For A Square, on his hundredth anniversary

Contents

Preface to the Dover Edition

Foreword by Martin Gardner

Preface

Preface to the
Dover Edition

WHEN I WAS GROWING UP in the 1950s, my main sources of esoteric information were mail-order paperbacks from Dover Books. I devoured Dover’s volumes on mazes, the fourth dimension, infinity, and language games—all of these to become lifelong interests.

    My Fourth Dimension has been one of my best-selling books, with translations appearing in France, Israel, Italy, Japan, Korea, Germany, the Netherlands, Portugal, and spain. I’m very pleased that Dover is republishing it in paperback and ebook editions.

    The fourth dimension is a universal skeleton key, applicable to any number of conundrums. I always start with the hallowed Flatland analogy: 4-D is to 3-D as 3-D is to 2-D. By now, good old A Square feels like a friend of mine, or perhaps an eccentric cousin. Eventually I went so far as to write a novel called Spaceland, featuring a man named Joe Cube. He begins in our normal world, encounters a four-dimensional being, and travels into higher space. It was an interesting challenge for me to visualize what he saw there. As usual, my starting point was Flatland.

    I’m glad to have had the fourth dimension in my life for so many years. And I’m happy to share this higher knowledge with you.

    Welcome to The Fourth Dimension!

Rudy Rucker

June 22, 2014

Foreword
by Martin Gardner

As WRITERS, mathematicians are notoriously inept. There are happy exceptions, of course, and at least one teacher of mathematics, Lewis Carroll, wrote immortal fantasy fiction. Eric Temple Bell not only wrote colorfully about mathematics for laymen, but under the pseudonym John Taine he churned out a raft of science fiction novels. In more recent times, several professional mathematicians have written science fiction with strong mathematical underpinnings. Now along comes Rudolf von Bitter Rucker — mathematician, novelist, cartoonist, rock music buff, and a thinker with the courage to explore dark and unfamiliar territory in what he likes to call the mindscape.

    Rucker’s major mathematical interests are transfinite sets (he has a doctorate in mathematical logic from Rutgers) and spaces of higher dimensions. After editing a selection of Charles Hinton’s writings about the fourth dimension, and writing a popular account of four-space and relativity, Rucker’s first big success came in 1982 with the publication of Infinity and the Mind. Science fiction fans had earlier known him for his wild, funny, sexually outrageous novels and stories in which higher spaces often play essential roles. His best-known novel, White Light, actually has the subtitle What Is Cantor’s Continuum Problem? The subtitle is the exact title of a paper by the eminent logician Kurt Gödel, with whom Rucker had the privilege of many stimulating conversations.

    The Fourth Dimension: Toward a Geometry of Higher Reality will be eagerly read by writers and readers of science fiction, but anyone with even a minimal interest in mathematics and fantasy will find the book as informative and mind-dazzling as Rucker’s book on infinity. After exploring the whimsical underworld of Flatland, he plunges into spaces above three with a zest and energy that is breathtaking. The book is interspersed with clever problems for the mathematically competent, and finally, in a grand Carrollian climax, Rucker boldly invades the infinite dimensions of Hilbert space.

    Science fiction? In part, yes, and Rucker pauses frequently to quote from some of his bizarre tales. But without the tool of infinite-dimensional spaces, modern physics would be almost impossible. Older books on quantum mechanics talk about how the measurement of particles and quantum systems “collapses the psi function.” More recent books prefer the language of a complex Hilbert space, an approach adopted by John von Neumann in his classic work on quantum theory. When you measure a quantum system you are said to “rotate the state vector,” an abstract line of precisely defined length and orientation that represents the system’s state in a set of coordinate systems that constitute a Hilbert space. Are such spaces “real”? Or are they no more than convenient fictions used by physicists to simplify their calculations?

    It is with such deep ontological questions that Rucker concerns himself throughout this book and especially in the final chapters. Here I am unable to go along with his “All is One” philosophy. He seems to have inherited a genetic fondness for the Absolute from his great-great-great-grandfather, the famous philosopher Hegel. Like William James, I do not know if ultimate reality is One or Many. Nor can I accept Rucker’s views on synchronicity, taken over from Jung and Koestler, and his seeming belief that consulting the I Ching is more likely to produce meaningful “coincidences” than consulting say, Homer, the Bible, or the works of Isaac Asimov.

    But no matter. We are each entitled to what fames called our “over beliefs.” Whether you agree or disagree with Rucker’s Tao-tinged metaphysics, you will find his speculations twisting your mind toward fundamental questions that refuse to go away no matter how hard pragmatists and positivists try to banish them.

Preface

    IN 1958, the Louisville Free Public Library had a single shelf of science fiction books. My friend Niles Schoening and I used to read and discuss them, wondering about time travel, wondering about the fourth dimension. This is where it all began.

    When I went off to college in 1963, my father, Embry Rucker, gave me a copy of Edwin Abbott’s Flatland. As an Episcopal priest, my father had already realized that the fourth dimension can serve as a symbol for higher spiritual realities.

    In the ensuing years, I puzzled over the relationship between the fourth dimension as higher reality, and the fourth dimension as time. When I got my first teaching job, at SUNY Geneseo, I began working out these connections in my lectures for a course on higher geometry. These lectures were published by Dover Publications in 1977, under the title Geometry, Relativity and the Fourth Dimension.

    In the seven years since then, I’ve learned a lot more about the fourth dimension. In writing The Fourth Dimension: Toward a Geometry of Higher Reality, I’ve tried to present a definitive and popular account of what the fourth dimension means, both physically and spiritually. I am grateful to Martin Gardner for lending me a number of rare books on the fourth dimension; to Thomas Banchoff for his help with the more technical research; to my editor, Gerard Van der Leun, for his encouragement; and to my illustrator, David Povilaitis, for his wit and weirdness.

    Most of all, I would like to thank my family, my friends, my students, and my correspondents. Enjoy the book!

Part I

THE
FOURTH
DIMENSION

1

A New Direction

This is a free online webpage edition, Copyright (C) Rudy Rucker 2016.
Purchase paperback or ebook editions from Dover Publications or Amazon.

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?

Fig. 1. A new answer for old questions.

    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 m which space is curved, and still another higher dimension may lead toward some utterly different universes existing parallel to our own.

Fig. 2. Where is it?

    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.

Fig. 3. Etch-A-Sketch drawing of A Square.

    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.

Fig. 4. Two degrees of freedom on a curved surface.

    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.

Fig. 5. One degree of freedom on a curved line.

    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:3-D::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.

Fig. 6. Plato’s 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.

Fig. 7. Plato’s Cave II.

    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?

Fig. 8. A Square.

    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.

2

Flatland

This is a free online webpage edition, Copyright (C) Rudy Rucker 2016.
Purchase paperback or ebook editions from Dover Publications or Amazon.

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.

Edwin Abbott Abbott (1838-1926)

    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 belauded Circles.”

Fig. 9. Eight Flatlanders: Woman, Soldier, Workman, Merchant, Professional Man, Gentleman, Nobleman, High Priest.

    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.

Fig. 10. Depth is simpler than dematerialization.

    A Square’s dimensional adventures begin when he has a dream, a dream of Lineland:

Fig. 11. Abbott’s drawing of Lineland.

    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!

Fig. 12. A Square moves through Lineland.

    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.

Fig. 13. A Square and his wife in a locked room.

    The next evening A Square and his wife are comfortably sealed up in the safety of their home, when suddenly a voice out of nowhere speaks to them. And then, a moment later, a circle appears in the confines of their tightly locked house. It is A Sphere, come to teach A Square about the third dimension.

Fig. 14. A Circle appears in A Square’s locked room.

    Reasoning by analogy, you can see that a four-dimensional creature would be able to reach into any of our rooms or cubbyholes, no matter how well they are sealed up. A four-dimensional creature could empty out a safe without cracking it, for the safe has no walls against the fourth dimension. A four-dimensional surgeon could reach into your viscera without breaking your skin. A four-dimensional creature could drink up your Chivas Regal without ever opening the bottle!

Fig. 15. Liquor thief from the next dimension.

    If only you had the muscles to twitch part of your arm up into the fourth dimension, you could reach in “around” the window at Tiffany’s and take out the biggest diamond on display. This would not be done by somehow having your arm turn into gas or a ray of light. The heist would be done by having your arm move up through the fourth dimension. The diamond would be brought out by lifting it up into the fourth dimension to get “around” the sheet of glass.

Fig. 16. The perfect crime.

    Returning to A Square, there he is, locked in his house, talking to what seems to be a circle, another two-dimensional creature. The Sphere objects to this flat characterization of himself:

Fig. 17. A Sphere moves through Flatland.

    The Sphere proceeds to demonstrate the third dimension by moving through A Square’s plane, just as A Square had moved through Lineland for the king. What A Square sees is a point that turns into a circle. The circle swells to a certain maximum size, then shrinks back to a point, which disappears. His great difficulty is in thinking of all these different circles as existing all together in the form of a sphere.

    Pause for a moment and try to imagine four-dimensional space. It is right next to you, but in a direction you can’t point to. No matter how well hidden you may be, a four-dimensional creature can see you perfectly well, inside and outside.

Fig. 18. A hypersphere moves through our space.

    What would you see if, right this moment, a four-dimensional hypersphere were to pass through the space near your head? Reasoning strictly by analogy, you would expect to see first a point, then a small sphere, then a bigger sphere, then a small sphere, and then a final point that disappears. Visually, it would be much the same as seeing a balloon that is first blown up and then deflated. Next time you have a balloon in hand, you might even try smoothly blowing it up and letting the air back out. That, basically, is what you would see if a hypersphere passed through the space of your room. A sphere is a three-dimensional stack of circles; a hypersphere is a four-dimensional stack of spheres.

    But it is very hard to see how to stack things up in a new dimension. A Square, far from believing that he has seen the cross sections of a sphere, shrieks, “Monster, be thou juggler, enchanter, dream or devil, no more will I endure thy mockeries,” and rams his hardest right angle against A Sphere’s cross section.

Fig. 19. Impaled by a policeman.

    An interesting question comes up here. If A Square were actually to cut into A Sphere, would it matter to the sphere? Would it be possible for the Flatlanders to keep a three-dimensional being captive? To make this question quite concrete, imagine that you had somehow found the real Flatland. You stick your hand through it and an enraged isosceles triangle spears right through your wrist. Then what?

    To answer the question, we have to decide exactly what Flatland is like. If the Flatlanders are truly two-dimensional, with no thickness at all, then they will be as immaterial as shadows or patches of light. In this case, having an isosceles intersect your wrist would not hurt, nor would it limit your freedom of movement. Indeed, it is debatable whether such an insubstantial Flatlander would actually be able to poke through your skin.

    One problem with the idea of truly two-dimensional Flatlanders is that it is a little hard to see how they could have any solidity or reality. If they were indeed just like shaded regions of the plane, then there would really be nothing to prevent them from freely moving through each other. One way out of this difficulty might be to have the “atoms” of a two-dimensional Flatland consist of little wrinkles or bumps in the plane of the Flatland space. Thus A Square might be a sort of mesa in the rubbery sheet of Flatland’s space. Here we could suppose that the space is infinitely thin, but it is easier to imagine that the “rubbery sheet of Flatland’s space” does itself have a slight thickness.

Fig. 20. A Square as a bump in the plane.

    So now what if the Flatlanders do have a slight thickness? Abbott himself opts for this alternative in the preface to the second edition of Flatland. Here he describes how A Square comes to believe that insofar as Flatland really exists inside some higher space, it must be that the Flatlanders have height as well as length and breadth. Since each of them has the same height, there is no way for them to notice it. A Square reports an unintentionally funny conversation in which he discusses this point with the ruler of Flatland:

    Even if all the Flat-landers were fully one inch high, they would not be able to notice this height, provided they still had no power of motion or variation in the third dimension. Of course, if there were Flatlanders of varying heights, then some kind of difference would be noticed, though the Fiat-landers’ description of the difference might be some vague, nongeometrical quality like charisma, force of personality, or “aura.”

Fig. 21. A Square is high with the boss.

    Now, if the Flatlanders did have an actual three-dimensional thickness, then having one of them cut into you would be like being cut with a knife blade. And if the creature was large enough, it might have enough mass to make your escape difficult.

    In Flatland, as it turns out, Sphere becomes angry with Square’s attempts to stab him, and finally grabs Square and lifts him up into space. A Square finds it an unsettling experience:

Fig. 22. What if the Sphere just got A Square’s skin}

    Here again, an interesting side issue comes up. Mightn’t it be harmful for A Square to be lifted out of his space? We had better assume that the Square has some thin membranes sealing off his upper and lower faces against the third dimension, for otherwise when A Sphere pulls on a corner he might get only the skin! And rather than thinking of A Square as resting on his space, let’s regard him as in a slightly thickened plane.

Fig. 23. A Square is really a thick part of a thick plane.

    The best-known two-dimensional world other than Flat-land is known as Astria. It is described in Charles H. Hin-ton’s 1907 book, An Episode of Flatland: Or How a Plane Folk Discovered the Third Dimension. Here is the passage where Hinton explains his idea for the world:

Fig. 24. Astrians on a small planet.

    A drawback with a world design like this is that the polygons have so much trouble moving past each other, building houses, and so on. Many of these difficulties are resolved in A. K. Dewdney’s 1984 book The Planiverse. As is The Fourth Dimension, Dewdney’s book is in some measure a celebration of Flatland’s centennial.

    One final plane-universe book worth mentioning is the 1965 book Sphereland, by the Dutch mathematician Dionys Burger. Burger describes a world that is something of a compromise between Flatland, with its great freedom of movement, and Astria, with its close resemblance to Earth. Burger’s notion is that, as in Astria, the 2-D creatures live near the surface of a disk planet. But he proposes that they be very light, and thus able to live in their planet’s atmosphere. It is as if people were able to live in clouds, clouds floating above tropical vegetation that is, in turn, floating on a sea that surrounds the planet’s dense core.

Fig. 25. Cities in the sky. (Drawing from Dionys Burger’s Sphereland.)

3

Pictures of the
Gone World

This is a free online webpage edition, Copyright (C) Rudy Rucker 2016.
Purchase paperback or ebook editions from Dover Publications or Amazon.

IMAGINE that you have been lifted up into hyperspace. What would our world look like from this vantage point? To begin with, note that our 3-D space would cut 4-D hyperspace into two regions, just as a 0-D point cuts a 1-D line in two, a 1-D line cuts a 2-D plane in two, and a 2-D plane cuts a 3-D space in two. (By the way, we speak of a point as zero-dimensional, 0-D, because someone whose entire space is limited to one point has no degrees of freedom in his or her motions.)

Fig. 26. An n-dimensional space cuts an (n + 1)-dimensional space in half.

    What shall we call the two regions of hyperspace determined by our space? Charles H. Hinton has suggested the words ana and kata, to be used more or less like the words up and down. Just to have something to think about, we might think of heaven as lying ana above our space, and hell as lying kata below. A 4-D angel expelled from heaven would tumble through our space like a man falling through Flatland: an exciting moment of grotesque, incomprehensible cross sections splitting and merging and falling about!

Fig. 27. Man falling through Flatland.

    Just as man’s Flatland cross section could be a number of irregular shapes with skin boundaries, a hyperbeing’s cross section in our space might be a bunch of bobbing globs of skin-covered flesh. Some of the blobs might have things like teeth or claws! Being “picked up” by a hyperbeing would probably involve a bunch of globs closing in on you like cross sections of a hand’s fingers.

    Once you are out in hyperspace, you can get some very strange perspectives on those you left behind. Consider how Flatland looks to us: we can see all four sides of A Square, and we can see every detail of innards. By the same token, a 4-D creature should be able to look down at me and, at one glance, see every square inch of my skin, the inside and outside of my stomach, the convolutions of my brain, and so on.

Fig. 28. Woman menaced by a creature from the fourth dimension.

    But how, you may ask, could a 4-D person “see” all sides of a 3-D object at once? A human being’s retina is a two-dimensional disk of nerve endings. By analogy, we would expect a 4-D creature’s retina to be a three-dimensional sphere of nerve endings. My seeing A Square consists of the excitation of a square-shaped pattern of nerve endings in my retina. A 4-D creature’s “seeing” me would consist of the excitation of a person-shaped pattern of nerve endings in the little ball of his retina. Each point in A Square’s body sends a light ray up to a single point in my retina. Each point in my body sends a light ray ana to a single point in the 4-D creature’s retina.

Fig. 29. A 2-D with an image of A Square, and a 3-D retina with an image of a person.

    It is a curious feature of 4-D space that we can connect two points in the interiors of two solid 3-D objects without piercing these objects’ surfaces. The trick is to use ana / kata motions to get in and out of the solid 3-D objects. If you’re inside a cubical room and move ana out of it, it’s as if you had suddenly dematerialized. You don’t go through the walls or the floor or the ceiling, you move over in the ana direction to a part of 4-D space where the room doesn’t exist at all.

Fig. 30. Connecting interior points without cutting through the boundary.

    So the reason that a 4-D creature can see all of me inside and out is that such a creature’s “retina” is able to form a completely detailed model of my body. But this is not really so puzzling or occult a phenomenon. The human brain is able to mimic such behavior … For do you not have a detailed 3-D mental image of your right hand? When you think of your hand you do not necessarily think of just the front or just the back. It is really possible to have the idea of a 3-D object seen from no particular direction — or from all directions at once.

    We can form especially good 3-D images of transparent objects such as paperweights, wine bottles, or glasses of water. Here, unlike with the hand, there is no difficulty in imagining the inside parts. Holding 3-D images in your mind is definitely something worth doing. Try, for instance, to think about your house — all of it, seen from no particular vantage point. Here you are getting close to a higher-dimensional experience.

    So a 4-D view of our 3-D world is not totally inconceivable. But what would it be like to look at 4-D objects? We saw with the hypersphere in chapter 2 that it is possible to get images of various 3-D cross sections of such a hyperobject, but how are we to combine these sections into a 4-D whole?

    Some people might say at the outset that it is hopeless to try to think of four-dimensional objects. For how can our 3-D brains ever hold images of 4-D objects? This argument has some force, but it is not really conclusive. Drawings use 2-D arrangements of lines to represent 3-D objects. Why shouldn’t we be able to build up 3-D arrangements of neurons that represent 4-D objects? More fancifully, perhaps our minds are not just 3-D patterns: maybe our brains have a slight 4-D hyperthickness, or maybe our minds extend out of our brains and into hyperspace!

    I’ll spend the rest of this chapter discussing two of the simpler 4-D shapes: the hypersphere and the hypercube. We’ll start with the hypersphere, though if you hate math you might want to skip up to where I talk about the hypercube.

