potential limitation, we hope and search for knowledge and understanding. Any insights we gain as we investigate structures in higher dimensions will be increasingly useful to future scientists, theologians, philosophers, and artists. Contemplating the fourth dimension is as startling and rewarding as seeing the Earth from space for the first time.

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