This is a highly readable, popular exposition of the fourth dimension and the structure of the universe. A remarkable pictorial discussion of the curved space-time we call home, it achieves even greater impact through the use of 141 excellent illustrations. This is the first sustained visual account of many important topics in relativity theory that up till now have only been treated separately.
Finding a perfect analogy in the situation of the geometrical characters in Flatland, Professor Rucker continues the adventures of the two-dimensional world visited by a three-dimensional being to explain our three-dimensional world in terms of the fourth dimension. Following this adventure into the fourth dimension, the author discusses non-Euclidean geometry, curved space, time as a higher dimension, special relativity, time travel, and the shape of space-time. The mathematics is sound throughout, but the casual reader may skip those few sections that seem too purely mathematical and still follow the line of argument. Readable and interesting in itself, the annotated bibliography is a valuable guide to further study.
Professor Rucker teaches mathematics at the State University of New York in Geneseo. Students and laymen will find his discussion to be unusually stimulating. Experienced mathematicians and physicists will find a great deal of original material here and many unexpected novelties. Annotated bibliography. 44 problems.
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Geometry, Relativity and the Fourth Dimension
By Rudolf v.B. Rucker
Dover Publications, Inc.Copyright © 1977 Rudolf v.B. Rucker
All rights reserved.
THE FOURTH DIMENSION
We live in three-dimensional space. That is, motion in our space has three degrees of freedom—no fewer and no more. In other words, we have three mutually perpendicular types of motion (left/right, forward /backward, up/down), and any point in our space can be reached by combining the three possible types of motion (e.g., "Walk straight ahead about 200 paces to the river, then go right about 50 paces until you come to a big oak tree. Climb about 40 feet up it. I'll be waiting for you there."). Normally it is difficult for us to perform up/down motions; space is more three-dimensional for a bird or a fish than it is for us. On the other hand, space is essentially one-dimensional for a car driving down a two-lane road, essentially two-dimensional for a snowmobile or a car driving around an empty parking lot.
How could there be a fourth dimension, a direction perpendicular to every direction that we can indicate in our three-dimensional space? In order to get a better understanding of what a "fourth dimension" might mean, consider the following sequence:
We take a 0-D point (Figure 1; from now on, we'll abbreviate "n-dimensional" by "n-D"), move the point one unit to the right (this produces a 1-D line segment, Figure 2), move this segment one unit downward (this, with the lines connecting the old and new segments, produces a 2-D square, Figure 3) and move the square one unit forward out of the paper to produce a 3-D cube (Figure 4).
Notice that we cannot actually draw a 3-D cube on this 2-D sheet of paper. We represent the third dimension by a line that is diagonal (rather than perpendicular) to the left/right and up/down dimensions. Now, we don't really know anything yet about the fourth dimension, but couldn't we try representing it by a direction on the paper that is perpendicular to the (diagonal) direction we used to represent the third dimension?
If we do so, we can continue our sequence by moving the cube one unit in the direction of the fourth dimension, producing a 4-D hypercube (Figure 5).
This design for the hypercube is taken from a little 1913 book, A Primer of Higher Space, by Claude Bragdon, an architect who incorporated this and other 4-D designs into such structures as the Rochester Chamber of Commerce Building.
It is also possible to consider a similar sequence of spheres of various dimensions. A sphere is given by its center and its radius; thus the sphere with center 0 and radius 1 is the set of all points P such that the distance between 0 and P is 1. This definition is independent of the number of dimensions your space has. There is no such thing as a 0-D sphere of radius 1, since a 0-D space has only one point. A 1-D sphere of radius 1 around 0 consists of two points (Figure 6).
A 2-D sphere of radius 1 can be represented by this figure in the xy plane (Figure 7).
A 3-D sphere of radius 1 in the xyz coordinate system looks like Figure 8.
Although, reasoning by analogy, a 4-D sphere (hypersphere ) can be seen to be the set of quadruples (x, y, z, t) such that x2 + y2 + z2 + t2 = 1 in the xyzt coordinate system, we cannot say that we have a very good mental image of the hypersphere. Interestingly, mathematical analysis does not require an image, and we can actually use calculus to find out how much 4-D space is inside a hypersphere of a given radius r.