    A sphere of any dimensionality is specified by giving its center and its radius. In any space, the sphere with center point 0 and radius r is the set of all points P whose distance from 0 is r. In 2-D space this definition leads to a circle of radius r, in 3-D space it gives a traditional sphere, and in 4-D space it gives a hypersphere.

Fig. 31. A circle is a 2-D sphere.

    Pick a point 0 in space near you and try to imagine a hypersphere centered on this point, a hypersphere with a five-foot radius. What points P lie on the hypersphere? First of all, there are the points in your space that are five feet from 0. But moving in our space is not the only way to move away from 0. What if one combines motion away from 0, as before, with a motion ana out of our space? We might, for instance, move four feet away from 0 in our space, turn at right angles, and then move ana three feet into hyperspace. (Those who remember the Pythagorean theorem, or the “distance formula” of analytic geometry, can check that this is true because 4² + 3² = 5².)

Fig. 32. 2-D plus “up” is like 3-D plus “ana.”

    What is interesting to notice here is that no matter which direction of our space the original four-foot displacement from 0 is taken in, the additional three-foot ana motion gives a point exactly five feet away from 0. So if we take all the points on a four-foot sphere around 0 and then move ana three feet, we will get a displaced sphere of points all belonging to the five-foot hypersphere around 0.

    Now we can see why the full hypersphere consists of a series of spheres, spheres that grow smaller as one moves ana or kata from the space where the center lies. Taken together, this family of spheres makes up a three-dimensional “hypersurface,” analogous to the two-dimensional surface of a sphere. The hypersurface of a hypersphere is a curved 3-D space located in 4-D space.

    This is an important concept because many scientists believe that the space of our universe is in fact the hypersurface of a very large hypersphere. Let’s try to understand it a little better.

    First of all, shouldn’t the hypersurface of a hypersphere be four-dimensional, not three? Not really. Consider the surface of an ordinary 3-D sphere such as the planet Earth. Although the surface is certainly curved in three dimensions, someone limited to the surface has only two degrees of freedom in moving: east/west or north/south. A Flat-lander sliding around on a 3-D sphere’s surface still feels himself to be in a 2-D space. It’s just that this space somehow curves back on itself.

    Now think of a little hyperfly who can move in hyperspace, but who has to stay exactly five feet from the point 0. If the fly starts out five feet away from 0 in our space then it has basically three kinds of motion open to it: east/west or north/south (around a five-foot sphere centered on 0 in our space) or an ana/kata motion (combined with a motion toward 0 to keep the distance at five feet).

    We’ll come back to the hypersphere later, but now it’s time to look at the hypercube.

    The hypercube, also known as the tesseract, is probably the best-known 4-D geometrical pattern. It arises in the following way:

Fig. 33. From point to cube.

    Start with a point and move it one unit to the right. This produces a one-dimensional line segment. Now move the line segment one unit down the page, producing a two-dimensional square. If we move the square one unit out of the page we get a three-dimensional cube.

    Now, we can’t really fit a three-dimensional object into the two-dimensional confines of this page. The standard convention, which we have used above, is to represent the third dimension as a direction diagonal to the first two. What if we were to use the other diagonal direction as the fourth dimension? If we move our image of the cube one unit in this “fourth dimension,” we get a picture of a four-dimensional hypercube.

Fig. 34. The hypercube.

    This figure is fun to look at … It has a certain mandala-like quality. If you are interested in drawing your own, note that the figure can be produced by constructing a square on each of the inner edges of a regular octagon. A regular octagon can be obtained by tearing down a STOP sign or, preferably, by dividing a circle into eight equal slices.

    The hypercube arises as the “trail” of a cube moving in four-dimensional space. A cube arises as the “trail” of a square moving in three-dimensional space. Any given cube can be generated in three different ways, depending which of three possible pairs of opposite squares are thought of as being the “start” and “finish” positions. The hypercube includes four pairs of cubes. Can you see them all?

Fig. 35. Squares on an octagon.

    A different way of drawing a hypercube is based on the idea that if you hold the wire skeleton of a cube close to your face, it will look like a small square inside a big square. Analogously, we can represent a hypercube by drawing a small cube inside a large cube. The idea is that the small cube is “farther away” in the direction of the fourth dimension. Surprisingly, such a hypercube shape can be naturally produced by dipping a cubical framework into soap solution, and then adding a bubble at the center.

Fig. 36. A cube as a square in a square, and a hypercube as a cube in a cube. (Hypercube drawing from D. Hilbert and H. Cohn-Vossen, Geometry and the Imagination)

    Yet another way of representing a hypercube is by unfolding one. As before, we reason by analogy. If you cut some of the edges of a paper cube, you can flatten the cube down into a connected two-dimensional pattern of six squares. This can be done in eleven essentially different ways.

Fig. 37. The eleven ways to unfold a cube.

    If a hypercube is cut in the proper way, then it can be unfolded and “flattened down” into a connected three-dimensional pattern of eight cubes. One unfolding of the hypercube produces a sort of three-dimensional cross. Salvador Dali actually uses this unfolded hypercube as a crucifix in his 1954 painting Christus Hypercubus. In his classic story “And He Built a Crooked House,” Robert Heinlein describes a house built in this pattern. The twist in Heinlein’s story occurs when there is an earthquake and the house folds up into a hypercube!

Fig. 38. One way to unfold a hypercube.

    As a last approach to the hypercube, let’s imagine a hollow stone hypercube, analogous to a hollow stone prison cell. What would we see if this hypercubical prison was pushed through our space?

    Well, what would A Square see if we pushed a hollow stone cube through the plane of Flatland? At the beginning the whole solid stone floor would intersect his space, later there would be hollow stone squares made up of cross sections of the walls, and finally there would be the solid stone cross section of the ceiling. He would see a solid square, followed by hollow squares, followed by a solid square. By the same token, if we were to push a Flatland prison cell — a hollow stone square — through Lineland, the Linelanders would see a solid segment, hollow segments, and then a solid segment.

Fig. 39. A 2-D prison cell passes through Lineland.

    Reasoning by analogy, it is not hard to see that if a hollow stone hypercube were to pass through our space, we would see first a solid cube of stone, then a series of hollow stone cubes, and finally a last solid stone cube. The eight solid “boundary cubes” of the hypercube would arise as the two solid cubes seen at beginning and end, and as the six “trails” of the floors, walls, and ceilings of the hollow cubes.

Fig. 40. Escaping from a sealed room.

    The hollow stone hypercube could serve as a prison cell in which a hyperbeing, say an angel, could be incarcerated. Think of the room you are in as having a solid stone floor, walls, and ceiling. Even an angel can’t get through that solid stone. Normally, however, an angel could escape the room by moving ana, over, and kata. But if your room is only a cross section of a stone hypercube, then when the angel moves ana he finds himself not in an empty space but still inside a stone-walled room. If the angel keeps moving ana he will run into one of the boundary cubes: a whole roomful of solid stone, impossible to move beyond.

Fig. 41. No escape.

    We have now discussed a variety of four-dimensional phenomena, but many readers may still feel that something is missing. One doesn’t want only to reason about the fourth dimension, one wants somehow to see it. Most of what we have learned about the fourth dimension so far has been by using Lineland and Flatland analogies. Although the analogies are very instructive, it is not unusual to have the feelings P. D. Ouspensky describes in Tertium Organum.

    In the next chapter we will look at a direct, though somewhat dangerous, path into the fourth dimension.

4

Through the
Looking Glass

This is a free online webpage edition, Copyright (C) Rudy Rucker 2016.
Purchase paperback or ebook editions from Dover Publications or Amazon.

THE FOURTH DIMENSION is essentially a modern idea, dating back not much further than the midnineteenth century. Although the vague notion of a higher world has always been with us, the scientific concept of a geometrical fourth dimension was late to develop.

    The first philosopher to seriously entertain the idea of higher-dimensional spaces was the great Immanuel Kant (1724-1804). In one of his earliest essays he writes longingly of such spaces: “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 God has somewhere brought them into being.”

    Later in his career, Kant proposed a famous puzzle that is related to the idea of the fourth dimension: If all of space were empty except for a single human hand, would it make sense to say that the hand was specifically a right hand? As it turns out, the answer is no. The concepts of “left” and “right” are meaningless in an empty space.

    To begin to understand why, imagine a big Plexiglas sign advertising a palmistry shop, a shop run by the famous fortuneteller Mom Oxo. The outline and wrinkles of a palm are drawn on the transparent Plexiglas. Now, if you look at the sign from one side you see a right palm; if you look from the other you see a left palm. Once you realize that it is possible to look from outside the two-dimensional plane of the sign, you see that it would make no sense to say the palm is really a right palm.

Fig. 42. Kant’s hand for Mom Oxo’s Palmistry.

    The same is true of three-dimensional space. Depending which four-dimensional “side” you look at space from, a hand can appear to be a left or a right. Another way of putting this is that a left hand can be converted into a right hand by lifting it into four-dimensional space and “turning it over.”

    Let’s dramatize this in terms of A Square. At the end of Flatland, Square’s trip into the third dimension ends with a big argument between him and A Sphere. Reasoning by analogy, A Square concludes that there must be a four-dimensional world beyond the sphere’s 3-D space, and asks to be taken there. But here, as sometimes happens, the student is seeing much further than the teacher has ever dared. Sphere is annoyed and finally angered by Square’s insistence that there must be a four-dimensional “Thoughtland” beyond Lineland, Flatland, and Spaceland. Yet A Square prattles on, dreaming of higher and higher dimensions. Sphere loses all patience and the trip is over:

    Now of course A Square wants to tell everyone about his revelations but — surprise, surprise — in Flatland it’s illegal to talk about higher dimensions. Square tries to hold himself back, but finally at a meeting of the Local Speculative Society,

    A Square is convicted and sentenced to life imprisonment. As time goes by he finds it harder and harder to think about the third dimension. Edwin Abbott’s Flatland ends on this downbeat note, with the increasingly depressed Square having served out seven years of his sentence.

    Nineteen eighty-four is the hundredth anniversary of Flatland’s publication. I am delighted to report that A Square is alive and flourishing. Not only is he alive, but he has consented to communicate an account of his further adventures. I will be quoting from my notes on The Further Adventures of A Square throughout the rest of this book. I will begin by quoting the first two paragraphs here, and will then proceed to some excerpts dealing with problems of left and right.

Fig. 43. A Cube visits A Square.

Fig. 44. A Square turns over.

    In his dream of Lineland, A Square imagines that once again he is talking to the king of Lineland, a segment with a bass voice at his left end and a tenor voice at his right end. Seized with a desire to really upset the king, A Square reaches down into Lineland and flips the king over, rotating him around his central point. The other Linelanders can hear that the king doesn’t sound right: he’s been turned into his own mirror image. In a frenzy, they set upon the king and tear him to shreds.

Fig. 45. The king of Lineland flips.

    When A Square wakes up the next morning, everything still looks backwards. And the guard who brings breakfast takes one look at our friend and begins to scream. It’s really true, that crazy Cube has turned A Square over, rotating him about his central axis.

    Normally, if a Flatlander’s eye is on his north side, then his mouth is facing east. But now A Square is the opposite.

    Just a totally flipped square.

Fig. 46. Flatland mirror.

    The Council met, pronounced A Square an “object of horror to the gods,” and decided to execute him.

    We’ll get back to A Square’s adventures in a while, but now let’s stop and ask what it would be like to get “turned over” in the fourth dimension. It would seem that a 4-D being could turn you into your own mirror image by rotating you, in the fourth dimension, around a plane that cuts through your body — say the plane that includes the tip of your nose, your navel, and your spine. That one plane of your body would stay in our space. Your right half would move ana, let us say, and your left half would move kata. The two halves would, in their parallel spaces, move past the plane of rotation, and then they would swing back into our space. Rotation about a plane is hard for us to imagine … But just remember how hard it is for a Flatlander to think of rotation about a line.

    While you were in the process of rotating you would really look strange, for all of you that would remain in our space would be a cross section something like a microtomed tissue. If you were moved up and down at right angles to our space, we could see each of your cross sections in turn, and really get a good 3-D knowledge of your inner workings.

Fig. 47. Before and after.

Actually the diagnostic tool known as a CAT scan consists of a process something like this: the building up of a 3-D model of the body by looking at a series of cross-sectional x-rays.

    Mysterious as a four-dimensional rotation into one’s own mirror image may seem, it is actually possible for us to watch such a rotation taking place, as we will see shortly. The method, which provides genuine insight into the fourth dimension, has to do with A Cube.

Fig. 48. A Cube.

    A Cube, you will notice, has typically schizoid features. The right half of his face, which is controlled by the analytic and sociable left brain, has a friendly smile and a precise triangular eye. The left half of his face, controlled by the dark and intuitive right brain, has a lax expression and a wandering round eye. Clearly A Cube and A Cube’s mirror image are totally different. A Cube has his triangle eye on the right, but A has his triangle eye on the left.

Fig. 49. A Cube and his mirror image.

    Just as no amount of sliding around in the plane ever could have turned A Square into his own mirror image, no 3-D contortions could ever turn A Cube into A. But, as the mathematician August Ferdinand Möbius discovered in 1827, it is in fact possible to turn a 3-D object into its mirror image by an appropriate rotation through four-dimensional space.

    Surprisingly enough, it is actually possible for our minds to perform such a rotation. Many readers will be familiar with the “reversing cube diagram,” also known as the Necker cube. If one stares at a skeleton drawing of a cube for a while, the mind’s 3-D interpretation of the figure flip-flops back and forth between the two versions shown. If you have trouble getting the figure to “do” this, it might help to focus your attention near the middle of the drawing and try mentally “pushing” or “pulling” at one of the corners.

    What makes the Necker cube reversal so important here is that the two possible 3-D interpretations of the original skeleton drawing are in fact mirror images of each other!

Fig. 50. The Necker cube and its two interpretations.

    We can see this if we draw A Cube’s features in on the Necker cube, and imagine that A Cube’s body is clear as glass. If we take the features to be on the side of the cube closer to us, then we are looking just at A Cube, who has a triangle eye on his right. But if we interpret the features as being on the side of the cube further away from us, then we are looking at a rear view of A, A Cube’s mirror image, who has a triangle eye on his left. If A Cube is transparent, then the original ambiguous figure can be seen as either the Cube or the Cube’s mirror image.

    The point of this is that the sort of twinkling rearrangement that takes place when a Necker cube reverses is equivalent to a rotation through the fourth dimension. It happens so fast it’s hard to catch, but if you watch it for a while it begins to feel like maybe you’re getting a peek into the fourth dimension. The hypercube drawing, by the way, is particularly good to look at in terms of Necker cube reversals. Stare at it as at a mandala and the thing fairly seethes with activity, doing its best to get hyper.

Fig. 51. A Cube changing into .

Fig. 52. The Hypercube.

    An especially dramatic version of the Necker cube can be created by folding together the Neck-A-Cube model depicted in figure 53. When it is folded, the paper is in the shape of A, as seen from the inside of his hollowed-out head. After Necker reversal pulls the corner out at you, it looks like A Cube should. But when you break the illusion (by opening your other eye, or by turning the model far enough to show its back), then the thing “topples” through the fourth dimension back to the hollowed-out A form.

1. Trace figure.
2. Cut out around outline.
3. Crease line AC, and then crease line DE. Each time crease by folding the marked surfaces together.
4. Slit from A to B.
5. Slide one of the upper “squares” behind the other to make something like the corner of a room where walls meet ceiling. Cup the object in your right hand.
6. Close one eye and stare at the corner. “Pull” at the corner till Necker cube reversal takes place.
7. Once the object is solidly reversed, try moving your hand around.
8. If you have trouble getting the illusion, make sure that the model is uniformly lit (so that shadows don’t provide depth cues); and make sure to hold it still till Necker reversal has taken place.

Fig. 53. Design for the Neck-A-Cube.

    Once the cube is reversed, it seems to move in an un-natural way. If you have it reversed and cradle it in your hand, then it will seem to stick up in the air and rock in the opposite way that your hand moves. An even more remarkable effect is obtained if you set the cube down, or glue it to the wall, and then reverse it. Now, if you can manage to hold it reversed as you move your head around, you will see the cube apparently turning so as to always stare right at you!

    I should mention here that my model is based on a similar design by jerry Andrus, a brilliant illusionist living in Albany, Oregon. Andrus calls his reversing pattern a Parabox.

    Staring at the Neck-A-Cube too much before bedtime can lead to really unpleasant dreams. Although the fourth dimension is of great theoretical and philosophical interest, there is something frightening and mind-warping about it when it starts to get too real. I remember in particular a series of dreams I had in 1976, soon after I carne to understand that left and right are concepts as relative as up and down, or front and back.

    In these dreams I would be walking down a street with someone on either side of me — my wife, Sylvia, let us say, on my left, and my friend Greg on my right. I would move out of my body and watch the three of us from a distance, first from a point in our space, then from a point entirely outside of space. What struck me was that depending on which half of hyperspace I looked from, the order of the three people would be Sylvia-Rudy-Greg, or Greg-Rudy-Sylvia. In my dream of walking down the street, I would begin to internalize such a shift, and the whole city around me would change back and forth between itself and its own mirror image. Some mornings I would wake up convinced that I and the whole universe had been replaced by our mirror images overnight. The worst day of all was when I sat up in bed and actually saw the reversal taking place: the whole room, furniture and all, rambunctiously do-si-do-ing about into mirror reversal. But, of course, I couldn’t prove anything … I’d turned over, too.

    After these experiences I often wondered if there could be any objective fact correlated to a person’s feeling that the world has turned into its mirror image. One possible idea that carne to mind was that a person might have a 4-D component of mind that projects out of the brain and into hyperspace. If this “hyperbrain” material were to shift from being primarily on, let us say, the ana side to being primarily on the kata side, then one would be justified in saying that the world has changed — in the same way that the palm drawn in figure 42 changes when an observer moves from one side to the other.

    Sometimes I get so confused about left and right that all I’m really sure about is what is next to what. I get this feeling if I go to England … or just watch an English movie … where everyone drives on the left side of the road. A very commonplace instance of left-right confusion arises if you happen to study your teeth in the mirror while feeling them with your tongue. Left and right can get hopelessly mixed up in short order. Charles Hinton, whom we will talk about in the next chapter, felt that deliberately inducing such a confusion about left and right was the best path to four-dimensional thought.

5

Ghosts from
Hyperspace?

This is a free online webpage edition, Copyright (C) Rudy Rucker 2016.
Purchase paperback or ebook editions from Dover Publications or Amazon.

SPIRITUALISM — the belief that spirits of the dead are nearby and eager to contact us — has never been so popular as in the late nineteenth century. Throughout the United States, England, and Europe, amateur or professional mediums were organizing séances. A group of people would sit around a table in a darkened room, the medium would moan and mutter, and then the spirits would manifest themselves.

