"This is an invigorating book, a short but spirited slalom for the mind." — Timothy Ferris, The New York Times Book Review
"Highly readable. One is reminded of the breadth and depth of Hofstadter's Gödel, Escher, Bach." — Science
"Anyone with even a minimal interest in mathematics and fantasy will find The Fourth Dimension informative and mind-dazzling... [Rucker] plunges into spaces above three with a zest and energy that is breathtaking." — Martin Gardner
"Those who think the fourth dimension is nothing but time should be encouraged to read The Fourth Dimension, along with anyone else who feels like opening the hinges of his mind and letting in a bit of fresh air." — John Sladek, Washington Post Book World
"A mine of mathematical insights and a thoroughly satisfying read." — Paul Davies, Nature Magazine
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About the Author
Rudy Rucker is an American mathematician, computer scientist, philosopher, and author of 31 books of nonfiction, science fiction, and short stories. He is Professor Emeritus of Mathematics and Computer Science at San Jose State. His other Dover books are Geometry, Relativity and the Fourth Dimension, and Mind Tools: The Five Levels of Mathematical Reality.
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The Fourth Dimension
Toward a Geometry of Higher Reality
By Rudy Rucker, David Povilaitis
Dover Publications, Inc.Copyright © 2014 Rudy Rucker
All rights reserved.
A New Direction
Is this all there is? Struggle, loneliness, disease, and death ... Is this all there is? Life can seem so chaotic, so dreary, so grindingly hard. Who among us has not dreamt of some higher reality, some transcendent level of meaning and peace?
There actually is such a higher reality ... And it is not so very hard to reach. For many, the fourth dimension has served as a gateway into it. But what is the fourth dimension?
No one can point to the fourth dimension, yet it is all around us. Philosophers and mystics meditate upon it; physicists and mathematicians calculate with it. The fourth dimension is part and parcel of many respected scientific theories, yet it is also of great use in such disreputable fields as spiritualism and science fiction.
The fourth dimension is a direction different from all the directions in normal space. Some say that time is the fourth dimension ... And this is, in a sense, true. Others say that the fourth dimension is a hyperspace direction quite different from time ... This is also true.
There are, in fact, many higher dimensions. One of these higher dimensions is time, another higher dimension is the direction in which space is curved, and still another higher dimension may lead toward some utterly different universes existing parallel to our own.
At the deepest level, our world can be regarded as a pattern in infinite-dimensional space, a space in which we and our minds move like fish in water.
Ordinarily, of course, we say that we live in three-dimensional space. What precisely is meant by this? Why three? Watch the wheeling flight of swallows chasing gnats at dusk. Mathematically speaking, these lovely sweeping curves are of great complexity. But it is possible to break any such space curve down into three types of motion: east / west, north / south, and up / down. By combining the three mutually perpendicular types of motion, one can trace out any possible curve in our space. No more than three directions are needed, and no less than three directions will do — hence we call our space three-dimensional.
This fact is illustrated in two dimensions by a toy that was popular a few years ago, the Etch-A-Sketch. The underside of the Etch-A-Sketch's glass screen is covered by some silvery dust. Turning the knobs moves a stylus under the screen, and the stylus scrapes off dust, leaving dark trails. The left-hand knob moves the stylus in the left / right direction, and the right-hand knob moves the stylus in the up / down direction. If one twists the two knobs at the same time, one can draw any two-dimensional curve at all.
It is not too hard to imagine a three-dimensional Etch-A-Sketch that would, let us say, move a brightly flaring sparkler about in a dark room. As the image of a sparkler stays on the retina for a few seconds, one could thus have the experience of seeing three-dimensional curves generated by twiddling three different knobs: left / right, up / down, and back / forth.
Speaking of sparklers, there is a nice picture inside the cover of the Rolling Stones album Black and Blue. It is a time exposure of the five Stones waving sparklers. Bill Wy-man traces a flat, tightening spiral. Ronnie Wood produces a messy figure eight. Charlie Watts slowly and patiently draws a big letter O. Keith starts out high and lets the sparkler fall in a tired zigzag. And Mick ... Ah, Mick ... Mick traces the only truly three-dimensional curve in the group: a complex rodeo pattern of swoops and loops. Waving sparklers in the dark is a good way to really savor our space's three-dimensionality.