The 1-D space inside a 1-D sphere of radius r is the length 2πr.
The 2-D space inside a 2-D sphere of radius r is the area πr2.
The 3-D space inside a 3-D sphere of radius r is the volume 4/3 πr3.
The 4-D space inside a 4-D sphere of radius r is the hypervolume 1/2 π2r4.
One of the most effective methods for imagining the fourth dimension is the method of analogy. That is, in trying to imagine how 4-D objects might appear to us, it is a great help to consider the analogous efforts of a 2-D being to imagine how 3-D objects might appear to him. The 2-D being whose efforts we will consider is named A. Square (Figure 9) and he lives in Flatland.
A. Square first appeared in the book Flatland, written by Edwin A. Abbott around 1884. It is not clear if Abbott was actually the originator of this method of developing our intuition of the fourth dimension; Plato's allegory of the cave can be seen as prefiguring the concept of Flatland.
A. Square can move up/down or left/right or in any combination of these two types of motion, but he can never move out of the plane of this sheet of paper. He is completely oblivious of the existence of any dimensions other than the two he knows, and when A. Sphere shows up one night to turn A. Square on to the third dimension, he has a rough time.
The first thing A. Sphere tried was to simply move right through the space in A. Square's study. When A. Sphere first came into contact with the 2-D section of his 3-D space which was Flatland, A. Square saw a point (Figure 10). As A. Sphere continued his motion the point grew into a small circle (Figure 11). Which became larger (Figure 12). And then smaller (Figure 13). And finally shrank back to a point (Figure 14), which disappeared.
A. Square's interpretation of this strange apparition was, "He must be no Circle at all, but some extremely clever juggler." And what would you say if you heard a spectral voice proclaim, "I am A. Hypersphere. I would teach you of the fourth dimension, and to that end I will now pass through your space," and if you then saw a point appear which slowly inflated into a good-sized sphere which then shrank back to a point, which winked mockingly out of existence. We can compare A. Square's experience and yours by putting them in comic-strip form, one above the other (Figure 15).
The difference between the two experiences is that we can easily see how to stack the circles up in the third dimension so as to produce a sphere, but it is not at all clear how we are to stack the spheres up in the fourth dimension so as to produce a hypersphere (Figure 16).
We can, however, work out some possible suggestions. One is that the spheres might be just lined up like pearls on a string, and that a hypersphere looks like Figure 17.
We can see that this suggestion is foolish, because if you line circles up like Figure 18 you certainly don't get a sphere. You only get some sort of 2-D design. Similarly, lining the spheres up like a string of pearls will merely give you a 3-D object, when a 4-D object is what you're after. A further objection against the string-of-pearls model is that it is discontinuous; that is, it consists of a finite, rather than an infinite, collection of spheres. A final objection is that the radii of the spheres in the "string" are not scientifically determined in our drawing.
Let's deal more closely with the last objection. It seems reasonable that the length of the "string" should be equal to the diameter of the largest sphere. The idea is that we will have a sphere moving along this length, starting as a point, then expanding to the size of the largest sphere, and then contracting back to a point. To get the picture, let's talk for a minute in terms of turning a 3-D sphere into a 2-D figure. Imagine slicing a 3-D sphere up into infinitely many circles. Then imagine simultaneously rotating each of these circles around its vertical diameter through 90°. The sphere will thus be turned into a 2-D figure consisting of infinitely many overlapping circles. The process can be compared to what happens when you pull the string on a venetian blind to turn all the slats from a horizontal to a vertical position. The resulting 2-D figure looks like Figure 19.
Notice that the radius of each of the component circles of this "closed venetian blind" version of the 3-D sphere is equal to the vertical distance between its center and S0, the circle whose radius is the same as that of the 3-D sphere (Figure 20).