    And what would said spirits do? They would make noises: sharp raps on the table. They would move things: turning or rocking the table and, occasionally, even lifting the table up into the air. They would send messages: often by causing a piece of pencil to scratch some words on a slate held under the table. They would materialize as white fogs, or sometimes as spirit hands that would reach up over the table’s edge.

    Spiritualism mediums were, of course, under heavy suspicion of fraud from the very beginning. People are so eager to believe in life after death that their reports of what they experience at a séance are not very reliable. No accumulation of colorful anecdotes can ever take the place of a really scientific investigation. The few scientists who believed in spiritualism began trying to find some kind of solid theory to support their belief in ghosts.

    Quite abstractly, there seem to be two possibilities if spirits exist. Either they are in our space, but very insubstantial, or they are somehow out side our space altogether. The idea of spirits as tenuous forms in our space was popular with some early spiritualists: they held that the spirits were made of “ectoplasm” or, even less concretely, of “vibrational energy.” One difficulty here was that if the spirits were so very insubstantial, how could they do things like lifting a heavy séance table?

Fig. 54. Occult forces.

    No such difficulty arises if we think of spirits as solid and substantial beings, but here of course we have the question of why one doesn’t notice spirits very often if they really are so solid. The answer to this is to assert that the home of spirits is somehow outside our space entirely. How best to have spirits live outside space? One could have them be infinitely far away, but then one would have the problem of how they can get here so rapidly when summoned up by a medium. A much more satisfactory explanation is to say that spirits live in the fourth dimension.

    The beauty of this idea is that the spirits are wholly outside of our normal, material space yet they are … at the same time … right next to us, waiting just a few feet ana or kata. Although the idea of spirits as 4-D beings had its greatest popularity in the nineteenth century, it had been hinted at some two hundred years before, by the Cambridge Platonist Henry More (1614-1687). Like the scientific spiritualists, More was opposed to the idea that spirits, angels, and Platonic forms could exist as insubstantial abstractions. He felt that if spirits really exist, then they must actually take up space. Yet, if a person’s soul takes up space, we have the question of how it can fit into the person’s solid physical body. In 1671, More carne up with the suggestion that spirits should be four-dimensional. He phrased this in terms of an occult quality he called spissitude, meaning something like “denseness of substance.” His idea seems to have been that the differences between the physically identical bodies of a dead person and a living person would be that the living body has more spissitude, and that spissitude is physically unobservable because it corresponds to a certain hyperthickness in the direction of the fourth dimension.

Fig. 55. Body and soul.

    The person who really popularized the spiritualist notion of ghosts from the fourth dimension was Johann Carl Friederich Zöllner (1834-1882). Zöllner was a professor of astronomy at the University of Leipzig, the same university where August Möbius made his 1827 discovery that it would be possible to turn a 3-D object into its mirror image by a hyperspace rotation, and the same university where Gustav Fechner wrote his 1846 essay “Why Space Has Four Dimensions.” Zöllner got his interest in spiritualism from an 1875 trip to England, where he visited William Crookes, inventor of the cathode-ray tube.

    Crookes was very committed to spiritualism, and was the champion of the American medium Henry Slade. When Slade’s stay in England ended in arrest and conviction for fraud, the medium went to visit Zöllner, who was eagerly waiting for someone to help him prove that spirits are four-dimensional. According to Zöllner’s Transcendental Physics of 1878, the experiments were an immediate success.

    The first thing Slade did was to tie four simple overhand knots in a string Zöllner provided. What made this feat so striking was that the string was originally unknotted, with its ends joined together by a glob of sealing wax impressed with Zöllner’s own seal. Of course, in reality Slade undoubtedly managed to switch the strings, but if he had not cheated, his trick would have been genuinely four-dimensional.

    Why? Well, if a spirit could move a little segment of the string ana out of our space, then it would be, for all practical purposes, like having a gap in the string so that one could move it “through” itself to get a knot. Once the string is in the proper position, you move the displaced segment back kata to our space and you’ve tied a knot without moving the ends of the string.

    That’s one way to do it. An easier way is to tie the knot first and then seal the ends together. In and of itself, a knotted string with its ends sealed together does not immediately make one believe that the knot was tied by a four-dimensional spirit. Zöllner was of course aware of this, and he designed several interesting tests by which Slade’s spirit friends could have established lasting proof of their four-dimensionality. Three of the tests are described in Transcendental Physics:

Fig. 56. Zöllner’s string, before and after.

    So the idea was that Slade’s spirits should link the two wood rings, turn some snail shells into their mirror-image forms, and produce a knot in a closed loop of skin cut from a pig’s bladder. Did it work?

Fig. 57. Evidence of 4-D ghosts? (Engraving from J. C. F. Zöllner, Transcendental Physics.)

    In other words, no. Instead of doing what Zöllner had wanted, the spirits put the rings around a table leg, moved a snail shell from the tabletop to the floor beneath the table, and burned a spot on the bladder band.

    Very few scientists were convinced by Zöllner’s experiments. Although Zöllner himself may well have been an honest man, he was almost unbelievably gullible — an unworldly scientist easily taken in by a professional charlatan like Slade. One might think that now, a century later, scientists could no longer be fooled by cheap conjuring tricks, but this does not seem to be the case. Just a few years ago, Uri Geller — who is almost certainly a fraud — was able to obtain the backing and endorsement of a number of physicists at the Stanford Research Institute. A glance at any supermarket tabloid makes it clear that the public’s interest in ghosts is greater than it has ever been.

Fig. 58. Where the balls went in Poltergeist.

    Most modern ghost tales do not bring in the fourth dimension. An interesting exception is Steven Spielberg’s movie Poltergeist. The 4-D aspect of this movie arises when balls that are thrown into the closet of one room appear from the ceiling of another room … indicating a route through the fourth dimension!

    The main effect of Zöllner’s work was that the fourth dimension began to seem disreputable and unscientific. Yet his basic message, spirits live in the fourth dimension, did not fall on totally deaf ears. The notion of beings living in an unseen hyperspace world was taken up by turn-of-the-century Protestant clergymen all over England.

    The best known of these clergymen is, of course, Edwin Abbott, but there were many others who enthusiastically adopted the notion that heaven, hell, our souls, the angels, and God himself could be comfortably lodged in some higher dimension. The basic credo of these Christian spiritualists can be found in A. T. Schofield’s 1888 book, Another World:

Fig. 59. “Aided by the most powerful telescopes.”

Fig. 60. God the Father?

    To a sophisticated thinker, there is something just a little too crude and materialistic about having souls and angels and God be literal chunks of hypermatter in four-dimensional space. Why precisely four?. If any kind of higher-space view of higher reality is reasonable, I would think it to be the notion that our souls are patterns infinite-dimensional Hilbert space. A second objection to the idea of spirits and angels swarming about our space is the feeling that, at higher levels, the whole concept of separately acting individuals might well fall away.

    Let me illustrate this last thought with a Flatland picture, figure 61.

Fig. 61. The One and the Many.

    What seems to be a collection of many individuals may turn out, at some higher level, to just be parts of one higher entity. There is something lacking in a world view that leaves out the notion of higher unity and has diverse spirits poking at our world like pool sharks at a game of eight ball.

    Another failing in spiritualism, Christian or not, is the reliance on anecdotes about miracles and psychic feats. If higher dimensions have any real meaning for us, that meaning should be part of our ordinary life. I don’t believe in miracles; I don’t disbelieve in them. The whole subject just leaves me cold. You can walk on water? You can bend spoons? So what. You’ve proved that nobody really walks on water? You’ve proved that nobody really bends spoons? I couldn’t care less. What I want to know is much simpler: What is it like to be alive?

Fig. 62. Bend that spoon.

    “We live in a three-dimensional space” is really quite a complex statement, and not a statement that every person would automatically think of as true. Explaining to someone why it is that space is three-dimensional really involves convincing them that they should think of the world around them in a certain way. Imagine an alien culture whose architecture is not based on cubes. Without the inevitable three perpendicular lines in each corner of one’s room, the notion of reducing all experiences to the oscillations of a three-knobbed Etch-A-Sketch might seem a bit strange. What about feelings, what about levels of thought, what about dreams?

    What I am trying to suggest here is that perhaps we ourselves are, in some very real sense, beings of more than three dimensions. P. D. Ouspensky wrote something very interesting about this in his 1908 essay ’The Fourth dimension.”

    For Ouspensky, the fourth dimension was not only a spatial concept but a type of consciousness, an awareness of greater complexities and higher unities. For him, the mathematical study of the fourth dimension led quite naturally to a belief in the teachings of mysticism. In their simplest form, these teachings are but two: All is One and The One is Unknowable.

Fig. 63. All is One.

    What, you may ask, do these two high-sounding precepts have to do with the fourth dimension? Quite briefly, the first idea is that higher space can be viewed as a background of connective tissue tying together the world’s diverse phenomena. If one moves toward higher and higher conceptions of space, one is tending toward some ideal “Superspace” in which everything — near and far, past and future, big and small, real and imagined — is together in some great Unity.

    In his 1885 essay “Many Dimensions” the great hyper-space philosopher Charles H. Hinton speaks very clearly and colorfully about his identification of the abstract idea of “space” with the “One” of mysticism:

    So for Hinton, space itself is the medium by which one is to grasp the world’s unity. And what of the second teaching of mysticism, The One is Unknowable? Here we must be clear on the sense in which the word “know” is to be used. Mystically speaking, the One is knowable, knowable in the sense that we can feel space around us, knowable in the sense that we can open our hearts to feel life and beauty and love. It is only for the rational mind that the One is un-knowable.

    Fig. 64. The One is Unknowable.

    No finite string of human symbols can really represent the ultimate reality: call it God, the One, the Absolute, Everything, or what you will. The situation here is a bit analogous to trying to understand an infinite set such as the set N of all whole numbers (1, 2, 3, …). Given the idea of number, we do have a pretty good understanding of what N is — this higher understanding is to be likened to the mystical knowing of the One. But if we insist on a complete, explicit listing of members, then the set N will always lie beyond our grasp — this is comparable to the rational mind’s inability to fully grasp the One.

    In the following long excerpt (also from “Many Dimensions”), Hinton makes clear the sense in which space is, at the deepest level, to be known by the heart and not by the brain.

    Fig. 65. “The least details of the visible world.”

Fig. 66. “The simplest acts of human goodness.”

    The author of this strange yet compelling passage led — in the context of his time — an equally strange life.

    Charles Howard Hinton, known to his family as Howard, was born in London in 1853. His father, James, was a well-known author who over the course of his life moved from being an ear surgeon to a religious philosopher to the champion of a new sexual morality. In his declining years, James Hinton managed to surround himself with a circle of admiring women, most of whom were sexually intimate with him. One of his sayings was, “Christ was the Saviour of men, but I am the saviour of women, and I don’t envy Him a bit!”

    Howard obtained his B.A. from Oxford in 1877, and soon after he married Mary Boole, daughter of the famous logician who invented “Boolean algebra.” In 1880 Hinton obtained a post as science master at the Uppingham School, meanwhile continuing work for a master’s degree in mathematics.

    Despite all of his education, Hinton felt somehow adrift, as if he could not get hold of any really concrete knowledge. For who knows what reason, he hit on the idea of memorizing a cubic yard of one-inch cubes. That is, he took a 36-by-36-by-36 block of cubes, assigned a two-word Latin name (for example, Glans Frenum) to each of the 46,656 units, and learned to use this network as a sort of “solid paper.” Thus, when he wished to visualize some solid structure, he would do so by adjusting its size so that it fit into his cubic yard. Then he could describe the structure by reviewing the list of occupied cells. (Although this sounds incredible, it is not really impossible — Hinton had a system that reduces the number of brute facts to 216.)

    As it turned out, this seemingly insane idea was a fantastic source of inspiration for Hinton. For what he had in effect done was to create within his mind the kind of “three-dimensional retina” that a 4-D being would have (as we discussed in chapter 3). Some mysterious inspiration drove Hinton forward on the correct path, and he next had the idea of learning his block of cubes in each of its twenty-four possible orientations (six choices for the bottom face times four choices for the front face).

    The result of this was that Hinton was now able to take a 3-D object and see its parts simply in terms of what is next to what, thus freeing himself from our spacebound concepts of front-back and up-down. Shortly after this, he took the final step and let the concept of left-right fall away, thus arriving at a perfectly four-dimensional view of the world (as we discussed in chapter 4). Now Hinton could, without difficulty, visualize all the cross sections of a hypercube, undisturbed by the way in which these cross sections can become their own mirror images depending on where the hypercube lies relative to our space.

    As his understanding of the fourth dimension grew, Hinton set to writing about it. His first published essay, “What Is the Fourth dimension?,” appeared in 1880 in the Dublin University Magazine, was reprinted in the Cheltenham Ladies’ College Magazine of September 1883, and finally was released as a pamphlet, with the crassly commercial subtitle “Ghosts Explained,” by Swann Sonnenschein & Co. in 1884. In the period 1884-1896, Swann Sonnenschein published nine pamphlets by Hinton, which were then collected in the two-volume set Scientific Romances. Some of the pieces were essays on the fourth dimension, and some were what we would now call science fiction stories.

Fig. 67. Hinton was a bigamist.

    Hinton might have settled into a comfortable life like his father’s: the life of a writer respected by intellectuals and loved by many women. But in 1885 disaster struck. Charles Howard Hinton was arrested for bigamously marrying a woman called Maude Weldon, with whom he had spent a week in King’s Cross Hotel, and by whom he was the father of twins. The headmaster at Uppingham had, till this time, been under the impression that Maude Weldon was Hinton’s sister — apparently she was a regular guest of Howard and his legitimate wife, Mary. Of course, he lost his job. His career was ruined. When he carne up for sentencing in 1886 the magistrate let him off with a token three days in jail. Soon after this, Howard, Mary, and their children fled to a teaching post at a middle school in Yokohama, Japan.

    The torment Hinton must have undergone during this period shows in two of his “Scientific Romances": “Stella” and “An Unfinished Communication.” “Stella” tells of a girl whose guardian has made her invisible, so that she will not fall into the traps of vanity and physical love. Yet Stella does find a lover, a man who forces her to wear clothes and make-up so that all can see her. It is easy to imagine that Maude Weldon is the model for Stella.

    “An Unfinished Communication” is a very strange story indeed. It tells of a desperate man who seeks the services of an Unlearner, a man who can make him forget his horrible past. The Unlearner stresses the importance of living openly and sharing one’s secrets, something Hinton must have been frantic to do as he sank deeper and deeper into the morass created by his attempts to live out his father’s free-love philosophy.

    When Hinton left for Japan, he left a manuscript in the hands of two friends. This book, A New Era of Thought, appeared in 1888. Here Hinton outlines in detail his system for learning to think four-dimensionally by manipulating a set of eighty-one colored cubes, cubes that are to stand for the eighty-one parts of a 3-by-3-by-3-by-3 hypercube. Hinton is at his most confident here, and he asserts that “the particular problem, at which I have worked for more than ten years, has been completely solved. It is possible for the mind to acquire a conception of higher space as adequate as that of our three-dimensional space, and to use it in the same manner.”

    In A New Era of Thought Hinton also explains why it should be possible for us to form 4-D thoughts. People sometimes feel that since our brains are made up of 3-D nets of neurons it is futile to hope to model 4-D patterns in our brains. But, replies Hinton, is it not possible that our space has some slight 4-D hyperthickness? If this is indeed the case, then the small bits of matter that code up our thoughts may be free to form actual 4-D patterns, and 4-D thought is easily achieved.

Fig. 68. Thick Square with mental image of A Cube.

    It was at one time possible to purchase sets of Hinton cubes from his publisher. I once made myself a set of them, and was somewhat disappointed. Playing with the colored cubes may not really be the best way to start thinking four-dimensionally.

    Eighteen ninety-three found Howard Hinton at Princeton University, as an instructor in the Mathematics Department. Oddly enough, Hinton devoted most of his energies while at Princeton to the invention of a “baseball gun.” The point of having such a gun was that during the week’s practice sessions the Princeton team could practice hitting really fast balls — without the pitchers having to wear out their arms. The gun could shoot a baseball at speeds of up to seventy miles per hour, and two rubber-coated steel fingers attached to the muzzle made it possible to shoot curves as well.

    Hinton was fired before long, and found his next job at the University of Minnesota. In 1900 he left Minnesota for Washington, D.C., where he worked first at the Naval Observatory and then at the Patent Office. His faithful wife Mary followed Howard through all these peregrinations, bearing him four sons. She was known in Washington as a lecturer on poetry.

    Hinton died suddenly and dramatically at the age of fifty-four, in 1907. A contemporary news article on his death, headlined SCIENTIST DROPS DEAD, describes how Hinton fell dead after complying with the toastmaster’s request for a toast to “female philosophers,” at the annual banquet of the Washington, D.C., Society of Philanthropic Inquiry.

    Charles Howard Hinton lived a wild and varied life. I would like to think that what sustained him through it all was that mystical vision of space he enjoyed as a young man.

    I hope that by now most readers will have some feeling for the fourth dimension as a concept interesting in its own right. In parts II and III of this book we will proceed to see how the fourth dimension can be used to understand the true nature of space, matter, time, and mind.

Part II

SPACE

6

What We’re Made Of

This is a free online webpage edition, Copyright (C) Rudy Rucker 2016.
Purchase paperback or ebook editions from Dover Publications or Amazon.

WE ARE ACCUSTOMED to thinking of the world as made up of lumps of matter floating in empty space. Matter is something, and space is nothing. But is this really correct? In the past, many powerful thinkers have held that the space between visible objects is filled with a subtler material, a smooth and continuous substance, a “universal plenum,” or aether.

    One does not hear much about the aether these days. But in the nineteenth century the concept was as commonplace as “fields” are now. In his article on aether for the ninth edition of the Encyclopaedia Britannica, James Clerk Maxwell (1831-1879), founder of the modern theory of electromagnetism, wrote:

    But why bother filling space up with aether? What’s wrong with having space just be the empty background against which matter moves? One problem is that if space is really empty, then it’s hard to understand how gravitational force can be transmitted. Isaac Newton set up his famous law of gravity as a quantitative description of the gravitational force, but he was keenly aware that he lacked any understanding of how this force could act across the empty distance of space. Writing in the late seventeenth century, Newton complains:

Fig. 69. How do we send signals through the void?

    As we will see shortly, the modern theory of gravity embodied in Einstein’s general theory of relativity does use a sort of “space-filling aether” to explain how gravitating bodies can influence the motions of faraway objects. For Einstein, the continuous aether is simply space itself, and it is the bending of this aether in higher dimensions that produces gravitational attraction! But I’m getting ahead of myself.

