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Do a little armchair time-travel, rub elbows with a four-dimensional intelligent life form, or stretch your mind to the furthest corner of an uncharted universe. With this astonishing guidebook, Surfing Through Hyperspace, you need not be a mathematician or an astrophysicist to explore the all-but-unfathomable concepts of hyperspace and higher-dimensional geometry.
No subject in mathematics has intrigued both children and adults as much as the idea of a fourth dimension. Philosophers and parapsychologists have meditated on this mysterious space that no one can point to but may be all around us. Yet this extra dimension has a very real, practical value to mathematicians and physicists who use it every day in their calculations. In the tradition of Flatland, and with an infectious enthusiasm, Clifford Pickover tackles the problems inherent in our 3-D brains trying to visualize a 4-D world, muses on the religious implications of the existence of higher-dimensional consciousness, and urges all curious readers to venture into "the unexplored territory lying beyond the prison of the obvious." Pickover alternates sections that explain the science of hyperspace with sections that dramatize mind-expanding concepts through a fictional dialogue between two futuristic FBI agents who dabble in the fourth dimension as a matter of national security. This highly accessible and entertaining approach turns an intimidating subject into a scientific game open to all dreamers.
Surfing Through Hyperspace concludes with a number of puzzles, computer experiments and formulas for further exploration, inviting readers to extend their minds across this inexhaustibly intriguing scientific terrain.
With this astonishing guidebook, Surfing Through Hyperspace, you need not be a mathematician or an astrophysicist to explore the all-but-unfathomable concepts of hyperspace and higher-dimensional geometry.
An unspeakable horror seized me. There was a darkness; then a dizzy, sickening sensation of sight that was not like seeing; I saw a Line that was no Line; Space that was not Space; I was myself, and not myself. When I could find voice, I shrieked aloud in agony, "Either this is madness or it is Hell." "It is neither," calmly replied the voice of the Sphere, "it is Knowledge; it is Three Dimensions; Open your eye once again and try to look steadily."
—Edwin Abbott Abbott, Flatland
Even the mathematician would like to nibble the forbidden fruit, to glimpse what it would be like if he could slip for a moment into a fourth dimension.
—Edward Kasner and James Newman,
Mathematics and the Imagination
May I pass along my congratulations for your great interdimensional breakthrough. I am sure, in the miserable annals of the Earth, you will be duly inscribed.
—Lord John Whorfin in The Adventures of Buckaroo Banzai Across the 8th Dimension
A man who devoted his life to it could perhaps succeed in picturing himself a fourth dimension.
—Henri Poincaré, "L'Espace et la géométrie"
To a frog with its simple eye, the world is a dim array of greys and blacks. Are we like frogs in our limited sensorium, apprehending just part of the universe we inhabit? Are we as a species now awakening to the reality of multidimensional worlds in which matter undergoes subtle reorganizations in some sort of hyperspace?
—Michael Murphy, The Future of the Body
A place is nothing: not even space, unlessat its heart—a figure stands.
—Paul Dirac, Principles of Quantum Mechanics
Traveling through hyperspace ain't like dusting crops, boy.
—Han Solo in Star Wars
degrees of freedom
FBI Headquarters, Washington, D.C., 10:00 A.M.
You have returned from Cherbourg and are relaxing in your office at the Washington Metropolitan Field Office of the Federal Bureau of Investigation located on 1900 Half Street, Washington, D.C. Very few people know your office exists because its door is cleverly disguised as an elevator bearing a perpetual "out-of-order" sign.
Inside, on the back of your door, is the colorful FBI seal and motto "Fidelity, Bravery, Integrity." The peaked beveled edge circumscribing the seal symbolizes the severe challenges confronting the FBI and the ruggedness of the organization.
Below the motto is a handmade sign that reads
I BELIEVE THE FOURTH DIMENSION IS REAL.
Sally follows you into your high-ceilinged office crammed with books and electrical equipment. Lying scattered between the sofa and chairs are three oscilloscopes, a tall Indian rubber plant, and a Rubik's cube. A small blackboard hangs on the wall. The bulletproof, floor-to-ceiling windows give the appearance of a room more spacious than it really is.