Now, if you take Figure 19 and replace each of its circles by a sphere, you get something that is a solid made up of infinitely many hollow 3-D spheres. Recall that the way in which we turned the 2-D figure (an area made up of infinitely many circles) into a 3-D sphere was by rotating each of its component circles 90° around its vertical axis. So it seems that the way to turn the 3-D solid that we have imagined into a 4-D hypersphere is to rotate each of its component spheres 90° around the plane that cuts the poles and is perpendicular to this sheet of paper. How do you rotate a sphere around a plane? As we'll see in a while, this isn't too hard if you can move through the fourth dimension. What's left of a sphere after you rotate it in this way? Well, half the sphere goes into the part of 4-D space "under" our 3-D space and half goes into the part of 4-D space "over" our 3-D space. And what's left in our space? Just a great circle, the part of the sphere that lay in the plane we rotated around. This is strictly analogous to what happens when you rotate a circle in 3-D space 90° out of this paper. All that remains on the paper is two points of the circle, a 1-D circle.
This all requires some real thought to digest. But read on, read on. It'll get easier in a couple of pages.
Let's return for a moment to the idea, mentioned a few lines above, that our 3-D space divides 4-D space into two distinct regions.
A point on a line cuts the line in two.
A line in a plane cuts the plane in two.
A plane in a 3-D space cuts the space in two.
A 3-D space in a 4-D hyperspace cuts the hyperspace in two.
People used to view the Earth as an infinite plane dividing the 3-D universe into two halves, the upper or heavenly half and the lower or infernal half. If we assume that the 3-D space we occupy is flat (in a sense that we will make clear in a later chapter), then we can conceive of Heaven and Hell as being two parts of 4-D space which are separated only by our 3-D universe. Any angel thrown out of Heaven has to pass through our space before he can get to Hell.
Now, if a hypersphere has been placed so that its intersection with our 3-D space is as large a 3-D sphere as possible, it will be cut into a heavenly hemihypersphere and a hellish hemihypersphere. We can use this idea to get a new way of imagining the hypersphere.
If you take a regular sphere and crush its northern and southern hemispheres into the plane of the equator, you get a disk, or solid circle. Similarly, we can imagine crushing the heavenly and infernal hemihyperspheres into the space of the hypersphere's largest component sphere to get a solid sphere. The solid sphere can be turned back into a hypersphere if we can somehow pull its insides in two directions perpendicular to all of our space directions. How do you do this? Well, how would you pull a solid circle out into a sphere? Imagine that the inner concentric circles belong, alternately, to the northern and the southern hemispheres. You can pull them in opposite directions without having them pass through each other (Figure 21). So to decollapse our solid sphere we pull its concentric spheres alternately heavenward and toward the infernal regions.
In this discussion of the hypersphere I've drawn on some new ideas about the fourth dimension: One is that you can rotate a 3-D object about a plane to leave only a plane cross section of this object in our space. Another is that you can "move through obstacles" without penetrating them, by passing in the direction of the fourth dimension. To clarify these, and other ideas, let's get back to good old A. Square.
After the sphere showed himself to A. Square, A. Square remained unconvinced. So A. Sphere did some more tricks. First he removed an object from a sealed chest in A. Square's room—without opening the chest and without breaking any of its walls. How was this possible? A chest in Flatland is just a closed 2D figure, such as a rectangle (Figure 22). But we can reach in from the third dimension without breaking through the trunk's "walls" (Figure 23).
The analogy is that a 4-D creature should be able to, say, remove the yolk from an egg without breaking the shell, or take all the money out of a safe without opening the safe or passing through any of its walls, or appear in front of you in a closed room without coming through the door, walls, floor or ceiling. The idea is not that the 4-D being somehow "dematerializes" or ceases to exist in order to get through a closed door. Your finger does not have to cease to exist for an instant in order for you to put it inside a square. The idea is that since the fourth dimension is perpendicular to all of our normal 3D space directions, our enclosures have no walls against this direction. Everything on Earth lies open to a 4-D spectator, even the inside of your heart.