    Late in the eighteenth century, Thomas Young and Augustin Fresnel successfully advocated the theory that light is a type of wave motion, rather than a stream of particles. A variety of optical effects, such as diffraction and polarization, seemed to indicate that light must be a vibration in some underlying medium. It was assumed that outer space is filled with a “luminiferous aether,” which transmits the wave motion that is light. As the nineteenth century wore on, this aether carne to be regarded as the transmitter of magnetic and electrical forces as well.

Fig. 70. Wave-particle duality.

    Nowadays we are comfortable with the quantum-mechanical idea that light is both particle and wave. A photon is a wavy yet solid little package that can zip through empty space without the benefit of any invisible vibrating jelly underfoot. Just to grasp how alien the notion of a luminiferous aether has become, try to picture a nineteenth-century drawing room from which all the luminiferous aether has been evacuated. The gas jets are burning but no one can see them! A light source with no luminiferous aether to ferry the light’s vibrations would be like a bell pounding silently in an evacuated bell jar.

    The idea of a room emptied of aether is, of course, quite unrealistic, for most aether theorists felt that ordinary matter offered an open structure to aether. In Thomas Young’s words, aether should be able to pass through solid objects “like wind through a grove of trees.” If this is the case, then no room’s walls could ever keep the aether out.

    The idea of an aether wind blowing right through your body is sort of refreshing. Sometimes, standing on the breezy crest of a hill, I have the illusion that the air is blowing through me, dusting off my molecules. If we do think of aether as being the same as space, it would seem to make sense to think of the aether as actually blowing through us as we move through space.

    But does this really make sense? Certainly we can move relative to other objects, but is space the kind of thing one can move relative to? The answer, as it turns out, is 220. Sometimes one hears that modern physics has proved there is no aether. What this actually means is that modern physics has proved that it is meaningless to speak of motion relative to empty space. How did this come about?

    In the late nineteenth century, physicists and astronomers became interested in determining the Earth’s absolute motion. The idea was that space is filled with stagnant, motionless aether, so that as our Earth and our solar system and our galaxy tumble about, we should feel some kind of aether breeze on our faces. The direction in which the breeze feels strongest would then be the direction in which we are really moving.

Fig. 71. A light source, with and without aether wind.

    “Feeling the breeze” is of course a metaphorical way of speaking: the actual experimental technique was to measure the speed of light in various directions. Light would go fastest when traveling with the aether wind, and slowest when traveling directly against it.

    A number of attempts to thus measure the motion of the earth through the aether were made, the most famous of these being the well-known Michelson-Morley experiment of 1887. The results were that a light signal always travels at the same speed, regardless of which direction it goes in. Feeling for the aether wind by measuring speeds of light rays is like sticking a flag out of a spaceship to see which way you’re going. Nothing happens: there’s no air in outer space, and no wind.

Fig. 72. No air, no wind.

    After the null result of the Michelson-Morley experiment, a few physicists still hoped to find some alternative way of measuring the earth’s motion through space and the aether. But most scientists began to suspect that there is, even in principle, no possibility of detecting an aether wind. In 1905 Albert Einstein made just such an assumption, and created his enormously powerful special theory of relativity. This theory, which we will discuss in more detail in chapter 9, is based on two assumptions: (1) The speed of light is always the same, and (2) There is no way to detect absolute motion.

    The first assumption was, by 1905, pretty well established on the basis of experiments like Michelson and Morley’s. The second assumption is the one that represents a dramatically new way of looking at things. According to Einstein, there is no possible way in which one can detect motion through empty space. He jumps here from an experimental fact (that measuring speeds of light rays does not reveal an aether wind) to a blanket hypothesis (that no possible experiment can ever detect an aether wind).

    Most of us smile when we hear of the nineteenth-century aether concept. But it is important to realize that insofar as we believe in space as something existing in its own right, we too have an aether concept. Empty space is aether, aether is empty space. But what, then, is the meaning of Einstein’s second assumption vis-à-vis aether? Here is what Einstein wrote on this question in his 1920 essay “Ether and Relativity.”

Fig. 73. Space is like a sea’s surface.

    The reader may wonder why Einstein bothers to go through such intellectual contortions. If the aether isn’t made up of distinct trackable parts, then why bother with it? Why not just go ahead and say that empty space is pure nothingness? Why pretend that space is some continuous substance called aether? Let’s go back to Flatland for an answer.

Fig. 74. A Square about to get guillotined.

    The Chief Circle decides to let his wife execute A Square … She’s a bloodthirsty segment, ready and willing to cut poor Square in half. Our hero is shackled in a heavy box with one opening. The Queen surges forward, her sharp end aglitter. She thrusts her point into the box’s small opening, thrusts again, and thrusts once more for good measure. But when they open the guillotine box, A Square is as good as new. What happened?

    To understand what happened, we should start thinking of the space of Flatland as being somewhat like a rubber sheet or, even better, something like a huge unbreakable soap film. If A Cube seizes a piece of Flatland’s space and pulls up, then he can stretch a bit of the space to be bigger than one would expect. And this is just what he did. Cube grabbed the bit of space inside the guillotine box and stretched that space for all he was worth. The Queen’s twelve-inch body was not long enough to reach over the “space bump” and get to A Square. Here is how it seemed to Square (I quote again from the imaginary Further Adventures of A Square):

Fig. 75. A Cube stretching Flatland’s space.

Fig. 76. Lumpy space.

    The point of this story is that if we think of space as being made up of a continuous aether jelly, then it becomes meaningful to speak of stretching or distorting space. Although, as Einstein stresses, we must not think of space as made up of particles, it is meaningful to think of space as having bumps and undulations. There is no absolute sense in which one might say that a given bump is moving this way or that, but one can certainly notice how the bumps move relative to each other.

    Bumps in space (spacetime, strictly speaking) can be used to explain gravitational attraction. Einstein’s 1915 general theory of relativity incorporates a theory of gravity that can best be understood as saying that (1) Matter and energy distort space, and (2) The distortions of space affect the motions of matter and energy. Aether, or space, thus serves as the medium for transmitting gravitational effects. Mass affects space, space affects mass. Let’s see how.

    We must imagine that the space around any massive body is stretched. The denser the mass, the greater the stretching. A good image for this is of a cue ball resting on a rubber sheet. A sheet sags down around the ball. Or we could think of a helium balloon under the sheet, making it sag up (as drawn in figure 77). The “up” or “down” doesn’t matter here; the point is just that the presence of matter stretches space.

Fig. 77. A massive body makes a bump in space.

    Now let’s try to see how the curvature of space affects the motion of particles in space. A particularly clear-cut example arises if we let our moving particle be a photon, a tiny piece of light.

Fig. 78. The shortest path from A to B.

    Ordinarily we think of light as traveling along straight lines. But if space is curved, there is no such thing as a really straight line in space. Nevertheless, light does travel along the straightest possible lines. Equivalently, we can say that a light ray from point A to point B will always go along the shortest possible path from A to B.

    If there is a big bump in space between A and B, then the shortest path will not be directly over the bump. The shortest path will be the path that compromises between going right over the bump and looping way out around it. This is easy enough to understand if, in figure 78, we think of A and B as villages separated by a mountain. The natural, shortest trail between the two is along the wavy line.

Fig. 79. The shortest path from A to B, seen from above.

    If we look down at this path from above, it looks for all the world as if the gravitational attraction of the massive object had bent the light ray by pulling on it. Yet what has really happened is that the mass stretched the space so that the shortest path from A to B has a bulge in it. The gravitational bending of the paths of material objects can also be explained in this way, although there the situation gets a little more complicated.

    So gravity can be explained by assuming that matter curves space. But why should matter do this? Why should matter curve space?

    One explanation is that space curvature is what matter is. William K. Clifford first proposed this theory in an 1870 paper called “On the Space Theory of Matter":

    I hold in fact

1. That small portions of space are in fact of a nature analogous to little hills on a surface which is on the average flat; namely, that the ordinary laws of geometry are not valid in them.
2. That this property of being curved or distorted is continually being passed on from one portion of space to another after the matter of a wave.
3. That this variation of the curvature of space is what really happens in that phenomenon which we call the motion of matter, whether ponderable or ethereal.
4. That in the physical world nothing else takes place but this variation, subject (possibly) to the law of continuity.

Fig. 80. Three bits of matter in space.

    This is a very interesting view of matter, a view that the contemporary physicist John Wheeler has called geometrodynamics. Traditionally, people thought of matter as a solid substance floating in empty space. But under the geometrodynamic viewpoint, space is not really empty and matter is not really solid. Space is an aether, a continuous substance that is curved in higher dimensions. And matter is a sort of patterning in the aether.

    This idea is intellectually satisfying because it represents the completion of a dialectic triad. Before we had as thesis the concept of solid matter, and as antithesis the concept of utterly empty space. Matter versus space; something versus nothing. The synthesis is to regard space and matter both as a continuous aether substance: when the aether is flat it looks like empty space, when it is sharply curved it looks like matter. The old thesis and antithesis are simply different aspects of the higher synthesis!

Fig. 81. A vortex thread.

    Clifford’s notion of building up matter out of pure curved space was a very bold step forward. A few years earlier, William Thomson had taken a partial step in this direction. Rather than taking matter to be higher-dimensional “bumps” in aether or empty space, Thomson proposed that matter is made up of three-dimensional vortex rings in the aether.

    A “vortex ring” is something like a smoke ring, a circle of substance that swirls around and around on itself. Thomson’s theory was inspired by Hermann von Helmholtz’s 1857 proof that in a perfect fluid, any whirlpools or vortices must be centered on lines that either go out to the boundaries of the fluid or curve back on themselves to make circles. In watching the water drain out of a bathtub, one often observes a vortex line of the first type: a wobbly, threadlike whirlpool running from the water’s surface to the drain below. The funnel of a tornado is a similar example of a vortex thread. Now if the vortex thread bends back on itself to make a circle, one gets a vortex ring. The remarkable thing about vortex rings is that they consist of a closed-off region of the underlying fluid. This can be seen in watching a smoke ring. For a while at least, a smoke ring does not gain or lose any air … It consists of the same smoky air circling around and around on itself.

Fig. 82. A vortex ring.

Fig. 83. A smoke-ring projector. (Engraving from A. E. Dolbear, Matter, Ether, and Motion.)

    Some nineteenth-century investigators actually built smoke-ring projectors and spent hours watching how smoke rings move, vibrate, and bounce off each other. The hope was that the various properties of matter could all be explained by treating atoms as vortex rings in some perfectly frictionless underlying aether. A particularly nice feature of the vortex-ring theory was that it explained how atoms could have a measurable size, yet be indivisible: a smoke ring has a certain radius, but if you try to cut the ring in half you get nothing but some rapidly dispersed air currents.

Fig. 84. The title page from Balfour Stewart and Peter Guthrie Tait’s The Unseen Universe.

    The vortex-ring theory of matter failed to lead to any testable predictions, and was eventually abandoned. One of the last books to advocate the theory was a curious work called The Unseen Universe (1875). The book seems to say that the soul exists as a knotted vortex ring in the aether. By way of illustrating this notion, the authors (Balfour Stewart and Peter Guthrie Tait) put a picture of a knot on the title page and the binding of their book.

    Although it is not immediately evident, the image of a knotted vortex ring involves the fourth dimension. The reason for this is that it is impossible to pass a vortex ring through itself; it has a kind of integrity or solidity akin to that of an actual ring of material substance. Just as Slade and Zöllner claimed that higher-dimensional spirits could knot their sealed cords, the authors of The Unseen Universe held that God creates our immortal souls by putting intricate knots and braids into vortex rings of aether. James Clerk Maxwell was amused and fascinated by this notion, and wrote a poem about it:

Fig. 85. “My soul is an entangled knot.”

    As the nineteenth century drew to a close, more and more bizarre theories involving matter, aether, and the fourth dimension began to appear — theories that make one think of baroque, overdone ornamentation, of layer after layer of Victorian corsets and petticoats, of opulent, fin de siècle decadence.

Fig. 86. Water squirts.

    In 1892, for instance, Karl Pearson proposed in his Grammar of Science that the aether is a four-dimensional fluid seeping into our three-dimensional space — sort of like water welling up through the holes in the bottom of a boat:

    A fourth dimension is involved in Pearson’s theory, for this higher dimension is needed to store the “machinery of the taps;/ that bring the flow of aether to each atom. Two such aether squirts will indeed attract each other because the aether between them is moving more rapidly than in the other parts of space. Let me briefly explain why. There is a law of fluid dynamics called Bernoulli’s law, which says that the more rapidly a fluid moves, the lower its pressure is. Since the aether between the squirts moves rapidly, there is a low-pressure region between them, and the rest of the aether will tend to compress this low-pressure region. Thus the two squirts will move together as if drawn by gravitational attraction.

Fig. 87. Two Flatland aether squirts.

    Crazy theories … but none of them crazy enough to be right!

    In chapter 11 we’ll look at the modern way of building matter up from space-aether. Clifford’s idea that matter is a bump in space is basically correct — but he was too conservative in assuming the underlying space to be three- or four-dimensional. According to modern quantum mechanics, a bit of matter is a “bump” in an infinite-dimensional Hilbert space!

8

Magic Doors
to Other Worlds

This is a free online webpage edition, Copyright (C) Rudy Rucker 2016.
Purchase paperback or ebook editions from Dover Publications or Amazon.

JUST SOUTH OF BALTIMORE there’s a highway exit labeled “Brooklyn.” Wouldn’t it be great if the exit led right into New York? Even better, wouldn’t it be nice to have a special kind of superdoor leading, say, from your living room to the Tuileries gardens in Paris? Or, most exciting of all, how about a hyperdoor leading out of your space and into a totally different universe?

    People have always enjoyed thinking about such magical doors. The perfect symbol of the mind’s freedom from the body’s spatial limitations, magic doors occur throughout fantastic literature, from Lewis Carroll to C. S. Lewis to Robert Heinlein. As a rule, writers of fiction have been very vague about how magic doors might actually be constructed; at best one hears them explained as “tunnels through hyperspace.” But as it turns out, modern cosmologists have developed some good ways of thinking about magic doors (also known as Einstein-Rosen bridges or Schwarzschild wormholes).

    To get the picture, we turn as usual to A Square. Suppose that Flatland is a plane, and parallel to it is another plane called Globland. Ordinarily, there is no way that an inhabitant of one of these two-dimensional universes can get to the other universe. But suppose that somehow a flap of space from each world has been snipped out, and say that the two flaps are sewn together. Now the Globbers can visit Flatland, and the Flatlanders can visit Globland. I quote once again from that imaginary classic, The Further Adventures of A Square:

Fig. 112. A space strip connecting Flatland and Globland.

    The problem with the world-to-world hookup illustrated in figure 112 is the presence of lethal space edges. But there is a much better way to connect two planes.

After a day’s thought, A Cube reappeared in my study. Three highly placed Women were visiting me, and the sudden materialization of the Cube’s cross section sent them into an ecstasy of fear. In their eyes I was a magician, the Cube my familiar spirit. I was eager to dazzle these lovely Segments; accordingly I played the wizard’s part.

Fig. 114. A Square showing off for the ladies.

    No one had yet ventured through the altered Other-Door. Desirous of assuring my conquest of Una, I courageously pressed forward to the mysterious Disk. It gave the strange appearance of a circular Mirror, such as ornament our Trees at festival time. Peering into it I could make out smaller and smaller Globbers, ever-dwindling toward the inconceivably distant central Point. I feared to enter, feared that I might be crushed by Shrinkage. But Una was vibrating at my side, urging me on with her low, melodious voice. “Come, Una,” I said, and slid forward into the weird Disk which somehow contained all of Globland.

Fig. 115. A Square and Una at the space tunnel’s mouth.

    The Globbers had appeared quite shrunken and distorted before we entered the Door. But now, as we pressed forward, they took on their familiar, albeit Irregular, appearance. Could it be that we had shrunken to their size? All around us lay the endless expanse of Globland. Was this really the interior of some magical Disk? My thoughts were interrupted by Una’s excited cries.

    UNA: Oh, look, dear Square, Flatland is now a Disk itself!

    I (turning to look back): Indeed. Perfect Symmetry prevails. As seen through the Space Tunnel, Flatland is a Disk in Globland, and Globland is a Disk in Flatland. All this have I wrought for your pleasure, my Lady.

    UNA: It is well, my Lord. My Husband cannot disturb us here in this enchanted Land.

    I: Then let us dally, fairest Una.

    UNA: Freely, my Lord … yet look into the Disk of Flatland. There slides my Mate, A Hexagon!

    I: He is small and puny. He is an ant.

    UNA: But, oh, dear Square, he waxes as he nears the Disk’s edge!

    I (to a nearby Globber): How now, sirrah, wilst grant me a boon?

    GLOBBER: Blub, yubba, gloop.

    I (seizing him with my Mouth): Just stretch yourself (mmpf) like this (mmpfmmp) and this, dear friend. And in such wise do curtain our Seraglio.

    The deed was done, and Una and I were free to take our pleasure. My perfect content was marred only by one question: How was it that passing through the Space Tunnel turned Inside to Outside, and Outside to In?

Fig. 116. A Globber blocks A Hexagon’s view of Una’s trans-dimensional tryst.

    Looking at the space tunnel from outside the planes of Flatland and Globland, we can see the answer to A Square’s question. The throat of the “wormhole,” or space tunnel, is bounded by a circle in either world. A Flatlander looking at this circle sees light from every part of Globland … Thus it seems to him that Globland is somehow compressed to fit inside a circle. By the same token, a Globlander will see light from all over Flatland as coming from the circular throat of the wormhole.

Fig. 117. A Hexagon thinks Globland is a disk.

Fig. 118. Hyperspace tunnel to another universe.

    Now, as we have done so many times before, let’s imagine raising everything by one dimension. Imagine that there is another 3-D universe, “parallel” to ours in 4-D space. If we could move ana through hyperspace we could get to the other universe. But we find it very hard to move in the fourth dimension. How, then, could we ever get to the other universe? By traveling through a hyperspace tunnel, a so-called Einstein-Rosen bridge. What would such a hyperspace tunnel look like? The entrance to it would look like a sphere that contained a whole other universe, incredibly shrunken and distorted. If you dived headfirst into this sphere, you would feel as if you passed right through it. But then when you looked around you would realize that you were in the other universe now, and looking back at the hyperspace tunnel, you would see a sphere that seemed to contain your whole original universe, incredibly shrunken and distorted.

    There is actually a very familiar object that looks just like the mouth of an Einstein-Rosen bridge: a glass Christmas tree ornament. Such a spherical mirror reflects, in principle, the entire universe around it. The farther an object is from the mirror’s surface, the closer its image seems to lie to the mirror’s center. Of course, if you were looking at the mouth of a hyperspace tunnel to another universe, you would not see the mirror image of our universe, you would see what looks like the mirror image of another universe.

Fig. 119. A collapsing star could form an Einstein-Rosen bridge.