Sally eyes the electrical hardware. "What's all this?"
She nearly knocks an antique decanter from a table onto your favorite gold jacquard smoking jacket slung comfortably over a leather chair.
You don't answer her immediately but instead slip a CD into a player. Out pours Duke Ellington's "Satin Doll."
Sally snaps her fingers to get your attention.
You turn to her. "About France—"
"I'd like to apologize for not rescuing you sooner. In less than a minute, I found you lying beside a tombstone. You weren't harmed."
She nods. "I still don't understand what happened to me."
"That's why we're here. I'm going to teach you about the fourth dimension and show you things you've never dreamed of."
She rolls her eyes. "You sound like my ex-husband. All talk."
You hold up your hand in a don't-shoot pose. "Don't worry, you'll like this. Take a seat."
You take a deep breath before starting your lecture. "The fourth dimension corresponds to a direction different from all the directions in our world."
"Isn't time the fourth dimension?"
"Time is one example of a fourth dimension, but there are others. Parallel universes may even exist besides our own in some ghostly manner, and these might be called other dimensions. But I'm interested in a fourth spatial dimension—one that exists in a direction different from up and down, back and front, right and left."
Sally lowers herself evenly into a chair. "That sounds impossible."
"Just listen. Our ordinary space is three-dimensional because all movements can be described in terms of three perpendicular directions." You remain standing and gaze down at Sally. "For example, let's consider the relative position of our hearts. You do have a heart, Sally?"
From your desk drawer, you remove a tape measure and hand one end of it to her. "I can say that your heart is about four feet south of mine, one foot east of mine, and two feet down from mine. In fact, I can specify any location with three types of motion."
She nods, apparently growing more interested in your talk. Outside there is a crack of lightning. You look toward the sky and then back at Sally. The gathering clouds and mists reflected in her eyes make them seem like gray puffs of smoke.
A fly enters your office, so you close a small vent by the window before any other insects can take refuge. "Another way of saying this is that motion in our world has three degrees of freedom." You write three words in big letters on the blackboard:
DEGREES OF FREEDOM
"Yes, I understand." Sally's hand darts out at the fly and captures it. You never knew she could move so fast. "This fly has three different directions it can travel in the room. Now that it is in my hand, it has zero degrees of freedom. I'm holding my hand very still. The fly can't more. It's just stuck at a point in space. It would be the same if I placed it in a tiny box. Now, if I stick the fly in a tube where it can only move back and forth in one direction, then the fly has one degree of freedom" (Fig. 1.1 and 1.2).
"Correct! And if you were to pull off its wings—"
"—and let it crawl around on a plane, it would have two degrees of freedom. Even if the surface is curved, it still lives in a 2-D world with two degrees of freedom because its movement can be described as combinations of two directions of motion: forward/backward and left/right. Since it can't fly, it can't leave the surface of the paper" (Fig. 1.3).
"The surface is a curved 3-D object, but the fly's motion, confined to the surface, is essentially a 2-D motion."
"Sally, you've got it. Likewise, a fly in a tube still lives in a 1-D world, even if you curve the tube into a knot. It still has only one degree of freedom—its motion back and forth. Even an intelligent fly might not realize that the tube was curved."
You trip over a Rubik's cube that you had left on the floor and bump into Sally's chair.
She pushes you away. "Ugh, you made me crush the fly." She tosses it into a wastebasket.
You wave your hand. "It doesn't matter." You pause and return to the discussion. "As I'll show you later, the space we live in may also be curved, just like a twisted tube or a curved piece of paper. However, in terms of our degrees of freedom, we are living in a 3-D world."
Sally holds her fist in front of you, as if about to strike you. "Let me see if I get this. My fist can be described by three numbers: longitude, latitude, and height above sea level. We live in a 3-D world. If we lived in a 4-D space, I would have to specify the location of my fist with a fourth number. In a 4-D world, to find my fist, you could go to the correct longitude, latitude, and height above sea level, and then move into a fourth direction, perpendicular to the rest."