Another way of expressing all this is to say that motion in our space has three degrees of freedom. At any instant, a bird has three essentially different ways to alter its flight: speed up / slow down, wheel left / wheel right, climb / dive. Although we can wave our sparklers with just as much freedom, we cannot really move our bodies around in this way. Someone hiking in the hills moves up and down with the roll of the land ... Yet in terms of control, he has only two degrees of freedom: forward / backward, and left / right. One can, of course, jump up and down a bit, but because of gravity, the effects of this are more or less negligible.
The point I am making here is that in terms of degrees of freedom, motion on the Earth's bumpy surface is basically two-dimensional. The surface itself is a curved three-dimensional object, granted. But any motion that is confined to this surface is essentially a two-dimensional motion. It could be that mankind's perennial dream of flight is a hunger for more dimensions, for more degrees of freedom. The average person only experiences three-dimensional body motion when he or she swims underwater.
Driving a car involves sacrificing yet another degree of freedom. One speeds up or slows down (possibly even reversing direction), but that's all. The road itself is a space curve in three-dimensional space, but motion that is confined to this particular curve is basically one-dimensional.
As we will see later, the space we live in is also curved: curved like a hillside, twisted like a mountain road. But, in terms of degrees of freedom, it is clear that our space is three-dimensional. Another way of expressing this is to point out that we can give any location above the Earth's surface by using three numbers: longitude, latitude, and height above sea level. Again, if we're both in a city, I might typically tell you how to find me by giving you three bits of information. "Walk four blocks uptown, turn right and go two blocks crosstown, then go in the building there and ride the elevator to the twenty-fifth floor."
Now, if our space were four-dimensional, such instructions would usually need a fourth component. "Get out of the elevator and shift through six levels of reality." just as there are many floors over a given location on a city's two-dimensional street grid, we can abstractly imagine there being many different "reality levels" available at each of our space locations. In a sense this is true ... Even though we're in the same room, I might ask, "Where's your head at?" in an effort to get into closer contact.
Let us pursue this line of thought a bit further. Imagine that objects in space could exist at different reality levels, and to make it quite concrete, suppose that each level has its characteristic color — ranging from red through green to blue. Assume that objects interact only with objects of the same color. A person on the twenty-fifth floor will not stumble over someone on the second floor; we propose that a blue person can pass right through a green person.
In this example, reality level or color would constitute a fourth dimension. The three space dimensions plus the color dimension would make up a sort of four-dimensional space. A typical person would probably exist on several levels at once. Waving a four-dimensional sparkler here would involve having the light's color, or reality level, change in some complicated way. This would be one way of beginning to think about a four-dimensional space.
Another, somewhat similar, approach is to propose using time as a fourth dimension. If, after all, I really want to see you, it is not enough to tell you how many blocks and how many floors to travel. I need to tell you how soon to show up. Maybe I will not be at the rendezvous for another hour ... And maybe I'll then stay there for only fifteen minutes. To really specify an event, it is not enough to give its longitude, latitude, and height above sea level. One must also state when it occurs. Just as a blue person can walk through a green person, a 2:00 A.M. person can walk through a 6:00 P.M. person. In terms of the waving sparklers, the dimension of time comes into play when one notes how rapidly the sparkler moves along each part of its path.
But somehow it misses the mark to represent the fourth dimension by reality level, by color, or by time. What is really needed here is the concept of a fourth space dimension. It is very hard to visualize such a dimension directly. Off and on for some fifteen years, I have tried to do so. In all this time I've enjoyed a grand total of perhaps fifteen minutes' worth of direct vision into four-dimensional space. Nevertheless, I feel that I understand the fourth dimension very well. How can this be? How can we talk productively about something that is almost impossible to visualize?
The key idea is to reason by analogy. The fourth dimension is to three-dimensional space as the third dimension is to two-dimensional space. 4-D:3D::3-D:2-D. This particular analogy is one of the oldest head tricks known to man. Plato was the first to present it, in his famous allegory of the cave.