    Is there any way — short of the miraculous intervention of a higher-dimensional being — that an Einstein-Rosen bridge could actually come into existence in our cosmos? Yes. If there are indeed some other 3-D universes parallel to ours, it might be that a sufficiently dense object could bulge our space out enough to touch another space. And the two spaces might join together like two soap films that have been brought into contact.

    In order to easily illustrate this, let’s just work with a cross section of two parallel Flatlands. As has been discussed, the presence of matter causes space to bulge out. Now, just as a woman’s high heel will dent a rubber mat more than a man’s larger heel, it turns out that the denser the matter, the greater the space distortion. If our sun could be compressed to a much smaller size, then it would distort space much more.

    The sun is basically a ball of hot gas. The gravitational attraction of the sun’s particles for one another works to try to make the sun smaller. The thermal agitation of the hot gas particles works to try to make the sun bigger. The two forces balance out with the sun just the size it is. Eventually, however, the sun will grow cooler. As it cools there will be less outward pressure, and gravity can work to make the sun get smaller and denser. This compression heats the sun back up for a while, but eventually it cools again, and contracts even more.

    All stars go through this stepwise contraction process as they age. Depending on the star’s starting mass, various final outcomes are possible. If at some point a star contracts too rapidly, it explodes and makes a nova or a supernova. If the star is not too massive to start with, it may contract down to form a solid, glowing lump of metal. If it is a bit more massive, it squeezes down further by collapsing the metal atoms. Protons combine with electrons to give neutrons, and one gets a fantastically dense “neutron star.” These stars are made up of a substance called neutronium, which masses about one billion kilograms per cubic centimeter.

    The contraction is most dramatic if the star’s mass is so great that even the neutronium ends up getting crushed. In such cases the star shrinks down to a tinier and tinier volume … possibly even down to point size! Such super-dense collapsed stars are the “black holes” that one so often hears about. The reason for the name is that if a star is dense enough, then its gravitational attraction becomes so powerful that light cannot escape the star. In other words, a star that becomes dense enough takes on the appearance of a black, lightless sphere in space — a region that emits no light at all. Obviously it is difficult to “see” a black hole, but a variety of indirect observations seem to indicate that there really are quite a few black holes floating around in space.

Fig. 120. A black hole absorbs light.

    As indicated in figure 119, if a massive star or black hole distorts space enough, it is possible that an Einstein-Rosen bridge to another universe could be created. Flying into the right kind of black hole might pop you out into a different world. The theme of black-hole-as-gateway-to-other-realities is amusingly used in the Walt Disney movie The Black Hole. At the end of the movie, the good guys and the bad guys all fall into a huge black hole. The hole turns out to be an Einstein-Rosen bridge with two exits: heaven and hell! This type of idea harks right back to the hyperspace theologians of Abbott’s time.

    It is also possible to have a wormhole, or E-R bridge, which leads back to the same space it starts from. This could be very important. Here’s why.

Fig. 121. The wormhole is a short cut from A to B.

    According to Einstein’s special theory of relativity, nothing can travel faster than light. This has always been a severe limitation for conscientious science fiction writers. It takes light four years to get to Alpha Centauri, our sun’s closest neighbor … And any conversation or cultural exchange that is riddled with four-year gaps is going to be pretty dull to read about. If you’re interested in traveling to another galaxy, the situation is much worse: our nearest galactic neighbor, the Great Magellanic Cloud, is well over ten thousand light-years away!

    SF writers have often avoided this problem by supposing that (1) our space is folded back on itself, and (2) there are E-R bridges or wormholes connecting the various folds. By finding the right hyperspace tunnel, a very long journey can be cut down to manageable size. One of the first writers to use this device was Robert Heinlein, the father of modern SF. The analogy is to an ant on a silk scarf. Normally it takes the ant a long time to crawl from one corner to the other, but if the scarf is all crumpled, then by leaving the material’s surface the ant can find a short and direct path through 3-D space.

Fig. 122. A hyperspace hop.

    Of course, being able to take a desired short cut depends on there being a usable E-R bridge at the right location. Some SF writers avoid this problem by having their space travelers create E-R bridges as necessary. In Piers Anthony’s mind-boggling Macroscope, the method of travel is to get inside a large object (say the planet Neptune), use some miraculous ray to make the object collapse to black-hole size, and then zip through the black hole to come out somewhere different!

    If we could really manipulate the curvature of space at will, then there would be an interesting alternate way of traveling through hyperspace. Instead of building a tunnel or wormhole leading to another space, one could pinch a small hypersphere off our space and just float away. This could be risky, of course, as you’d have no way of predicting where and when your little space bubble might meet another universe. But one nice thing about leaving space by pinching off a closed piece of it is that this doesn’t leave a hole behind.

Fig. 123. A piece of space pinching off.

    So far in this chapter we have discussed how one might travel through hyperspace to other universes, and we have mentioned how this type of travel might also be useful for finding short cuts from one region of our space to another. One question we have not touched on yet is whether or not there really are any other universes.

    As was hinted at in the last chapter, it is abstractly possible to treat the gravitational distortion of space as a type of stretching and shrinking of flat space — as opposed to the bulging out of flat space into some higher dimensions. Many scientists feel that “curved space” is just a colorful phrase, and that there really is nothing outside our three space dimensions. For these somewhat cautious thinkers, the visible universe is all that exists, and any talk about alternate universes is just empty dreaming.

Fig. 124. A Square traveling on a space sphere.

     But if we take the fourth dimension quite seriously, then it seems natural to suppose that there might be other universes. All these universes taken together make up a much grander entity, variously known as the cosmos or as super-space. In traditional Christian doctrine, the cosmos has three parallel layers: heaven, our world, and hell. The theosophists hold that the cosmos has seven layers, six of these being “astral.” A common notion in science fiction is that there are endlessly many parallel universes — with each possible universe existing somewhere. A variation of this last idea has actually been incorporated into modern quantum mechanics, and we will return to it in later chapters.

Fig. 125. Parallel worlds.

    Probably the least interesting viewpoint on the question of how many universes there are is that which says: “The whole question is meaningless. No one has any idea how to detect another universe. Since statements about other universes cannot be subjected to immediate scientific testing, these statements don’t really say anything at all.”

    Such a viewpoint combines two assumptions: (1) Seeing is believing, that is, if something is real we can find a way to observe it; and (2) There’s nothing new under the sun, that is, we’ve already observed every type of thing we ever will observe. The first assumption is central to the philosophical school of logical positivism, a modern outgrowth of traditional British empiricism. For the positivist or empiricist, the world is basically equivalent to the sum total of all possible sensory experience. I have no problem with this — indeed, I will advocate a similar position in part III. It is the second assumption I object to. No one has yet found a way to observe the other universes, granted. But this does not automatically prove that no one will ever find a way to “see” the other worlds.

    People speculated about atoms centuries before there was any hope of detecting an individual atom. And if no one had ever talked about atoms, the means to detect them might never have been developed. Talking about other universes would be a more respectable pastime if we could already see them. But unless we go ahead and try to imagine ways in which this might happen, the day will never come. We have already seen that something like Einstein-Rosen bridges may exist as actual pathways to other universes. What I want to do now is to think of some other ways in which these universes might make themselves known.

    Just so as not to be lost in a sea of possibilities, let’s limit ourselves to one fairly reasonable model for the cosmos: four-dimensional hyperspace with a number of hyper-spheres floating in it. Each of the hyperspheres makes up a single universe. We might think of these hyperspheres as being something like bubbles in a fluid; alternatively, we might think of them as being like planets floating in space. In terms of two-dimensional beings, we are thinking of a bunch of Spherelands.

Fig. 126. Hyperspherical universes floating in hyperspace.

Fig. 127. Two spaces merging.

    If we are limited to the 3-D hypersurface of our hypersphere, is there any way in which we can become aware of the other hyperspheres? One very dramatic way in which this could happen would be if one of the other universes happened to bump into ours. Imagine what a Spherelander would see if his space bubble were suddenly to bump into and join up with another space bubble. The effect would be, in our terms, as if all the visible stars were to move out toward the horizon, leaving room for a whole lot of new stars at the zenith! Of course, if the other space was considerably smaller than ours, the effect would be less dramatic. If a small hypersphere merges with ours we might perceive this joining-up process as the occurrence of an unusually bright spot in the sky. Conceivably, the very distant and very bright light sources known as quasars (for “quasi-stellar objects”) are spots where small energy-filled hyperspheres are in the process of joining up with our own hypersphere.

Fig. 128. Quasars?

    But are there any less large-scale and less obvious effects that might point to the existence of other hyperspherical spaces? In thinking about this question, it is helpful to imagine the situation of a man blind from birth. Suppose that he takes it into his head that the sun, the moon, the other planets, the stars, and so on do not exist. Suppose he insists that space is a vast emptiness containing but one object: the spherical planet Earth. How might you convince him he is wrong?

Fig. 129. Blind and stubborn.

    Offhand I can think of three approaches: (1) You might teach him to be sensitive enough to heat radiation so as to “feel” the sun’s passage across the sky. Or you might couple a telescope to a photocell that controls the volume of a little buzzer. Moving this telescope around, the man would learn to perceive the stars as “loud” spots. (2) You could get him to notice the rise and fall of the tides, and explain to him that this is caused by the gravitational pull of the moon. (3) You could get him to notice the various effects of the Earth’s rotation: the equatorial bulge, the so-called Coriolis force, and the existence of poles. And then you could argue that if the Earth is really rotating, it must rotate relative to some other celestial objects.

    Let’s consider the higher-dimensional analogues of these three sorts of ways to notice other worlds.

    1. Unless another hypersphere actually touches ours, there is no way for light to leave its space and come to ours. So we cannot hope to see it. So far as we know, any other kind of radiation would also be confined to the 3-D space where it originates. So it does not seem very likely that any kind of training or equipment can help us “see” the other hyperspheres. Our position is, after all, really like that of a polygon on a Sphereland … not like that of a blind man on Earth. And there seems to be no reason that any one Sphere-land would send radiation to other Spherelands. A second difficulty here is that even if some higher-dimensional radiation from other spaces did fall on our space, the radiation would not be focused at any particular spot. At best, those regions of space that lie nearer to the other hyperspheres would be observed to have more radiation in them.

    2. Gravity is not so much a type of radiation as a condition of space. It is conceivable that there could be a higher-dimensional analogue of gravitation, according to which various four-dimensional objects would distort the hyperspace they float in. Just as the moon’s motion around the Earth causes a bulge to travel around the Earth’s surface, we can suppose that a nearby hypersphere might distort the shape of our own hypersphere. But present-day scientific apparatus is not even sensitive enough to measure our hypersphere’s radius — let alone “tidal” variations of the same radius.

    3. This approach is probably the most important one. Given that nonrotating spheres are virtually unheard of in our universe, it seems rather likely that the hypersphere that makes up our space is itself rotating. And, as I will now argue, if our space is rotating, then it is almost certain that there are spaces outside our own. This last move will not seem immediately obvious: it is based on a fairly unfamiliar notion called Mach’s principle. Ernst Mach (1838-1916) formulated this principle to account for objects’ inertia, their tendency to resist being moved. The point Mach makes is that if an object were totally alone in empty space, then it would be meaningless to say the object is rotating or being accelerated. An object alone in empty space would, in effect, have no inertia, no heft, no resistance to being moved. Therefore, argues Mach, the fact that an object on Earth has heft is a consequence of the existence of all the distant stars and galaxies. By the same token, the fact that we notice the Earth’s rotation is also the result of the existence of the distant stars. Generalizing Mach’s principle to hyperspace, we conclude that if we can find evidence that our hyper-spherical universe is rotating, then we have good reason to believe that there are other universes, relative to which we are rotating. O.K. Now the question is this: What kind of evidence for the rotation of our universe might we hope to find?

Fig. 130. Spinning globe with hurricanes.

     Well… How can the blind man tell that Earth is rotating? A major effect of the Earth’s rotation is that the Earth, instead of being a perfect sphere, bulges at the equator and is flattened at the poles. But, as was already mentioned, we can’t even measure our universe’s curvature yet — let alone the deviation of this curvature from perfect hypersphericity. Another very important effect of the Earth’s rotation is the existence of the Coriolis force. It is the Coriolis force that causes hurricanes to rotate clockwise in the Northern Hemisphere and counterclockwise in the Southern Hemisphere.

Fig. 131. The Coriolis force in everyday life.

    How does the Coriolis force work? To see it in operation, go to a playground that has one of those hand-pushed merry-go-rounds. Get on, crank it up, face the center, and let your right leg swing back and forth. What you will notice is that when your leg moves toward the center, some mysterious force pushes it to the right, and when your leg moves away from the center, a force pushes it to the left. This is a Coriolis force, resulting from the merry-go-round’s rotation relative to the rest of the universe. If the merry-go-round were the only object in the universe it would be meaningless to say it is rotating and you would, in fact, feel no Coriolis force. Returning to the swinging leg for just a moment, note that the net result of the Coriolis force is to turn a simple swinging motion into a clockwise circling motion. It is the same force, on a larger scale, that causes air masses to spiral in a clockwise fashion in Earth’s Northern Hemisphere.

    The fact that hurricanes tend to turn different ways in different parts of the world is a definite asymmetry. If we could find some pervasive asymmetry in the behavior of matter throughout space, we might suspect that this asymmetry results from some overall property of the universe. I have no very specific suggestions as to what kinds of asymmetries we should look for, but one general principle is that we should expect to find the crucial asymmetry in the domain of very small particles. Why? My feeling is that if our space has a slight thickness in the fourth dimension, then very small particles may enjoy a certain four-dimensional freedom of motion, and thus be more susceptible to influences from hyperspace. The situation is complicated by the fact that the rotation of a hypersphere in hyperspace is more complex than a sphere’s rotation in space.

    But enough of these subtleties. Whether there really are other universes is not nearly so important as the realization that we are integral parts of this one. The fabric of space joins us all together; we are all ripples on the bosom of the aether sea. Space is not some dead abstraction, but rather a living and moving thing. We are patterns in space, and, strangely enough, it is not beyond our abilities to speculate on the overall shape of space itself.

Part III

HOW TO
GET THERE

9

Spacetime Diary

This is a free online webpage edition, Copyright (C) Rudy Rucker 2016.
Purchase paperback or ebook editions from Dover Publications or Amazon.

Monday, November 15, 1982

IF IT WASN’T for time, I could live forever. Does that make sense? If it wasn’t for space, I could be everywhere. Is there a difference? I want to go back to my happy college days. I want to be a newlywed again. I want to be three feet tall and sit on my mother’s lap. I don’t want to die. I want to see the future. Time won’t let me. Let’s kill time. Let’s get past time. Let’s reach through time and grab hold of eternity. Now there’s no time. There’s no time now.

Fig. 132. Let’s kill time.

     Later. Do you hate time? Alarm clocks, sure. Changing the clocks for daylight-saving time is the worst. How can they just take an hour away like that? Remember in 1973 when Nixon took away two hours for the oil companies?

    “The older I get, the faster time goes” my mother told me. “The years just fly by. Every time I turn around, it’s Christmas or Thanksgiving.” Party time speeds up and slows down like an out-of-control movie. Ten minutes lasts for two hours, but the next time you look at your watch, it’s three in the morning. Airport time. Sex time. Street time. Fast or slow, it all passes.

Fig. 133. Fast and slow.

    That was my big realization twenty years ago. It all passes. Here I am at the bathroom door, and how can I ever get to the sink? How can high school ever end, how can I ever finish college, how can I ever be married? But then I’m at the sink, I’m back out the door, I have a Ph.D., I’m married with three kids, and twenty years have passed. Here I am alive, and how can I ever die? But I will, I know I will, I know it in my soul.

    Death. It’s like the basic puzzle issued to each of us at birth. Hi, you’re alive now, isn’t it nice? Someday you’ll die and it’ll be over. What are you going to do about it? It’s awful, it’s terrifying, it’s enough to make a person commit suicide!

Fig. 134. It all passes.

    If time didn’t pass, I’d always be here now, writing this chapter. I’m scared of dying. I’d like to think that time doesn’t really pass. What I’m going to do in this chapter is to present some scientific justifications for the belief that the passage of time is an illusion.

    People ordinarily think of the world as being a three-dimensional space that changes with the passage of time. The past is gone, the future doesn’t exist yet, and only the present is real. But there is another way of looking at the world: we can regard the world as a block universe. When we think of the world as a block universe, we put all of space and time together to make a single huge object. The block universe is made up of spacetime. Spacetime is four-dimensional: three space dimensions plus one time dimension. To look at spacetime from the outside is to stand outside of history and view things sub specie aeternitatis.

    “Spacetime” may sound like something technical and far removed from ordinary life. But I would argue that it is really a more natural concept than “space that changes with time.”

    Suppose you work in an office miles away from your house. At 7:00 you see your bedroom; at 10:00 you see your desk. One day at 10:00 you sit in your office and wonder what is real. If you believe the world consists of a space that changes with time, then you are more or less committed to the view that the past is gone. So you will feel that your 10:00 bedroom exists, but your 7:00 bedroom does not exist. Yet your 10:00 bedroom is not something you can see, sitting there in your office. Wouldn’t it be more reasonable to believe that the 7:00 bedroom (which you saw and can well remember) is real, and that it is the 10:00 bedroom whose existence is doubtful?

Fig. 135. What’s real!

    My world is, in the last analysis, the sum total of my sensations. These sensations can be most naturally arranged as a pattern in four-dimensional spacetime. My life is a sort of four-dimensional worm embedded in a block universe. To complain that my lifeworm is only (let us say) seventy-two years long is perhaps as foolish as it would be to complain that my body is only six feet long. Eternity is right outside of spacetime. Eternity is right now.

    This is not a new idea by any means. The teaching that all history is an eternal Now is central to the classic mystic tradition. In one of his sermons, the fourteenth-century priest Meister Eckhart expressed the basic idea as vividly as anyone before or since:

    Whether or not we share Eckhart’s beliefs about God, the image of spacetime being created all at once is a powerful one. Whenever I read his words I get an image of a big old man with a white beard flinging a bucket of paint at a barn wall. Splat: there’s all of spacetime, created all at once, created right now.

Fig. 136. “God makes the world and all the things in this present now.”

Tuesday, November 16, 1982

Space is made up of locations; spacetime is made up of events. An “event” is just what it sounds like: a given place at a given time. Each of your sense impressions is a little event. The events you experience fall into a natural four-dimensional order: north/south, east/west, up/down, sooner/later. When you look back at your life, you are really looking at a four-dimensional spacetime pattern. So there is nothing very strange or confusing about spacetime, as long as we are looking at it from the “inside.”