You nod. "Excellent. At each location in my office you could specify different distances in a fourth spatial direction that currently we can't see. It's very hard to imagine such a dimension, just as it would be hard for creatures confined to a plane, and who can only look along the plane, to imagine a 3-D world. Tomorrow I want to do some more reasoning from analogy, because the best way for 3-D creatures to understand the fourth dimension is to imagine how 2-D creatures would understand our world."
Sally taps her hand on your desk. "But how does this explain my encounter at Cherbourg?"
"We'll get to that. By the time your lessons are finished, we're going to see some horrifying stuff...." You reach for a seemingly empty jar on the shelf and hold it in front of Sally's sparkling eyes.
She examines the jar cap, which bas been sealed securely to the jar using epoxy. "There's nothing in here."
Your grin widens. "Not yet."
She shakes her head. "You scare me sometimes."
The Science Behind the Science Fiction
Future historians of science may well record that one of the greatest conceptual revolutions in the twentieth-century science was the realization that hyperspace may be the key to unlock the deepest secrets of nature and Creation itself.
—Michio Kaku, Hyperspace
If we wish to understand the nature of the Universe we have an inner hidden advantage: we are ourselves little portions of the universe and so carry the answer within us.
—Jacques Boivin, The Single Heart Field Theory
Early Dreams and Fears of a Fourth Dimension
Look at the ceiling of your room. From the corner of the room radiate three lines, each of which is the meeting place of a pair of walls. Each line is perpendicular to the other two lines. Can you imagine a fourth line that is perpendicular to the three lines? If you are like most people, the answer is a resounding "no." But this is what mathematics and physics require in setting up a mental construct involving 4-D space.
What does it mean for objects to exist in a fourth dimension? The scientific concept of a fourth dimension is essentially a modern idea, dating back to the 1800s. However, the philosopher Immanuel Kant (1724-1804) considered some of the spiritual aspects of a fourth dimension:
A science of all these possible kinds of space would undoubtedly be the highest enterprise which a finite understanding could undertake in the field of geometry.... If it is possible that there could be regions with other dimensions, it is very likely that a God had somewhere brought them into being. Such higher spaces would not belong to our world, but form separate worlds.
Euclid (c. 300 B.C.), a prominent mathematician of Greco-Roman antiquity, understood that a point has no dimension at all. A line has one dimension: length. A plane had two dimensions. A solid had three dimensions. But there he stopped—believing nothing could have four dimensions. The Greek philosopher Aristotle (384-322 B.C.) echoed these beliefs in On Heaven:
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.
Aristotle used the argument of perpendiculars to prove the impossibility of a fourth dimension. First he drew three mutually perpendicular lines, such as you might see in the corner of a cube. He then put forth the challenge to his colleagues to draw a fourth line perpendicular to the first three. Since there was no way to make four mutually perpendicular lines, he reasoned that the fourth dimension is impossible.
It seems that the idea of a fourth dimension sometimes made philosophers and mathematicians a little nervous. John Wallis (1616-1703)—the most famous English mathematician before Isaac Newton and best known for his contributions to calculus's origin—called the fourth dimension a "monster in nature, less possible than a Chimera or Centaure." He wrote, "Length, Breadth, and Thickness, take up the whole of Space. Nor can fansie imagine how there should be a Fourth Local Dimension beyond these three."