The only way in which A. Sphere could finally convince A. Square of the reality of the third dimension was to actually lift him out of Flatland and show him what it was like to move in three dimensions. Is there any hope of this happening to us? Is it likely that there are 4-D beings who, if summoned by the proper sequence of actions, will lift us out of our cramped three dimensions and show us the "real stuff"? A lot of people used to think so at the time of the Spiritualist movement around 1900. The idea was that spirits were 4-D beings who could appear or disappear at any point, see everything, and so on. A fairly reputable astronomer, a Professor Zöllner, even wrote a book, Transcendental Physics, describing a series of seances he attended in an attempt to demonstrate that the "spirits" were actually 4-D beings. He seems, however, to have been hopelessly gullible, and his book is totally unconvincing. In general, the idea of a fourth dimension seems to precipitate authors into orgies of occultist mystification, rather than to lead to clear-sighted mathematical inquiry. The fact that something is difficult does not mean it has to be confused. The best of the books on the fourth dimension written from a mystical point of view is Tertium Organum by P. D. Ouspensky, who also has a good chapter on the fourth dimension in his book A New Model of the Universe.
In any case, Abbott's Flatland ends shortly after A. Square takes his "trip" into the third dimension. The Flatlanders lock him up and throw away the key. It has been this author's great good fortune to come into the possession of the true chronicle of the rest of A. Square's life.
A. Square had been in jail for about ten years when suddenly his old friend A. Sphere turned up again as a circle of variable size in poor Square's cell. "What's happening, baby?"
"Ah, noble Sphere, would that I had never seen you, would that I had been of too small an angularity to grasp your message."
"Man, you ain't seen nothing yet! You want me to lift you out of this jail and put you back in your wife's bedroom? Though I oughtta tell you, there's another mule kicking in your stall, a big sharp Isosceles."
"Sphere, Sphere, if only they'd believe me! There's no use letting me out. They'd just lock me up again, maybe even guillotine me. No, I knew you'd return and I have a plan. Turn me over. Turn me over and then my very body will be proof that the third dimension exists."
A. Square then explained his idea. He had been thinking about Lineland some more. Lineland was a world which Square had seen in a dream once, many years ago. Lineland consisted of a long line on which segments (Linelanders), with sense organs at either end, slid back and forth (Figure 24).
A. Square thought of Lineland in the same way in which we think of Flatland. He confronted his difficulties with the third dimension by imagining the Linelanders' difficulties with the second dimension. In jail for having preached the subversive doctrine of the third dimension, A. Square was understandably concerned with having A. Sphere create some permanent change in Flatland that would attest to the reality of the third dimension. (Note here that Prof. Zöllner was also concerned with getting the spirits to do something that would provide a lasting and incontrovertible proof of their four-dimensionality. His idea was a good one. He had two rings carved out of solid wood, so that a microscopic examination would confirm that they had never been cut open. The idea was that spirits, being free to move in the fourth dimension, could link the two rings without breaking or cutting either one. In order to ensure that the rings had not been carved out in a linked position, they were made of different kinds of wood, one alder, one oak. Zöllner took them to a seance and asked the spirits to link them, but unfortunately, they didn't).
In his cell A. Square had pondered on the kind of permanent change he could create in Lineland if he were back there. He could, of course, remove one of the segments, but this would probably just be termed a mysterious disappearance. He recalled that each Line-lander had a voice at each end, a bass on the left and a tenor on the right. If he turned one over, the voices would be reversed and everyone could observe this in Lineland (Figure 25). Now, if he could rotate a segment around a point, shouldn't A. Sphere be able to rotate a square around a line (Figure 26)? And everyone in Flatland would be able to tell, since everyone was built so that if the eye was toward the north, the mouth was toward the east. A. Sphere could turn A. Square into his own mirror image (Figure 27)!
Excerpted from Geometry, Relativity and the Fourth Dimension by Rudolf v.B. Rucker. Copyright © 1977 Rudolf v.B. Rucker. Excerpted by permission of Dover Publications, Inc..
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Table of Contents1: The Fourth Dimension
2: Non-Euclidean Geometry
3: Curved Space
4: Time as a Higher Dimension
5: Special Relativity
6: Time Travel
7: The Shape of Space-Time