    Looking at spacetime from the “outside” is a little harder: four-dimensional things are always difficult to visualize. Let us, once again, think about Flatland. Imagine that A Square is resting alone in an empty field, and that shortly after noon his father, A Triangle, slides up to him and then slides off. If we take time to be a third dimension perpendicular to the plane of Flatland, then we can illustrate these events by a spacetime diagram as shown in figure 137. Here A Square and A Triangle are wormlike patterns in spacetime. Their brief encounter at 12:05 is represented as a bending together of their lifeworms. Nothing really moves here; this is just an eternal pattern in spacetime. At 12:05 A Triangle is next to A Square; this is an eternal fact, a fact that can never change.

    Try to imagine a picture like figure 137 that encompasses the entire space and time of Flatland. This vast tangle of worms and threads would make up what we call the Flat-land block universe. You could think of making a model of the Flatland block universe by standing above Flatland and filming the action as the polygons move around. If you then cut apart the film’s frames and stacked them up in temporal order, you’d have a good model of part of the Flatland block universe.

Fig. 137. A region of Flatland’s spacetime.

Fig. 138. Flatland’s spacetime is like a stack of film frames.

    Before going any further, I should stop to answer a question that some of you may be asking. If we’re going to think of time as a fourth dimension, does that mean that all the things we’ve said about the fourth dimension are really about time? The answer is no. Just as there is no one fixed direction in space that we always call “width,” there need be no one fixed higher dimension that is always called “time.” All our talk about the fourth dimension has enabled us to think of a variety of higher dimensions: a direction in which one can jump out of space, a direction in which space is curved, a direction in which one moves to reach alternate universes. We can, if we like, insist that the past/future axis of time is the fourth dimension. And then we pretty well have to say that the ana/kata axis out of space is the fifth dimension, and the sixth dimension is the direction to other curved spacetimes. But there’s no point being so rigid about it. Nobody goes around saying width is the second dimension and height is the third dimension. Instead we just say that height and width are space dimensions. Rather than saying time is the fourth dimension, it is more natural to say that time is just one of the higher dimensions.

Fig. 139. Space + Time + Hyperspace, for Lineland.

    If we think of Lineland as being a one-dimensional (east/west) world whose space bulges into hyperspace (ana/kata) wherever matter is present, and whose space is different at different times (past/future), then we get a 3-D space-hyperspace-time diagram as drawn in figure 139. Of course, if 1-D Lineland were beefed up to a full 3-D space, the diagram would have to be 5-D. Staying with figure 139 for just another second, note that what is shown is two segments merging to make one large segment, and an extra-large segment giving birth to a smaller segment. This picture makes up part of what we might call the Lineland block universe.

    O.K. So now I’ve made the point that although time is one of the higher dimensions, there are many other possible dimensions as well. By the end of this book I’ll be raving, and out-of-it, and saying space is infinite-dimensional, no doubt. But there’s still a lot more to say about spacetime and the concept of the block universe.

    Many philosophers argue that it is wrong to say our reality is a block universe. They do not want to represent our past-present-future universe as a static 4-D spacetime pattern.

Wednesday, November 17, 1982

Another day has passed, and here I am trying to claim that the passage of time is an illusion. What could be more ridiculous? I remember, about five years ago, visiting my father in the hospital. He was having heart trouble, and felt depressed. I tried to cheer him up by explaining the block universe to him, and by pointing out that one’s life is a permanent unchanging pattern in spacetime. “Rudy,” he said wearily, “all I know about time is that you get old and then you die.”

    It certainly feels like time is passing; I’d be foolish to argue otherwise. But I want to show you that this feeling is a sort of illusion. Change is unreal. Nothing is happening. The feeling that time is passing is just that: a feeling that goes with being a certain sort of spacetime pattern.

    Let me illustrate my thoughts with another excerpt from that imaginary classic, The Further Adventures of A Square.

     I: What must I do to escape?

    SPHERE: Ask the villain Cube, when next you see him. Mayhap he knows some ruse to extend your Span. But it is all foolishness, your race is Mortal. The teaching I bring to this Vision is beyond your squalid struggle for more Time. My Teaching is that Time is Unreal, and Eternity is Now.

    I: What kind of Death do you foresee?

    SPHERE: Silence, fool. Behold!

    And there before us I seemed to see a strange and intricate Form of three dimensions. It was like a Cube, but transparent and patterned all within. Tubes and Worms and Threads ran from this strange Cube’s bottom to its top; some of the Tubes were round in cross section, others were Triangular or Square. The uppermost surface of the Cube looked familiar to me, and suddenly I realized it was my World.

Fig. 140. A tangled tale.

    There slumbered my Square form, there was my Father’s familiar Triangle, and in the distance cowered a chastened Una. A Hexagon and an Isosceles were near my sleeping form, evidently bent on Violence. Only my loyal Father’s intercession was keeping them at bay.

    All this I saw atop the Cube. Moving my attention downward I could trace out the whole tangled history of Love and Hate.

The Isosceles’s Point attracted my notice above all else, and I begged the Sphere for aid.

    I: You will save me, will you not, most noble Sphere?

    SPHERE: Your salvation is not mine to grant. What is this object that we gaze upon?

    I: An ingenious model of some part of Flatland. There on the top is my sleeping Form, my Father, and …

    SPHERE: What of the Cube’s interior, Square?

    I: You have stacked up many models of my world, O Teacher. Each plane cross section of the Cube displays a different instant of my recent (and regretted) Career. It is indeed a clever Construct, an inspired use of Higher Space.

    SPHERE: Suppose I were to tell you this is no Construct? What you see is a higher level of Reality. What you see is your Space and Time. This is your World.

    I: My Liege is pleased to be merry. This dead, unmoving Construct is to replace the passionate bustle of Flatland life? One could as well say that a painting breathes, or that a statue weeps!

    SPHERE: The Teaching is strange, but it is no Jest. The block you see is a region of Flatland’s Spacetime.

    I: Spacetime, my Lord?

    SPHERE: Space plus Time, thou Dullard. Hear the words of a great Spaceland thinker: Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality. Space is a shadow, Time’s passing is an illusion; only Spacetime is real.

Fig. 141. The moving “Now.”

    I: Again I must protest, O Round One. Life consists of change and Motion. Where in this Construct of Flatland’s Spacetime is there Motion?

    SPHERE: You may imagine the Motion as follows. Suppose that a plane were to move up through the Cube of Spacetime. Think of the plane as a moving “Now.” Fix your attention upon it, and you will see your Form dancing its sorry Jig.

    I: You are saying, then, that my conscious Mind lights up a cross section of Spacetime, and that the passage of Time is the upward motion of my Mind?

    SPHERE: I say no such thing. There is no motion in spacetime. Your Mind, such as it is, extends the length of your Span. More truly spoken, the Mind is everywhere, and you have no Mind at all.

    I: I do not understand you, my Lord.

    SPHERE: Nor do I understand myself.

Fig. 142. Time as the motion of the mind’s eye.

Thursday, November 18, 1982

The purpose of the last dialogue was to present what the physicist David Park has called “the fallacy of the animated Minkowski diagram.” A spacetime diagram such as figure 140 is called a Minkowski diagram in honor of Russian mathematician Hermann Minkowski (1864-1909), who was the first to think of drawing such pictures. A Square says that such a diagram is lacking something: the passage of time. We do not experience childhood, adolescence, and maturity all at once. We live through one stage, then the next, then the next, and so on. A Square feels that figure 140 is just a model, but that the reality would be best represented by moving an illuminated plane up through the spacetime solid. First one cross section would be lit up, then the next, then the next, and so on. In this manner the static Minkowski diagram could be animated, or brought to life. If we think of the spacetime diagram as being like a movie, it is as if A Square is saying that the movie needs to be taken out of the can and projected. If we think of the diagram as being like a novel, it is as if A Square is saying that the novel needs a reader who goes through it page by page.

    But there are a lot of problems with the notion of an animated Minkowski diagram. One difficulty is that if we think of a static spacetime, and then imagine an external Mind, which sort of moves a searchlight up along it, we have introduced a second level of time: the time that lapses as the Mind moves its attention through spacetime. Now, if spacetime is to be everything, then it seems awkward and wrong to have a second kind of time lapsing external to it. With something like a novel, this poses no problem: the book incorporates its own time pattern, and the time it takes me to read the book is something else entirely. But we do not stand outside the universe like a reader outside a book; we are in our spacetime.

    That’s just my opinion, of course. I have, in fact, met people who do hold to the belief that spacetime is something like a novel being “read” by their soul, the “soul” being some kind of eye or observer that stands outside spacetime, slowly moving its gaze up along the time axis. I find this unsatisfying.

Fig. 143. Three animations of the same life.

    If your life is like a novel that your soul is “reading,” then the past is not so real after all. Even worse, death is just as real as ever: death is when the soul runs out of spacetime pattern to “read.”

    Another confusing point the animated Minkowski diagram raises is this: what if the diagram is animated more than once? What if, in other words, there is a second “reader of the novel,” or a third, or a fourth, or infinitely many? What if a whole series of souls moves through your spacetime, living your life over and over? What if a whole infinite continuum of souls is moving up through your life, one at each instant at all times? But then your whole life is “lit up,” so why say any thing’s moving at all?

Friday, November 19, 1982

What I want to say is that each of us is a certain spacetime pattern in the block universe. Today, or the day of my birth, or the day of my death — all are equally real, all are different pieces of the block universe. I will never stop living this instant. This instant will never cease to exist; this instant has always existed.

    About ten years ago I had a chance to meet the great logician Kurt Gödel. I used to phone him up and ask him questions about philosophy. In my book Infinity and the Mind I have a section that describes a conversation I had with Gödel about the passage of time:

    By a “given,” Gödel means a person’s sensations at some particular time. At any moment the world “gives” us a collection of sights, sounds, smells, and so on. By a more or less unconscious process, we organize these sensations into a stable framework. This background framework, which everyone agrees on, is reality: a continuum of three space dimensions and one time dimension. When I am in my office, I do not doubt the existence of my home. By the same token, when it is 10:00, I do not doubt the existence of 7:00. I do not think of a person as an object adrift in space; a person is a certain kind of pattern in spacetime.

    A human body changes most of its atoms every few years. Daily one eats and inhales billions of new atoms, daily one excretes, sheds, and breathes out billions of old ones. Physically, my present body has almost nothing in common with the body I had twenty years ago. Since I feel that I am still the same person, it must be that “I” am something other than the collection of atoms making up my body. “I” am not so much my atoms as I am the pattern in which my atoms are arranged. Some of the atom patterns in my brain code up certain memories; it is the continuity of these memories that gives me my sense of personal identity.

Fig. 144. People are spacetime braids of atom threads.

    In figure 144 we have drawn a picture to represent the fact that people are persistent spacetime patterns. To simplify things we represent a person as a braid of three atom threads (where an “atom thread” is the spacetime trail of an individual atom). Note that in neither of the two braid-people in the picture’s center do we have any single atom thread going the whole length. Note also that an atom can leave one person’s braid and become part of someone else’s.

    What I find most striking about figure 144 is that the gray boxes enclosing individual lives are purely imaginary. The simple processes of eating and breathing weave all of us together into a vast four-dimensional tapestry. No matter how isolated you may sometimes feel, no matter how lonely, you are never really cut off from the whole.

    I find this insight a great comfort. Instead of thinking of myself as a decaying bag of meat, I can think of myself as a part of eternal spacetime. This is a way to cheat death. Instead of identifying myself with my specific body pattern, I identify myself with the block universe as a whole. I am, as it were, an eye that the cosmos uses to look at itself. The Mind is not mine alone; the Mind is everywhere. If I do not exist, then how can I die?

Monday, November 22, 1982

“How can I die?” Well, I could have fallen off that broken viaduct I was drunkenly standing at the edge of, Saturday night, just to show off. Ow. Why, at my age, do I still do things like that? To prove that I have free will.

    Every so often, one likes to do something foolish or unexpected. Much of life is quite predictable, but it is the crazy zigzags that give human life its peculiar savor. It’s not really necessary to do something stupid and dangerous … Taking your wife out for dinner on a Wednesday night can be unusual enough.

    If we are indeed spacetime patterns in a block universe, then the future already exists. Does this conflict with the notion that we have free will?

    Not really. When I say the future is already there, I am not saying that the future can be predicted. When you are halfway through a detective novel, the ending is already there, printed on the last pages. But this does not mean that you can always guess what the ending will be. I feel that my entire life exists as a timeless whole. But this does not mean that I can predict with certainty what I’ll write tomorrow, or where I’ll live next year.

    Sometimes when I try to explain the spacetime viewpoint to people they say, “If you’re just a pattern in spacetime and the future’s already fixed, then why don’t you kill yourself and get it over with? I mean, you’re going to die anyway, right? Why not shoot yourself?” The answer is simple: “Because I don’t want to.” I may enjoy dramatizing the choice between life and death by standing at the edge of a hundred-meter drop — but I’m damn careful not to fall off.

Fig. 145. I don’t want to die.

    It is a plant’s nature to grow toward the sun, to bloom, and to bear fruit. It is a person’s nature to live and love and work. In all likelihood, there is no big “Answer,” and life has no significance outside itself. But that’s enough. As Don Juan puts it in Carlos Castaneda’s A Separate Reality:

    I choose to live, and to laugh, not because it matters, but because that choice is the bent of my nature … A man of knowledge chooses a path with heart and follows it … Nothing being more important than anything else, a man of knowledge chooses any act, and acts it out as if it matters to him.

    The idea here is that your life is a whole, and the overall pattern is what counts. The unexpected kinks in the pattern correspond to places where you feel yourself to be making a free will decision.

    Some people object very much to this view. So strong is their belief in the importance of their own free will, that they feel the future does not exist at all. They may grant that the past exists, but they feel that the block universe is something that is growing upward as time goes on. In figure 146 we illustrate this viewpoint, along with the block universe viewpoint, and the viewpoint that gives reality only to the present instant.

Fig. 146. Differing world views.

    The great advantage the block universe has over the other viewpoints is that in the block universe there is no objectively existing “Now.” Nothing is moving in the block universe, and there is no need to try to find some absolute and objective meaning for the horizontal space sheet that the other two models depend on.

    As it turns out, it is actually impossible to find any objective and universally acceptable definition of “all of space, taken at this instant.” This follows, as we shall see, from Einstein’s special theory of relativity. The idea of the block universe is, thus, more than an attractive metaphysical theory. It is a well-established scientific fact.

Tuesday, November 23, 1982

Today I want to draw a lot of Minkowski diagrams: pictures of spacetime. To make things easier, we’ll think of space as a one-dimensional line, and we’ll think of objects as points moving back and forth on this line. The spacetime trail of a dot is called the dot’s world line. In figure 147 we see five different sorts of world lines. A represents a motionless point, and B represents a point that moves steadily to the right. C is a dot that starts out motionless but then begins moving faster and faster to the right. D makes an excursion to the right and then comes back to his starting point. E is in a steady state of right-left oscillation.

Fig. 147. Various kinds of motion.

    It is really a bit misleading to say that A is motionless, and B is moving to the right. If A and B are actual people, say astronauts floating in empty space, then all they can really be sure of is that they are moving apart from each other.

Fig. 148. Three descriptions of the same state of affairs.

Since it is impossible to make marks on the fabric of space, there really is no such thing as absolute motion. The only kind of motion one can hope to observe is the motion of one object relative to some other object. This is the content of Einstein’s principle of relativity: “The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of coordinates in uniform translatory motion.” In formulating his theory of space and time, Einstein makes one other assumption, the principle of the constancy of the speed of light: Whenever someone measures the speed of light, he or she will always come up with the same value c (= 29,979,245,620 centimeters per second ≈ one billion miles per hour). These two assumptions have strong empirical support. Taken together, they lead to a number of surprising consequences.

    Usually, in drawing Minkowski diagrams, one adopts a system of units so that the path of a light ray is represented by a 45° line. Light moves at about one billion miles per hour, so the idea is to mark off the space axis in units of billions of miles and mark off the time axis in units of one hour. In figure 149 we have marked the axes in this way, and we have drawn in the world line of a pulse of light. At time 1, A sends a light flash off to the right. One billion miles away stands M, patiently holding a mirror. After an hour’s travel time, the light hits M’s mirror and bounces back to the left. A gets out of the light’s way, and it continues traveling to the left indefinitely.

Fig. 149. World line of a photon.

    As far as we know, nothing travels faster than light. It is interesting to realize that you are never actually seeing the world right now. What you see is always slightly in the past, as it takes the light time to get to you; what you hear is even

Fig. 150. Sight is faster than smell.

further in the past; and smells travel even slower than sounds. You see the sky rocket’s flash, you hear its explosion, you smell the smoke. Your sensations, at any given instant, are all signals from various events in the past. Even what you feel and taste is not happening quite exactly now,

Fig. 151. One’s sensations all come from the past.

Fig. 152. Bug Bites, UFO Zooms, Moon Booms.

as it takes some time for nerve impulses to travel from your skin to your brain.

    To talk about “all of space, taken right now,” is really to speak very abstractly. You certainly don’t see all of space right now. The tree you see is really the tree of a ten-millionth of a second ago, the moon you see is the moon of two seconds ago, the light from the setting sun started out nine minutes ago, and the twinkling stars are scattered back hundreds and thousands of years in the past. In point of fact, you can’t see anything that’s happening right now. You have to wait for the light to get to you.

    It is, of course, possible for us to piece together some kind of mental image of space right now. Suppose, for instance, that it is midnight, and you are sitting outside, looking at the moon. A mosquito bites you, and then two seconds later you see the flash of some huge explosion on the moon. Since you know that it takes light two seconds to travel from moon to earth, you can conclude that the “Now” that had you being bitten by the mosquito also included the explosion on the moon.

    So far this does not seem to go against any of the viewpoints that hold that there really is a spacelike “Now” moving forward through time. But there is a problem with the way in which we construct our notion of “all space taken right now.” The problem is that if someone is moving relative to us, then he will construct his spacelike “Now” in a different way.

    Suppose, to be specific, that a flying saucer had drifted past the Earth-moon system on that fateful night when the mosquito bit you and you saw a flash on the moon. If the aliens happen to be moving from Earth toward the moon, then they will actually reach the conclusion that the explosion happened first. If they are moving from the moon toward the Earth, they will, in their frame of reference, say that the explosion happened second. The reasoning behind these two claims is not obvious, and I won’t try to give it here.

    The point is simply this: differently moving observers will differ on whether or not certain events happen at the same time. Nobody can really be at both the mosquito bite and the explosion. Whether these two events happen at the same time is really just a matter of opinion; nature stipulates no absolute answer to the question.

    The phenomenon just alluded to is called the relativity of simultaneity. The important consequence of the relativity of simultaneity is that differently moving observers will find different ways of slicing spacetime up into a stack of “Nows.” The three observers drawn in figure 153 could be peacefully living in three different galaxies that are moving relative to one another. What reason could we possibly have to say that one of the three ways of slicing up spacetime is correct, and the other two are wrong?