Similarly, throughout history, mathematicians have called novel geometrical ideas "pathological" or "monstrous." Physicist Freeman Dyson recognized this for fractals, intricate structures that today have revolutionized mathematics and physics but in the past were treated with trepidation:
A great revolution separates the classical mathematics of the 19th century from the modern mathematics of the 20th. Classical mathematics had its roots in the regular geometrical structures of Euclid and Newton. Modern mathematics began with Cantor's set theory and Peano's space-filling curve. Historically the revolution was forced by the discovery of mathematical structures that did not fit the patterns of Euclid and Newton. These new structures were regarded as "pathological," as a "gallery of monsters," kin to the cubist painting and atonal music that were upsetting established standards of taste in the arts at about the same time. The mathematicians who created the monsters regard them as important in showing that the world of pure mathematics contains a richness of possibilities going far beyond simple structures that they saw in Nature. Twentieth-century mathematics flowered in the belief that it had transcended completely the limitation imposed by its natural origins. But Nature has played a joke on the mathematicians. The 19th-century mathematicians may have been lacking in imagination, but Nature was not. The same pathological structures that the mathematicians invented to break loose from 19th-century naturalism turned out to be inherent in familiar objects all around us. (Science, 1978)
Karl Heim—a philosopher, theologian, and author of the 1952 book Christian Faith and Natural Science—believes the fourth dimension will remain forever beyond our grasp:
The progress of mathematics and physics impels us to fly away on the wings of the poetic imagination out beyond the frontiers of Euclidean space, and to attempt to conceive of space in which more than three coordinates can stand perpendicularly to one another. But all such endeavors to fly out beyond our frontiers always end with out falling back with singed wings on the ground of our Euclidean three-dimensional space. If we attempt to contemplate the fourth dimension we encounter an insurmountable obstacle, an electrically charged barb-wire fence.... We can certainly calculate with [higher-dimensional spaces]. But we cannot conceive of them. We are confined within the space in which we find ourselves when we enter into our existence, as though in a prison. Two-dimensional beings can believe in a third dimension. But they cannot see it.
Although philosophers have suggested the implausibility of a fourth dimension, you will see in the following sections that higher dimensions probably provide the basis for the existence of everything in our universe.
Hyperspace and Intrinsic Geometry
The fact that our universe, like the surface of an apple, is curved in an unseen dimension beyond our spatial comprehension has been experimentally verified. These experiments, performed on the path of light beams, shows that starlight is bent as it moves across the universe.
—Michio Kaku, Hyperspace
Imagine alien creatures, shaped like hairy pancakes, wandering along the surface of a large beach ball. The inhabitants are embedded in the surface, like microbes floating in the thin surface of a soap bubble. The aliens call their universe "Zarf." To them, Zarf appears to be flat and two-dimensional partly because Zarf is large compared to their bodies. However, Leonardo, one of their brilliant scientists, comes to believe that Zarf is really finite and curved in something he calls the third dimension. He even invents two new words, "up" and "down," to describe motion in the invisible third dimension. Despite skepticism from his friends, Leonardo travels in what seems like a straight line around his universe and returns to his starting point—thereby proving that his universe is curved in a higher dimension. During Leonardo's long trip, he doesn't feel as if he's curving, although he is curving in a third dimension perpendicular to his two spatial dimension. Leonardo even discovers that there is a shorter route from one place to another. He tunnels through Zarf from point A to point B, thus creating what physicists call a "wormhole." (Traveling from A to B along Zarf's surface requires more time than a journey that penetrates Zarf like a pin through a ball.) Later Leonardo discovers that Zarf is one of many curved worlds floating in three-space. He conjectures that it may one day be possible to travel to these other worlds.
Now suppose that the surface of Zarf were crumpled like a sheet of paper. What would Leonardo and his fellow pancake-shaped aliens think about their world? Despite the crumpling, the Zarfians would conclude that their world was perfectly flat because they lived their lives confined to the crumpled space. Their bodies would be crumpled without their knowing it.
This scenario with curved space is not as zany as it may sound. Georg Bernhard Riemann (1826-1866), the great nineteenth-century geometer, thought constantly on these issues and profoundly affected the development of modern theoretical physics, providing the foundation for the concepts and methods later used in relativity theory. Riemann replaced the 2-D world of Zarf with our 3-D world crumpled in the fourth dimension. It would not be obvious to us that our universe was warped, except that we might feel its effects. Riemann believed that electricity, magnetism, and gravity are all caused by crumpling of our 3-D universe in an unseen fourth dimension. If our space were sufficiently curved like the surface of a sphere, we might be able to determine that parallel lines can meet just as longitude lines do on a globe), and the sum of angles of a triangle can exceed 180 degrees (as exhibited by triangles drawn on a globe).
Around 300 B.C. Euclid told us that the sum of the three angles in any triangle drawn on a piece of paper is 180 degrees. However, this is true only on a flat piece of paper. On the spherical surface, you can draw a triangle in which each of the angles is 90 degrees! (To verify this, look at a globe and lightly trace a line along the equator, then go down a longitude line to the South Pole, and then make a 90-degree turn and go back up another longitude line to the equator. You have formed a triangle in which each angle is 90 degrees.)