Here Plato asks us to imagine a race of men who are chained up in an underground den, chained in such a way that all they can ever look at is shadows on their cave's wall. Behind the men is a low ramp, and behind that a fire. Objects move back and forth on the ramp, and the fire casts shadows of these objects on the cave's wall. The prisoners think that these shadows are the only reality ... They do not even realize that they have three-dimensional bodies. They talk to each other, but hearing the echoes bounce off the wall, they assume that they and their fellows are also shadows.
There are several interesting features in Plato's allegory. It is particularly striking that the prisoners actually think that they are their own shadows. This is interesting because it suggests the idea that a person is really some higher-dimensional soul that influences and watches this "shadow world" of three-dimensional objects.
To bring this particular idea home, let us update Plato's cave allegory a bit. Imagine a very large TV screen that displays full-color computer-generated images of people and objects moving about. Now imagine some people who have from birth been chained motionless in front of the giant tube. Electrodes run from their nervous systems to the image-generating computer, and for each person there is a particular TV personality he or she can control. These prisoners would mistake the flat phosphorescent TV screen for reality.
So, one conclusion to draw from Plato's allegory is that we should not be too sure that our everyday view of the world is the most correct and most comprehensive view possible. Common sense can be misleading, and there may be a great deal more to reality than meets the eye.
An even more important aspect of Plato's allegory is that this allegory introduces the notion of a two-dimensional world. Insofar as the prisoners in the cave really think that they are shadows on the wall, they are viewing themselves as two-dimensional patterns. What would it be like to be a two-dimensional being? Would a two-dimensional being be able to imagine a third dimension?
In the next chapter we will talk about an imaginary two-dimensional world known as Flatland; and we will study the adventures of A Square, Flatland's most famous citizen. A Square's path to an understanding of the third dimension is, we will see, a guide for our own attempts to understand the fourth dimension.CHAPTER 2
Flatland, first published in 1884, is the story of a square who takes a trip into higher dimensions. A century has passed and people are still talking about it. The author of Flatland was a Victorian schoolmaster named Edwin Abbott Abbott. Given the curious fact that his middle and last names were identical, it seems possible that Abbott might have been nicknamed Abbott Squared or A Squared. Thus, it may be that Abbott felt a considerable degree of identification with A Square, the hero of Flatland. After all, Abbott's life was, in some ways, as strictly regulated as the life of a two-dimensional Flatlander.
Edwin Abbott Abbott was born in London on December 20, 1838, the son of Edwin Abbott, head of the Philological School at Marylebone. Abbott attended the City of London School as a boy. He went on to Cambridge, was ordained a minister, was married, and at the age of twenty-seven returned to the City of London School as headmaster. He wrote a number of books on grammar and theology, books with titles like How to Parse and Letters on Spiritual Christianity. Flatland was his one venture into fantasy.
The book works on three levels. Most obviously, it is a satire on the staid and heartless society of the Victorians. "Irregulars" (cripples) are put to death, women have no rights at all, and when A Square tries to teach his fellows about the third dimension he is imprisoned. The second level of meaning in Flatland is scientific. By thinking about A Square's difficulties in understanding the third dimension, we become better able to deal with our own problems with the fourth dimension. Finally, at the deepest level, we can perhaps view Flatland as Abbott's circuitous way of trying to talk about some intense spiritual experiences. A Square's trip into higher dimensions is a perfect metaphor for the mystic's experience of higher reality.
Flatland is a plane inhabited by creatures that slide about. We might think of them as being like coins on a tabletop. Alternatively, we could think of them as colored patterns in a soap film, or ink spots in a sheet of paper.
In Flatland, the lower classes are triangles with only two sides equal. The upper classes are regular polygons, that is, figures with all sides equal. The more sides one has, the greater one's social standing. The highest caste of all consists of polygons with so many sides that they are indistinguishable from perfect circles.