    So relativity theory implies that there is no one preferred way in which to slice spacetime into a stack of “Nows.” Recall that relativity is based on the assumption that it is impossible to permanently mark a given space location. Another way of putting this is that “right here” has no real meaning over a period of time. We have seen that this assumption, oddly enough, implies that “right now” has no real meaning over a region of space. Since there is no one preferred way of defining “Now,” it must be that “Now” has no objective meaning. This means that there really is no moving present, and that the block universe viewpoint is the correct one. Spacetime is a single unified whole, and the passage of time is just an illusion.

Fig. 153. Three ways to slice spacetime.

Wednesday, November 24, 1982

The special theory of relativity is extremely well tested. It makes a number of definite predictions about the behavior of matter in motion. These predictions have been borne out thousands and thousands of times in experiments involving particle accelerators.

    Yesterday I was discussing one of the very attractive implications of relativity: the implication that we are eternal spacetime patterns in an unchanging block universe. But relativity also has some less pleasant consequences. The worst thing about relativity is that it implies that we can never travel faster than the speed of light.

    Some people get almost paranoid about this galling prediction. Without ever bothering to learn anything about relativity, they jump to the conclusion that Einstein was simply a dogmatic spoilsport not much different from the people who used to say that man will never fly. “Why shouldn’t we be able to go faster than a billion miles per hour? If you get a big enough rocket and run it long enough, you can go as fast as you like, right? What’s the matter with Einstein anyway? Wasn’t he a German or a Jew or something? Un-American for sure, and probably scared of progress!”

    There are, according to relativity theory, two problems with faster-than-light travel. The first problem is that the closer any object gets to the speed of light, the greater the object’s mass becomes. And the greater an object’s mass is, the harder it is to accelerate the object to a yet higher speed. It would require, in point of fact, an infinite amount of rocket fuel to accelerate a spaceship up to the speed of light.

Fig. 154. Past, Future, and Elsewhere.

    If I can never travel faster than the speed of light, then only a limited region of spacetime is accessible to me. Those events that I can still reach without traveling faster than light are called my “future.” This is a use of the word that is peculiar to relativity theory. Normally I think of my future as being those events I will actually experience, but here I am using “future” to mean all the events that I could conceivably attend or influence, without using faster-than-light travel. In a similar vein, I can take my “past” to be all the spacetime locations from which it would have been possible to travel, in order to reach me here and now. The rest of spacetime is called “elsewhere,” a lovely word. It always relaxes me to think about events as being “elsewhere.” Whatever our leaders in Washington may be doing right this instant doesn’t really matter: it’s “elsewhere.” Unfortunately, these important people’s actions don’t stay elsewhere — my “past” is larger at points further along my world line.

Fig. 155. “Elsewhere” today, “Past” tomorrow.

    The second problem with faster-than-light travel has to do with the relativity of simultaneity. Given any event at all in your “elsewhere,” there will be some observer to say it is happening at the same time as your “here and now.” The whole of “elsewhere” is a sort of smeared out “Now.” Usually we think of “Now” as a line between “past” and “future,” but in relativity theory, this “Now” smears out into the hourglass-shaped “elsewhere.” What this means is that one path into “elsewhere” is as good as any other. The trouble is that some kinds of paths into “elsewhere” seem too weird to be possible. In figure 156 I have drawn three paths into “elsewhere.” The top one is unproblematic; it simply corresponds to someone who travels two billion miles per hour, or twice the speed of light. The next path is horizontal and, relative to the initial frame of reference, seems to represent a person who travels billions of miles in no time at all. This corresponds to an infinite velocity. The bottom path is the worst. Here the traveler seems to go backward in time. If he were to return from such a trip along a horizontal path, he would be able to meet up with his own past self! In other words, faster-than-light travel can lead to time travel — and many scientists feel that time travel must be impossible. The trouble with time travel is that it leads to some really vicious paradoxes, paradoxes we’ll go into in the next chapter. For now, suffice it to say that so far as we know, no chains of cause and effect can proceed faster than light.

Fig. 156. Trips into Elsewhere.

    Well, that’s all I have time for. I have to go home early today, to help get ready for Thanksgiving. Seems like every time you turn around it’s Christmas or Thanksgiving.

Tuesday, November 30, 1982

Another holiday over with. My mother came, and my brother Embry with his wife and two kids. It was nice to see the five little cousins enjoying it: the treats, the wheel of the seasons, the eternal return.

    Borges has an interesting essay on the esoteric doctrine of “the eternal return.” This is the idea that everything repeats itself.

    Many things in our ordinary experience do repeat themselves. Exhale/inhale, day/night, summer/winter, parent/child: these are cycles that repeat themselves endlessly. Obviously, my 1982 Thanksgiving is not identical with all the other Thanksgivings. Yet, on some level, they are all the same, at least as regards the broadest features, the turkey and gravy, loud children, drunkenness, quarreling, laughter, and prayer. At some point during a long holiday’s welcome ordeal, most of us will experience moments, or even quarter-hours, of transcendence, of timelessness, of that full acceptance of time’s fabric, which, paradoxically, sets one free from time entirely. The very repetitiousness of human rituals provides some kind of glimpse into eternity; the endless line is bent into a circle.

Fig. 157. The Eternal Return.

    Despite all this, the eternal return is still a very unlikely notion. Can anyone really believe that at some future time the universe will be in exactly the same state as it is now? But this is what the doctrine of the eternal return asserts: there is some fixed length of time, some cosmic cycle, after which the entire history of the universe repeats itself. We might suppose a full cycle to last, say, twenty billion years.

Fig. 158. Endless, repeating time.

    There are two different ways of looking at a cosmos that has the property of repeating itself. We can imagine that time extends to infinity in both directions, and that space-time is made up of infinitely many identical horizontal “stripes.” Alternatively, we can imagine that time is bent into a finite, yet endless, circle. If a person is strongly committed to the “moving Now” viewpoint, then the first alternative will seem more attractive. If, that is, you believe that only the present has real existence, then you will feel that even if the universe were to repeat itself, the next cycle would be somehow different. But for someone who holds to the “block universe” view of spacetime, the second way of illustrating the eternal return will seem more natural. If spacetime is indeed a solidly existing thing, then we have no real problem with bending it into a “cylinder.”

Fig. 159. Circular time.

    The notion of “circular time” leads to some interesting paradoxes, which we’ll discuss in puzzle 10.5. But right now let’s go on and look at some of the other overall shapes that spacetime might have.

    Keep in mind that in all of these pictures, we think of the horizontal axis as being space, and we think of the vertical axis as being time. The horizontal axis is, if you will, our concept of “now,” and the vertical axis is our concept of “here.” The event that we call “here and now” is, of course, the point where the two axes intersect. One often “reads” these spacetime diagrams by imagining the horizontal axis to move upward as time goes on. But, as I have repeatedly pointed out, it is not always necessary or desirable to animate our Minkowski diagrams in such fashion.

    Figure 160 shows three kinds of spacetime. In each of them, space is infinite, stretching out endlessly in both directions; and in each of them, the future also is endless. The models differ in the way they depict the past. The picture in the middle shows a universe in which all of space springs into existence at once, and the left-hand picture shows a universe that has no beginning in time. The right-hand picture shows a much odder situation, a universe in which different parts of space come into existence at different times. In such a universe there might be, for instance, a big hole in space out between Neptune and Uranus, a hole that would eventually shrink and disappear. (In the picture I have labeled the event of such a hole’s disappearance with the letter D.)

Fig. 160. Three ways to start the universe.

Fig. 161. Circular space.

    Another type of spacetime model arises if we assume that space is “hyperspherical,” as was proposed in chapter 8. In terms of Lineland, this would mean that space is a circle, leading to the kind of picture drawn in figure 161. Figure 162 shows an alternate way of viewing a universe in which traveling far enough in any direction always leads to a region apparently identical with your starting point. Instead of showing space as curving back on itself, figure 162 shows space as extending to infinity in both directions, and with spacetime divided into infinitely many identical vertical strips. A mysterious harmony would have to be invoked to explain the repetition. Figure 161 is, of course, much more natural, just as a “circular time” model was a more natural way of representing the eternal return.

Fig. 162. Endless, repeating space.

Fig. 163. A pre-established harmony?

    One reason that I’ve been dating the sections of this chapter as I write them is simply to make a little fun of my argument that time is unreal. A more serious reason is that I feel like an Arctic explorer, keeping a log of the trip as he pushes further and further into the unexplored waste. At night the barking of the dogs, the mad splendour of the Aurora Borealis. The goal of my quest is a certain vision, a vision of the cosmos as a pattern in infinite-dimensional space. I want a comprehensive and absolute model for the world around me, a system that includes the realities of ordinary human life as well as the cold truths of science.

Fig. 164. Onward!

    Until today I’ve been unsure as to whether I can finish the journey. But this morning I caught sight of the goal. Reality is one, it is a thing that provides an answer to any of the infinitely many questions one might ask; it is, in other words, a coloring-in of infinite-dimensional space. Stick with me, we’ll get there yet!

11

What Is Reality?

This is a free online webpage edition, Copyright (C) Rudy Rucker 2016.
Purchase paperback or ebook editions from Dover Publications or Amazon.

STARTING with no preconceptions at all, what is the most reasonable model of the world that we can build up?

    Only two things seem really certain: one exists, and one has perceptions. I may be a meat machine, a soul, an eye of God, a collection of ideas, or who knows what — but I am certain that I exist. I am the thing that writes these words. Of course, you may doubt whether I am real — perhaps you are only dreaming that you read this book — but you do know for certain that you yourself exist.

    The fact that one has experiences is equally certain; to put it more neutrally, one cannot doubt that perceptions occur. In classical physics, one assumes that the perceptions are caused by objects in three-dimensional space; but, if you think about it, that is really quite an artificial assumption.

Fig. 190. What is reality?

    I do not perceive the world as stable objects, all equally present at all times. The world of immediate perceptions is all bits and pieces, rags and tags. The hiss of the radiator, a bad taste in my mouth, an ache in my hip, the red color of my typewriter, the clack of typing, the sky’s pearl-gray glow, my glasses resting on my nose, rain dripping through the ceiling in the next room, birds in the sky, a car’s tires on the street outside, the titles on my bookcase — this and that, now and again, a stew of perceptions, and stable reality is a mere construct! Concentrating now on my line of reasoning, I lose all awareness of the sky, rain, and cars; is it not reasonable to say that for this moment they cease to exist? Must an individual be like Atlas, who carries the whole world on his back?

Fig. 191. Immediate perceptions.

    The Irish philosopher George Berkeley (1685-1753) advocated an idealistic philosophy called immaterialism. No one has written more eloquently on Berkeley than Borges does in his essay “A New Refutation of Time.”

Fig. 192. "No elementary phenomenon is a phenomenon until it is an observed phenomenon.”

    It is surprising to learn that such a seemingly perverse world view is embraced by modern physicists. In the words of John Wheeler, one of the grand old men of physics, “No elementary phenomenon is a phenomenon until it is an observed phenomenon.” By this, Wheeler means that the rise of quantum mechanics has demolished the view that the universe sits “out there” while we sit back and observe it. The kinds of questions one asks — and the order one asks them in — has a profound influence on the answers one gets, and on the world view one builds up.

    What I would like to do now is build up a model of reality based on the notion that all that exists is the perceptions of various observers. The “three space plus one time dimension” story is just one particular framework for organizing our sensations. We can — and do — equally well order our thoughts and impressions according to many other systems. Thoughts and memories having to do with “food” fall into one area, for instance, and these food thoughts are in turn organized according to various kinds of criteria. There are the good-food/bad-food axis, the sweet/salty axis, the raw/cooked, mine/not-mine, home-made/purchased, cold/hot, red/green, and endlessly many other axes.

    What I am suggesting is that if we take sensations and thoughts as primary, then there is no reason to limit the “dimensions” of the world to the space and time dimensions involved in the motions of inanimate objects. Part and parcel of every object you see is what the object reminds you of, how you feel about it, what you know about its past, and so on. If we make an honest effort to describe the world as we actually live it, then the world grows endlessly more complicated than any simple 3-D picture. There is a feeling that the more we delve into the nature of reality, the more we will find. Far from being limited, the world is inexhaustibly rich.

    We are used to taking left/right, forward/backward, up/down, past/future, and — perhaps — ana/kata to be the only possible dimensions. But why shouldn’t cold/hot, nice/nasty, pretty/ugly, and all other distinctions have equal claim to being dimensions of reality? It is, after all, not enough to just tell a person where and when a given object is. One wants also to know if it is red, if it is good to eat, how much it costs, who else sees it, how much it weighs, and so on.

Fig. 193. Patterns of atoms.

    The traditional approach of science has been to explain sensory phenomena, such as color and warmth, in terms of the spacetime patterns of atoms. But who has ever really seen an atom? At best one might see some kind of electron-microscope photograph of a grainy dot. And one’s experience of seeing such a photograph is in fact a collection of sensory phenomena, involving color, brightness, and so on. Matter may explain our perceptions, but it is our perceptions that tell us about matter.

    I propose that we stop trying to explain our mental experiences in terms of invisibly tiny objects arranged in patterns in 3-D space. Instead let us take our actual thoughts and sensations as the truly fundamental entities. Think of a “dimension” as being any possible type of variation, category, or distinction. For each question you might ask about an object, there is a range of possible answers; we take each of these ranges to be an axis in the true “space” underlying our perceptions.

    What shall we call this space? Fact-space sounds good. An entity of any kind is a glob in fact-space. To the extent that an entity’s position on some axis is precisely given, the corresponding glob will have a narrow cross section in the direction of that axis. If an entity’s properties are vague or indeterminate, then its glob will be hazy and spread-out. The world — the collection of all thoughts and objects — is a pattern spread out through fact-space.

Fig. 194. One-dimensional fact-space.

    What is the pattern like? It will be helpful here to look at a lower-dimensional example. Imagine a Lineland inhabited by two Dots. Suppose for now that the Dots are at rest on their line of space, and suppose that they are so simple that all there is to know about any given Dot is its position. If we imagine looking at this Lineland from the outside, then it has a fact-space that is simply a line with two Dots on it. If for some reason the exact positions of the Dots are uncertain, then the globs representing their positions will be smeared out. In thinking of the smeared images, it is helpful to think of spots of light: take the smeared image to be a patch of light that is brighter at the center and shaded off at the edges.

    There are, in practice, no really distinct boundaries between things. In terms of our model, this can be represented by supposing that no point between A and B is totally dark: the whole line is slightly lit, but as it has two particularly bright regions we say there are two distinct Dots.

    If we suppose that the Dots have some properties other than position, then the fact-space takes on more dimensions. If we suppose, for instance, that the Dots can be at different temperatures and in different moods, then we get something like figure 195. Looking at it, we see that A is to the east of B, and that B, though colder than A, is in a better mood. Note also that the uncertainty about the Dots’ exact mood is greater than the uncertainty about their temperature. But hold it; whose uncertainty am I talking about?

Fig. 195. Three-dimensional fact-space.

    This is an important point. A fact-space like figure 195 is drawn as if some observer is standing outside the Dots’ universe and measuring the Dots’ properties. But this kind of attitude is not going to work if we want to talk about the actual world we live in. We cannot observe our world from the outside. If we take the “world” to be everything that exists, then there can be no outside observer!

    Suppose we drop any notion of observing the Dots from the outside, and assume that the only real facts in Lineland involve what the Dots know. Assume again that the only property the Dots have is position. In this case the fact-space will have two axes: one axis for where A thinks the two Dots are, and one axis for where B thinks the two Dots are. If there were ten Dots, we would need ten axes, one for each Dot’s opinion of the positions.

    In figure 196 I have drawn the fact-space for two Dots, A and B, who notice their mutual positions. The lower patch represents A, and the other patch represents B. If we drop down vertically from the A-patch we get a narrowly defined location on the position-according-to-A axis. A knows quite well where A is. Notice, however, that A does not have a very good idea where B is. Studying the diagram further, we can see that B is a less “self-centered” Dot than A is. That is to say, if we run across horizontally from the A-patch and the B-patch, we find that B knows both Dots’ positions with equal accuracy.

Fig. 196. Self-generated fact-space.

    It is interesting to note that the quality of A’s “self-centeredness” is not something to be found just by looking at the A -patch. We notice this quality only by looking at the whole pattern in fact-space. What, by the way, shall we call a pattern in fact-space? Suppose we call it a world state, “state” being used here in the sense of “state of being” or “state of consciousness.” Rephrasing the point just made, we might say that properties of an individual are bound up in the overall pattern of the world state.

Fig. 197. A has no idea where B is.

    If Dot A were to focus so entirely on its own location as to lose all sight of B, we would get a world state as shown in figure 197. For any individual at any given time, most of the world is going to be unknown or indeterminate, as B is relative to A here. Note, however, that the indeterminacy of B relative to A is still compatible with B’s having a good knowledge of where B and A are.

    It is worth noting here that this kind of model resolves one of the seeming paradoxes of immaterialism. Two people, let us say, are standing in a field. A rabbit runs past. One of the people sees the rabbit, the other doesn’t. For the first person the rabbit has definite existence; for the second person the rabbit does not exist. How can one and the same rabbit both exist and not exist? In our fact-space model there is no problem. The glob representing the rabbit is shaped so as to have a sharp projection on one person’s axis, and a fuzzed-out projection on the other person’s axis.

    Returning to our Lineland fact-space, note that if we were to add the properties of “temperature” and “mood” as before, then we would need a total of six axes: three axes for A’s ideas about position, temperature, and mood; and three axes for B’s opinions. In general, a world with P distinct properties and I distinct individuals will have P-times-I axes in its full fact-space. The state of the world can be thought of as a pattern of more or less distinct globs in a fact-space of very many dimensions.

Fig. 198. A great number of individuals.

    As the various individuals interact with each other, their knowledge about each other changes. The globs of light shift and merge, thicken and attenuate. We can think of the pattern as changing with the passage of time — alternatively, we can take each individual’s time peception as an axis and have time change frozen right into the world state.

    So far I have not discussed the issue of what kinds of things in our world are going to count as individuals. Just people? People and animals? People, animals, and plants? People, animals, plants, and robots? Should we include galaxies and rocks? Since we’re going to get an unmanageably large collection anyway, my inclination is to be generous about this. I’m prepared to go ahead and let anything you like be an individual capable of certain kinds of “knowledge.” A rock in a field doesn’t know much, but it does know that there’s something massive (the Earth) right under it. We can tell that it “knows” since if you lift it and let go, it falls right back down! Putting it a little more seriously, any given object, animate or not, “knows” or embodies information about a number of other objects.