Let's return to our 2-D aliens on Zarf. If they measured the sum of the angles in a small triangle, that sum could be quite close to 180 degrees even in a curved universe, but for large triangles the results could be quite different because the curvature of their world would be more apparent. The geometry discovered by the Zarfians would be the intrinsic geometry of the surface. This geometry depends only on their measurements made along the surface. In the mid-nineteenth century in our own world, there was considerable interest in non-Euclidean geometries, that is, geometries where parallel lines can intersect. When physicist Hermann von Helmholtz (1821-1894) wrote about this subject, he had readers imagine the difficulty of a 2-D creature moving along a surface as it tried to understand its world's intrinsic geometry without the benefit of a 3-D perspective revealing the world's curvature properties all at once. Bernhard Riemann also introduced intrinsic measurements on abstract spaces and did not require reference to a containing space of higher dimension in which material objects were "curved."
The extrinsic geometry of Zarf depends on the way the surface sits in a high-dimensional space. As difficult as it may seem, it is possible for Zarfians to understand their extrinsic geometry just by making measurements along the surface of their universe. In other words, a Zarfian could study the curvature of its universe without ever leaving the universe—just as we can learn about the curvature of our universe, even if we are confined to it. To show that our space is curved, perhaps all we have to do is measure the sums of angles of large triangles and look for sums that are not 180 degrees. Mathematical physicist Carl Friedrich Gauss (1777-1855)—one of the greatest mathematicians of all time—actually attempted this experiment by shining lights along the tops of mountains to form one big triangle. Unfortunately, his experiments were inconclusive because the angle sums were 180 degrees up to the accuracy of the surveying instruments. We still don't know for sure whether parallel lines intersect in our universe, but we do know that light rays should not be used to test ideas on the overall curvature of space because light rays are deflected as they pass nearby massive objects. This means that light bends as it passes a star, thus altering the angle sums for large triangles. However, this bending of starlight also suggests that pockets of our space are curved in an unseen dimension beyond our spatial comprehension. Spatial curvature is also suggested by the planet Mercury's elliptical orbit around the sun that shifts in orientation, or precesses, by a very small amount each year due to the small curvature of space around the sun. Albert Einstein argued that the force of gravity between massive objects is a consequence of the curved space nearby the mass, and that traveling objects merely follow straight lines in this curved space like longitude lines on a globe.
In the 1980s and 1990s various astrophysicists have tried to experimentally determine if our entire universe is curved. For example, some have wondered if our 3-D universe might be curved back on itself in the same way a 2-D surface on a sphere is curved back on itself. We can restate this in the language of the fourth dimension. In the same way that the 2-D surface of the Earth is finite but unbounded (because it is bent in three dimensions into a sphere), many have imagined the 3-D space of our universe as being bent (in some 4-D space) into a 4-D sphere called a hypersphere. Unfortunately, astrophysicists' experimental results contain uncertainties that make it impossible to draw definitive conclusions. The effort continues.
A Loom with Tiny Strings
In heterotic string theory ... the right-handed bosons (carrier particles) go counterclockwise around the loop, their vibrations penetrating 22 compacted dimensions. The bosons live in a space of 26 dimensions (including time) of which 6 are the compacted "real" dimensions, 4 are the dimensions of ordinary space-time, and the other 16 are deemed "interior spaces"—mathematical artifacts to make everything work out right.
—Martin Gardner, The Ambidextrous Universe
String theory may be more appropriate to departments of mathematics or even schools of divinity. How many angels can dance on the head of a pin? How many dimensions are there in a compacted manifold thirty powers of ten smaller than a pinhead? Will all the young Ph.D.s, after wasting years on string theory, be employable when the string snaps?
—Sheldon Glashow, Science
String theory is twenty-first century physics that fell accidentally into the twentieth century.