As mentioned above, Flatland is more than just a book about dimensions. In somewhat the same manner as Gulliver's Travels, it satirizes the attitudes of the society in which its author lived. In our Western culture, women have probably never been at such a disadvantage as in the nineteenth century. Accordingly, the women of Flatland are not even skinny triangles: they are but lines, infinitely less respected than the priestly circles. Abbott, of course, realizes the injustice of this. When a sphere from "Space-land" visits Flatland, he has this to say: "It is not for me to classify human faculties according to merit. Yet many of the best and wisest in Spaceland think more of the affections than of the understanding, more of your despised Straight Lines than of your be lauded Circles."
An initial question about Flatland is the problem of how these lines and polygons can see anything at all. If you were to put a number of cardboard shapes on a tabletop and then lower your eye to the plane of the table, you would really see just a bunch of line segments. How can Flatlanders tell a triangle from a square? How do they build up the idea of a two-dimensional world from their one-dimensional retinal images?
Abbott reports that the space of Flatland is permeated with a thin haze. Because of this, the glowing sides of the polygons shade off rapidly into dimness. If you are looking at the corner of a triangle and the corner of a pentagon, you can tell them apart because the triangle's sides shade off more rapidly.
This may seem a bit artificial, but just stop to consider this: our retinal images of the world are two-dimensional patterns, yet we can distinguish a wide range of three-dimensional objects. If I look, for instance, at a sphere and a flat disk, I can tell them apart by their shading. Another important way in which we notice our world's three-dimensionality is by the fact that objects can move behind and in front of each other. If, looking out a restaurant window, I see a person walk in front of my car, I do not assume that he is somehow dematerializing my car. I recognize that there is a third dimension of space, and that in this dimension the sidewalk is closer than the street. Just as we can build up a mental image of our three-dimensional world, the Flatlanders have adequate images of their two-dimensional world.
A Square's dimensional adventures begin when he has a dream, a dream of Lineland:
I saw before me a vast multitude of small Straight Lines (which I naturally assumed to be Women) interspersed with other Beings still smaller and of the nature of lustrous points — all moving to and fro in one and the same Straight Line, and, as nearly as I could judge, with the same velocity.
A noise of confused multitudinous chirping or twittering issued from them at intervals as long as they were moving; but sometimes they ceased from motion, and then all was silence.
Approaching one of the largest of what I thought to be Women, I accosted her, but received no answer. A second and a third appeal on my part were equally ineffectual. Losing patience at what appeared to me intolerable rudeness, I brought my mouth into a position full in front of her mouth so as to intercept her motion, and loudly repeated my question, "Woman, what signifies this concourse, and this strange and confused chirping, and this monotonous motion to and fro in one and the same Straight Line?"
"I am no Woman," replied the small Line: "I am the Monarch of the world."
Although all the Linelanders can see of each other is a single point, they have a very good sense of hearing, and can estimate how far away each of their fellows is. The men have a voice at either end: bass on the left, tenor on the right. By noting the time lag between the two voices it is possible to tell how long a given male Linelander is. The poor women, of course, are just points!
A Square tries to tell the king about the second dimension. The king doesn't understand, and asks A Square to move in the direction of the mysterious second dimension. A Square complies, and moves right through the space of Lineland. (In figure 12 I have labeled the king's bass and tenor ends.) Naturally enough, the king simply perceives this "motion" as a segment that appears out of nowhere, stays for a minute, and then disappears all at once. The king denies the reality of the second dimension, A Square loses his temper, the dream ends.
Excerpted from The Fourth Dimension by Rudy Rucker, David Povilaitis. Copyright © 2014 Rudy Rucker. Excerpted by permission of Dover Publications, Inc..
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Table of Contents
ContentsPreface to the Dover Edition,
Foreword by Martin Gardner,
I. The Fourth Dimension,
1. A New Direction,
3. Pictures of the Gone World,
4. Through the Looking Glass,
5. Ghosts from Hyperspace?,
6. What We're Made Of,
7. The Shape of Space,
8. Magic Doors to Other Worlds,
III. How To Get There,
9. Spacetime Diary,
10. Time Travel and Telepathy,
11. What Is Reality?,