—Edward Witten, Science
Various modern theories of hyperspace suggest that dimensions exist beyond the commonly accepted dimensions of space and time. As alluded to previously, the entire universe may actually exist in a higher-dimensional space. This idea is not science fiction: in fact, hundreds of international physics conferences have been held to explore the consequences of higher dimensions. From an astrophysical perspective, some of the higher-dimensional theories go by such impressive sounding names as Kaluza-Klein theory and supergravity. In Kaluza-Klein theory, light is explained as vibrations in a higher spatial dimension. Among the most recent formulations of these concepts is superstring theory that predicts a universe of ten dimensions—three dimensions of space, one dimension of time, and six more spatial dimensions. In many theories of hyperspace, the laws of nature become simpler and more elegant when expressed with these several extra spatial dimensions.
The basic idea of string theory is that some of the most basic particles, like quarks and fermions (which include electrons, protons, and neutrons), can be modeled by inconceivably tiny, one-dimensional line segments, or strings. Initially, physicists assumed that the strings could be either open or closed into loops, like rubber bands. Now it seems that the most promising approach is to regard them as permanently closed. Although strings may seem to be mathematical abstractions, remember that atoms were once regarded as "unreal" mathematical abstractions that eventually became observables. Currently, strings are so tiny there is no way to "observe" them. Perhaps we will never be able to observe them.
In some string theories, the loops of string move about in ordinary three-space, but they also vibrate in higher spatial dimensions perpendicular to our world. As a simple metaphor, think of a vibrating guitar string whose "notes" correspond to different "typical" particles such as quarks and electrons along with other mysterious particles that exist only in all ten dimensions, such as the hypothetical graviton, which conveys the force of gravity. Think of the universe as the music of a hyperdimensional orchestra. And we may never know if there is a hyperBeethoven guiding the cosmic harmonies.
Whenever I read about string theory, I can't help thinking about the Kabala in Jewish mysticism. Kabala became popular in the twelfth and following centuries. Kabalists believe that much of the Old Testament is in code, and this is why scripture may seem muddled. The earliest known Jewish text on magic and mathematics, Sefer Yetzira (Book of Creation), appeared around the fourth century A.D. It explained creation as a process involving ten divine numbers or sephiroth. Kabala is based on a complicated number mysticism whereby the primordial One divides itself into ten sephiroth that are mysteriously connected with each other and work together. Twenty-two letters of the Hebrew alphabet are bridges between them (Fig. 1.4).
The sephiroth are ten hypostatized attributes or emanations allowing the infinite to meet the finite. ("Hypostatize" means to make into or treat as a substance—to make an abstract thing a material thing.) According to Kabalists, by studying the ten sephiroth and their interconnections, one can develop the entire divine cosmic structure.
Similarly, physical reality may be the hypostatization of these mathematical constructs called "strings." As I mentioned, strings, the basic building blocks of nature, are not tiny particles but unimaginably small loops and snippets loosely resembling strings—except that strings exist in a strange, 10-D universe. The current version of the theory took shape in the late 1960s. Using hyperspace theory, "matter" is viewed as vibrations that ripple through space and time. From this follows the idea that everything we see, from people to planets, is nothing but vibrations in hyperspace.
In the last few years, theoretical physicists have been using strings to explain all the forces of nature—from atomic to gravitational. Although string theory describes elementary particles as vibrational modes of infinitesimal strings that exist in ten dimensions, many of you may be wondering how such things exist in our 3-D universe with an additional dimension of time. String theorists claim that six of the ten dimensions are "compactified"—tightly curled up (in structures known as Calabi-Yau spaces) so that the extra dimensions are essentially invisible.
As technically advanced as superstring theory sounds, superstring theory could have been developed a long time ago according to string-theory guru Edward Witten, a theoretical physicist at the Institute for Advanced Study in Princeton. For example, he indicates that it is quite likely that other civilizations in the universe discovered superstring theory and then later derived Einstein-like formulations (which in our world predate string theory by more than half a century). Unfortunately for experimentalists, superstrings are so small that they are not likely to ever be detectable by humans. If you consider the ratio of the size of a proton to the size of the solar system, this is the same ratio that describes the relative size of a superstring to a proton.
John Horgan, an editor at Scientific American, recently published an article describing what other researchers have said of Witten and superstrings in ten dimensions. One researcher interviewed exclaimed that in sheer mathematical mind power, Edward Witten exceeds Einstein and bas no rival since Newton. So complex is string theory that when a Nobel Prize-wining physicist was asked to comment on the importance of Witten's work, he said that he could not understand Witten's recent papers; therefore, he could not ascertain how brilliant Witten is!
Recently, humanity's attempt to formulate a "theory of everything" includes not only string theory but membrane theory, also known as M-theory. In the words of Edward Witten (whom Life magazine dubbed the sixth most influential American baby boomer), "M stands for Magic, Mystery, or Membrane, according to taste." In this new theory, life, the universe, and everything may arise from the interplay of membranes, strings, and bubbles in higher dimensions of spacetime. The membranes may take the form of bubbles, be stretched out in two directions like a sheet of rubber, or wrapped so tightly that they resemble a string. The main point to remember about these advanced theories is that modern physicists continue to produce models of matter and the universe requiring extra spatial dimensions.
In this book, I'm interested primarily in a fourth spatial dimension, although various scientists have considered other dimensions, such as time, as a fourth dimension. In this section, I digress and speak for a moment on time and what it would be like to live outside the flow of time. Readers are encouraged to consult my book Time: A Traveler's Guide for an extensive treatise on the subject.
Einstein's theory of general relativity describes space and time as a unified 4-D continuum called "spacetime." The 4-D continuum of Einstein's relativity in which three spatial dimensions are combined with one dimension of time is not the same as hyperspace consisting of four spatial coordinates. To best understand this, consider yourself as having three spatial dimensions—height, width, and breadth. You also have the dimension of duration—how long you last. Modern physics views time as an extra dimension; thus, we live in a universe having (at least) three spatial dimensions and one additional dimension of time. Stop and consider some mystical implications of spacetime. Can something exist outside of spacetime? What would it be like to exist outside of spacetime? For example, Thomas Aquinas believed God to be outside of spacetime and thus capable of seeing all of the universe's objects, past and future, in one blinding instant. An observer existing outside of time, in a region called "hypertime," can see the past and future all at once.
There are many other examples of beings in literature and myth who live outside of spacetime. Many people living in the Middle Ages believed that angels were nonmaterial intelligences living by a time different from humans, and that God was entirely outside of time. Lord Byron aptly describes these ideas in the first act of his play Cain, A Mystery, where the fallen angel Lucifer says:
With us acts are exempt from time, and we
Can crowd eternity into an hour,
Or stretch an hour into eternity.
We breathe not by a mortal measurement—
But that's a mystery.
A direct analogy involves an illustration of an "eternitygram" representing two discs rolling toward one another, colliding, and rebounding. Figure 1.5 shows two spatial dimensions along with the additional dimension of time. You can think of successive instants in time as stacks of movie frames that form a 3-D picture of hypertime in the eternitygram. Figure 1.5 is a "timeless" picture of colliding discs in eternity, an eternity in which all instants of time lie frozen like musical notes on a musical score. Eternitygrams are timeless. Hyperbeings looking at the discs in this chunk of spacetime would see past, present, and future all at once. What kind of relationship with humans could a creature (or God) have who lives completely outside of time? How could they relate to us in our changing world? One of my favorite modern examples of God's living outside of time is described in Anne Rice's novel Memnoch the Devil. At one point, Lestat, Anne Rice's protagonist, says, "I saw as God sees, and I saw as if Forever and in All Directions." Lestat looks over a balustrade in Heaven to see the entire history of our world:
... the world as I had never seen it in all its ages, with all its secrets of the past revealed. I had only to rush to the railing and I could peer down into the time of Eden or Ancient Mesopotamia, or a moment when Roman legions had marched through the woods of my earthly home. I would see the great eruption of Vesuvius spill its horrid deadly ash down upon the ancient living city of Pompeii. Everything there to be known and finally comprehended, all questions settled, the smell of another time, the taste of it....
If all our movements through time were somehow fixed like tunnels in the ice of spacetime (as in the eternitygram in Fig. 1.5), and all that "moved" was our perception shifting through the ice as time "passes," we would still see a complex dance of movements even though nothing was actually moving. Perhaps an alien would see this differently. In some sense, all our motions may be considered fixed in the geometry of spacetime, with all movement and change being an illusion resulting from our changing psychological perception of the moment "now." Some mystics have suggested that spacetime is like a novel being "read" by the soul—the "soul" being a kind of eye or observer that stands outside of spacetime, slowly gazing along the time axis.
Note 8 describes the metamorphosis of time into a spatial dimension during the early evolution of our universe. At the point when time loses its time-like character, the universe is in the realm of what physicists call "imaginary time."
|1||Degrees of Freedom||3|
|2||The Divinity of Higher Dimensions||23|
|3||Satan and Perpendicular Worlds||53|
|4||Hyperspheres and Tesseracts||81|
|6||The Gods of Hyperspace||141|
|App. A||Mind-Bending Four-Dimensional Puzzles||169|
|App. B||Higher Dimensions in Science Fiction||175|
|App. C||Banchoff Klein Bottle||185|
|App. E||Four-Dimensional Mazes||190|
|App. F||Smorgasbord for Computer Junkies||192|
|App. G||Evolution of Four-Dimensional Beings||196|
|App. H||Challenging Questions for Further Thought||199|
|App. I||Hyperspace Titles||212|
|About the Author||233|
A few years ago I was walking in a field when I came upon a large skull. It was probably from a bear, although I like to imagine it was part of the remains of a prehistoric mammal that once roamed Westchester County, New York. I'm a collector of prehistoric skulls. In my office, I have a skull of a saber-toothed tiger. This killing machine had huge, daggerlike canine teeth and a mouth that could open 90 degrees to clear the sabers for their killing bite.
When I run my fingers lingeringly over the skulls, I am sometimes reminded of stars in the heavens. Without stars, there could be no skulls. The elements in bone, like calcium, were first created in the hot stellar furnaces and then blown into space when the stars died. Without stars there would be no elements heavier than hydrogen and helium, and, therefore, life would never have evolved. There would be no planets, no microbes, no plants, no tigers, no humans.
In my book, you'll do a little armchair space travel, rub elbows with alien life forms, and glimpse the furthest corners of our uncharted universe. Stars have fascinated humankind since the dawn of history and have allowed us to transcend ordinary lives in our literature, art, and religion. Where did we come from? What is the universe's ultimate fate? Are there other universes we can never see? Was our universe designed by a god?
In about 5 billion years, the hydrogen fuel in our sun will be exhausted in its core, and the sun will begin to die and dramatically expand, becoming a red giant. At some point, our oceans will boil away. As Freeman Dyson once said, "No matter how deep we burrow into the Earth...we can only postpone by a few million years our miserable end." Where will humans be, a few billion years from now, at the end of the world? It seems so sad. However, I don't think we have to mourn for humanity. In a few billion years, humans will probably have downloaded their minds to computers, left the solar system in some great diaspora, and sought their salvation in the stars.
You can tell that my book is an amalgam of science fiction, science fact, religion, art and science. It's also a serious astronomy primer covering all the basics of stellar structure and evolution and also on creative theories about our place in this grand universe. I've touched on similar cosmic topics on the borderlands of science before, and I hope you'll take a look at some of my other books, for example: Black Holes: A Traveler's Guide, Time: A Traveler's Guide, Surfing Through Hyperspace, The Science of Aliens, and The Paradox of God and the Science of Omniscience. Please visit my web page, Pickover.com, and join our discussions on The Stars of Heaven and all my other books. I leave you with an appropriate Serbian proverb to get your mind in gear: "Be humble for you are made of dung. Be noble for you are made of stars." (Clifford Pickover)
Posted March 9, 2000
Everthing you've ever wanted to know about the 4th, 5th and nth dimensions but were afraid to ask! Did you know that you could squeeze a blue whale into a 24-D sphere with a radius of 2 inches? Well, you will after reading this. I finished it in one sitting. An absolutely mind-blowing book. My brain feels like it's been dragged through a 5-dimensional hedge backwards.
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