    This viewpoint — that everything is in some sense alive or conscious — has historically been known as hylozoism, or panpsychism. It is, of course, a viewpoint that is easy to ridicule. Can a garbage can feel love? Does a shoe think about mathematics? But the point is not that we are to think of objects as being “like people.” The point is only that we might perhaps think of objects as integrative centers in fact-space, existing entities in the same sense that we are existing patterns ourselves.

    So there are to be a great number of individuals in fact-space. And how many possible properties might there be? There really seems to be no end to the number of properties individuals might be sensitive to. If you start asking questions about some individual object, it seems as though you might really keep on asking questions forever. So if fact-space is to have as many dimensions as the number of individuals times the number of properties, I think we may as well go ahead and say that fact-space is probably infinite-dimensional. Our world is a pattern in an infinite-dimensional, or “∞-D,” space.

Fig. 199. The world is like a game of “Infinity Questions” — there’s no end to the number of questions you can ask.

Fig. 200. Turtles all the way down.

    The notion of ∞-D space is not something I just made up. The hyperspace philosophers of the late nineteenth century were well aware that the notion of higher dimensions leads to an infinite regress: Earth’s 2-D surface is part of our 3-D universe. Our 3-D universe may well be the hypersurface of a 4-D hypersphere. Our 4-D hypersphere is a cross section of a 5-D curved spacetime pattern. Our 5-D curved spacetime is perhaps only a layer in a 6-D stack of alternate space-times. The 6-D stack may itself be warped and twisted in 7-D space; various versions of the stack may nestle together in 8-D space. Perhaps the whole 8-D space can be thought of as evolving along a nine-dimensional axis of supertime. And so on. Where can it stop? Only at infinity.

    Hinton compared this to the kind of regress that arises if someone says the world rests on the back of a tortoise. What does the tortoise stand on? Another tortoise, which stands on another, which stands on another, ad infinitum. The theologian Arthur Willink found such an infinite regress of spaces exhilarating, and in his 1893 work The World of the Unseen, he argues that God lives in the ultimate ∞-D space:

    This is a very interesting passage, certainly one of the first philosophical uses of ∞-D space. A somewhat similar passage can be found in an 1886 essay, “On the Various Standpoints with Respect to Actual Infinity,” by the German mathematician Georg Cantor:

    It was Georg Cantor who first developed a mathematically rigorous treatment of infinity. Before Cantor, many mathematicians and philosophers had feared that infinity is basically a contradictory notion — but after Cantor, scientists were able to begin using infinities quite casually.

    Early in the 1900s, mathematician David Hilbert drew on Cantor’s work to develop a theory of infinite-dimensional spaces — spaces like the fact-space we have been talking about. Just as a point in a 3-D space can be mathematically represented by an ordered sequence of three coordinate numbers, a point in an ∞-D space is represented as an infinite sequence of numbers. There are various ways of going on to define things like angles and distances in these ∞-D spaces; the most commonly used type of mathematical ∞-D space is known as Hilbert space.

    For a decade or so, the work on Hilbert spaces seemed to be simply another example of mathematicians pursuing abstraction for abstraction’s sake. But in the 1920s, physicists Werner Heisenberg and Erwin Schrödinger discovered that the best way of interpreting quantum mechanics is to say that particles are patterns in ∞-D Hilbert space.

    Since that time, mathematicians and physicists have developed an elaborate quantum-mechanical theory of the world as a pattern in an ∞-D Hilbert space. One of the great problems with this theory has been to find some actual significance for the ∞-D space involved. When classical physics presents us with equations involving four variables, we are able to understand these equations as being about the three space dimensions plus the one time dimension. But the mathematics of quantum mechanics seems somehow to have developed in the absence of any good underlying picture. There is no doubt that the quantum theory as it stands makes correct predictions about definite experiments. But no one seems to have a good feeling for what Hilbert space really means.

    In describing our world of perceptions as a pattern in ∞-D fact-space, I have been trying to give the notion of ∞-D space some real content. The notion of fact-space is, in many ways, modeled on the quantum-mechanical Hilbert space. Given the protean nature of infinity, I would not be surprised to learn that the two spaces are really the same. It is important here to realize that the particular axes one lays down in a given space are quite arbitrary. Although a Californian and a New Yorker live in the same space, the directions they call “up” are slightly different. Space is given in itself, and without axes. Once we build up the world as a pattern in ∞-D fact-space, it could be that a simple redrawing of the axes would lead to the Hilbert space model. The important thing to remember is that the axes have no objective existence at all.

    What is reality?

    Take all your perceptions and all of mine, take everyone’s thoughts and all the visions. In an infinite-dimensional space there is room to fit them all together; each is a piece of the infinite-dimensional One, and this One is reality.

    Reality is indescribably rich and complex. Sometimes I forget this, and life goes gray. But the world is alive, and we are living parts of it. Thoughts are as real and important as objects. Every object is an endless source of wonder.

    We don’t know why we’re here — we don’t even know what we are. But we exist, and the world is going on. Our ordinary notions of space and time are just a convenient fiction. Higher dimensions are everywhere. There’s no need to work for enlightenment; enlightenment is here and now, as close as the fourth dimension.

Puzzle Answers

ANSWER 1.1

If the two car shapes are of different size, then they can move past each other. The third dimension of distance-from-the-window is, in terms of images, represented by size. For the two-dimensional shapes moving in the window glass, size is a higher dimension. Imagine living in a three-dimensional analogue of such a world: you would be able to change your size at will, and you would be able to move “through” people of different sizes.

ANSWER 2.1

Self-gripping gut.

One way to keep poor A Square from going to pieces would be to have the type of gut depicted here. The projections on the upper half grip the knobs on the lower half, keeping A Square’s body together. Food is passed down along the gut in the fashion of a barge moving through locks in a canal, with one after another of the barriers momentarily opening.

ANSWER 2.2

We would, in strict analogy to our world, expect two-dimensional creatures to crawl around on the rim of a disk: their planet.

ANSWER 3.1

If there were a 3-D space intersecting ours only in a plane, then we would see perhaps something like this: a plane of light slanting up from the ground into the sky, with odd-shaped globs drifting around in the plane, sliding up and down like holy Frisbees. The globs would be very thin and would feel solid to the touch.

ANSWER 3.2

A Square scanning Flatland.

A Square’s retina is a line segment designed to take in light from the plane of A Square’s body. Now it seems that, looking down at Flatland from the third dimension, A Square would really see just those objects of Flatland that intersect the plane of his vision. The situation would be exactly like that of a Flatlander in a perpendicular world, as described in puzzle 3.1.

    Now if A Square rocks back and forth, he can scan the various cross sections of Flatland and then mentally combine these sections to get a full 2-D image. By the same token, if you were in 4-D space looking at our world, you would see various planar cross sections of our world. With some effort, you would be able to combine these sections into a full 3-D image of everything, inside and out.

ANSWER 3.3

It is pretty easy to see that the number of corners will double each time we go up a dimension. But what about the other entries? How, without actually counting the lines in figure 34, do we know that the hypercube has 32 edges? The idea is that a hypercube is gotten by starting with a cube in initial position, moving the cube one unit ana, and then having the cube in final position. The initial and final cube each contribute 12 edges, and the cube’s eight corners each trace out an edge during the ana motion. 12 + 8 + 12 = 32. Similar reasoning will justify the remaining entries.

ANSWER 3.4

The very bottom face must join up with the very top face, and the side faces on the bottom cube connect to the four open faces of the main part. This is easy to see if we consider the analogous process of folding a crucifix up into a cube.

Crucifix into cube. (Answer 3.4)

ANSWER 3.5

The formula is S⁴, and the specific hypercube mentioned would have a hypervolume of 16 hypercubic feet.

    The Greeks thought of numbers as specific geometrical magnitudes. For a given length S, S² stood for a certain square’s area, and S³ stood for the volume of a certain cube. Since they had no notion of a fourth dimension, they did almost no work with formulas or equations involving powers higher than 3. Only after the Renaissance did mathematicians feel confident enough of their algebra to work with higher-degree equations in a formal way.

The pentahedroid. (From D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination.)

ANSWER 3.6

If we go into 4-D space, then it is possible to find a fifth point (by moving an a from the tetrahedron’s center), so that now all five points are the same distance from each other. These five points are the corners of a so-called pentahedroid.

    In looking at this picture, we are to imagine that the central point is actually a bit further away in the fourth dimension, so that all the edges are really of equal length.

    Just as a triangle is made up of three segments, and the tetrahedron is made of four triangles, the pentahedroid is made of five tetrahedra. Can you see all five?

ANSWER 4.1

In cross section, a cube can appear to be a square, a triangle, a rectangle, or a hexagon. This is illustrated in a drawing by Claude Bragdon.

Cross sections of cubes. (From Claude Bragdon, A Primer of Higher Space.)

ANSWER 4.2

This illusion was sent to me by Orville L. Parrinello of Brazoria, Texas. At first it is a little hard to get, but the idea is that if you let the folded “staircase” reverse back and forth, the bug will alternate between being visible on the floor in front of Kilroy, or being hidden on the ceiling of the space behind Kilroy. The illusion is particularly interesting as it raises the idea that Kilroy’s objective reality is not fixed at all. What if seemingly concrete facts about your life depended totally on how you look at things?

ANSWER 4.3

If A Square were transparent, then he could seem to reverse when seen edge-on, just as the transparent A Cube reverses in figure 51. The reversal would seem sort of as if the Square were “pulled through himself,” just as a left glove can be turned into a right glove by turning it inside out.

Is this A Square or (Answer 4.3)

ANSWER 5.1

Trail of a line.

A line can’t be knotted in 2-D space, because there’s no way to have a line in Flatland cross above itself. And a line won’t stay knotted in 4-D space because, as is discussed in chapter 5, the extra degree of freedom will cause any knot to slip through itself. Moving everything up a dimension, we expect that a plane can be knotted in 4-D space, but not in 3-D or in 5-D. How do you get a knotted plane? The idea is to start with a knotted line, and then imagine moving the knotted line ana out of space. The trail the line traces out will be a knotted plane. The plane, it is important to realize, is knotted, but does not intersect itself. Of course, if we move the knot in 3-D space, the trail does intersect itself, but since ana is perpendicular to every space direction, the 4-D trail will not come back on itself anywhere.

ANSWER 6.1

The turned-over Astrian would have his or her vibrator uselessly sticking up away from space. Such a person might seem to be without atmosphere or sensitivity, a zombie, a man without qualities, a burnt-out case. Is it interesting here to recall that some early thinkers, including Rene Descartes, thought that the pineal gland, located in the brain’s center, might serve as a sort of third eye that perceives other people’s auras or astral vibrations.

ANSWER 6.2

If an Astrian could dig his little 3-D thorn into underlying space hard enough, then he could oppose a force of gravity pulling him across space. In Charles H. Hinton’s 1907 novel, An Episode of Flatland, this is exactly what the Astrians do:

ANSWER 6.3

A Square executes a dangerous maneuver.

Cube snipped the piece of space containing A Square and turned the piece over. Now you might worry about the space of Flatland “popping” from having a hole in it. Well, perhaps A Cube took care of that by using something like embroidery rings to clamp down around the edges of the hole and of the cut-out piece. Or maybe there just was a hole in space for a while. What would it be like to have a hole in space? We’ll get back to the question in chapter 7.

ANSWER 6.4

No, for we cannot tell whether the hole is moving through space. This is analogous to the fact that a particular matter bump does not serve to mark out some particular region of the fabric of space: as a wave moves across the water, its component water bits are constantly changing. We might compare a moving space hole to a bubble floating up through a liquid. Although the bubble’s shape and size stay the same, the bits of liquid that lie at the bubble’s boundary are changing as the bubble moves. An offbeat idea suggested by these considerations is that perhaps the smallest components of matter are not vortices or space bumps but actual holes in space.

ANSWER 6.5

The idea is that the galaxy’s bump is on a line directly between us and the quasar. Light from the quasar to us can travel along two alternative shortest paths: one around either side of the galaxy’s big space bump. This splitting of a quasar’s image was definitely ob-served in 1979. (See Frederic Chaffee, “The Discovery of a Gravitational Lens,” Scientific American, November 1980.) The phrase “gravitational lens” is an exciting way of expressing the fact that space curvature can bend light. It is amusing to think of a vast supertelescope based on gravitational lenses millions of miles across.

Three bumps in a row, and two shortest paths.

Top view of the two paths.

Two kinds of singularity.

ANSWER 6.6

A point-sized mass can be most simply represented as a sharp cusp in space. Alternatively, one might imagine this cusp to be pulled all the way out to infinity.

ANSWER 7.1

A geodesic on a sphere is a so-called great circle, that is, a circle, such as the equator, which is as big as possible. Relative to the sphere’s surface, the equator is “straight” because it bends neither north nor south. A smaller circle, such as the Arctic Circle, can be seen to bend on the surface of the sphere and is not regarded as a geodesic.

ANSWER 7.2

Virtual image of an antipodal star.

One natural explanation of a massless star would be to say that space is hyperspherical, and that the massless star is the virtual image of a real star at the opposite end of the universe. Unfortunately, even if space really is a hypersphere, we are not likely to actually observe any such “fake stars.” The problem is that space is marred by medium-scale irregularities that will prevent the perfect focusing of the star’s light rays at the point furthest away from the star. Another difficulty is that space contains clouds of dust here and there, dust which will absorb most of a star’s light long before the light makes it halfway round the universe. Were it not for these two problems, we would in general expect to find a virtual image of any observed star at a diametrically opposite spot in the sky — provided, of course, that space really is hyperspherical.

ANSWER 7.3

Finite universe with “central” mass.

It would look like a light bulb, or an ice cream cone: a ball with a big bulge on it. It is conceivable that our space actually has such a lopsided structure. If we were just about on the opposite side from the superstar that makes the bulge, then we wouldn’t necessarily have to be able to see it directly. Space dust and intermediate space bumps could dissipate the monster’s image. Such a model for our space is discussed in Paul Davies, The Edge of Infinity, 1983.

ANSWER 7.4

A curved space with natural distances.

We would get something like this: a square surface with a peak in the middle. The first and second pictures are two different ways of representing the same fact: there is more space toward the middle of this surface than one would ordinarily expect to find.

ANSWER 7.5

Square goes around the Möbius strip.

He turns into his own mirror image!

ANSWER 7.6

An infinitely distant hole in space.

The space near the hole is stretched up into an endless “chimney.” No Flatlander ever gets to the end of the chimney; no Flatlander ever slides into the hole.

ANSWER 7.7

A finite area with infinite length.

The idea is to use an endless halving procedure to cut the square into infinitely many pieces: a piece of height 1/2, one of height 1/4, 1/8, 1/16, and so on. Zeno strikes again! If we line up the regions obtained, we get something that is infinitely long, but of unit area.

ANSWER 8.1

The two sheets of space might well snap apart. If the snap were too abrupt, it might do some damage to the Globber in question.

ANSWER 8.2

This is the type of connection represented in figure 112, although the holes in the spaces could be eliminated. The image is of a strip of space joining two distinct spaces together. The strip could, of course, be made very short by bulging the spaces down to meet each other at the surface of the mirror. (Keep in mind here that just as a mirror in our space is a piece of a plane, a mirror in Flatland is a piece of a line.) This kind of link between spaces is exactly what Lewis Carroll deals with in Through the Looking-Glass. Marcel Duchamp was also obsessed with the notion of mirrors as doors to alternative universes. He was struck by the fact that a point approaching a mirror has the choice, in principle, of either breaking through the mirror and continuing in normal space, or of moving out of our space and into the alternate space we see inside the mirror. Thus, for Duchamp, a mirror represented a sort of railroad switch where one chooses between two spaces: real space and mirror space. See Linda Dalrymple Henderson, The Fourth Dimension and Non-Euclidean Geometry in Modern Art, 1983.

A link between two worlds.

    A very vivid way to enhance the illusion that a mirror is a door into another world is to get a flashlight and go up to a mirror in a dark room. The laws of optics are such that if you shine the flashlight into the mirror, the image looks exactly as if the flashlight beam goes through the mirror and into a dark room on the other side of the mirror.

ANSWER 8.3

Install an Einstein-Rosen bridge in the basement, and have this hyperspace tunnel lead to an infinite space empty except for the endless plane of Flatland.

ANSWER 9.1

Take a small spherical balloon. Blow it up and then let the air out. The entire spacetime trail of the balloon’s surface and inside is a solid hypersphere. The trail of the surface alone is the hypersurface of the hypersphere.

    The speed with which the balloon should be inflated and deflated depends on what kind of conversion factor between space and time is adopted. In relativity theory the speed of light is used as a conversion factor; that is, a “meter of time” is taken to be the time it takes a light ray to travel one meter — about three ten-millionths of a second.

    Constructing a hypercube in 4-D spacetime is even simpler. As Professor Tom Banchoff of Brown University puts it, “A hypercube in spacetime is just a cube … for a while.”

ANSWER 9.2

The “melting future” world view corresponds to the notion that future events exist, stored up and waiting for us. A uniform “now” moves forward with the passage of time, and instant after instant is permanently used up. In this viewpoint, past events are totally nonexistent. It is not uncommon for people to feel this way about their lives. Life here becomes a scarce resource that is consumed, and once something is over it doesn’t matter at all. This is probably the least rewarding way possible to think about spacetime, as can be seen by thinking about the kind of personal philosophies inherent in the other three world views shown in figure 146. Whenever you cut yourself off from your past, you’re in an extremely rootless and vulnerable position. But if you do throw out the past, you might as well throw out the future, too, and get totally into the “now.”

ANSWER 9.3

Mr. Lee will put X halfway between A and C. The reason is that Lee will assume that it takes the light just as long to travel back from the other end of the platform as it took it to get there from his end. It is natural for him to think this, in view of the assumptions (1) and (2) mentioned in the puzzle. We, of course, feel that it really takes the light longer to get from A to B than it takes it to get from B to C … But Mr. Lee will say we just think that because we’re racing past him at half the speed of light!

A slanted time line causes a slanted space line.

    If we draw a dotted line connecting B with the event X (which Mr. Lee says is simultaneous with B), then we get one of Lee’s lines of simultaneity. This line corresponds to the space axis in a Minkowski diagram that Lee might draw; another way of putting it is that at the event X, this line will represent Lee’s conception of “now” — just as his world line represents his concept of “here.” It is possible to prove that in such a diagram, the observer’s line of simultaneity will always tilt up by the same angle at which his world line tilts over.

ANSWER 9.4

An expanding finite space.

The picture would be a sort of “conical” spacetime, as drawn here. The starting point is known as “the initial singularity,” or as the Big Bang. Whether or not our space will eventually contract back to a point is unknown at present. Apparently it depends on how much mass is actually in our universe: if there is enough mass, then the gravitational forces will pull things back together.

ANSWER 10.1

To get into the sealed room, travel far, far into the future until you reach a time where the room’s walls have crumbled. Step into the room’s space and then travel back in time. Note that this is exactly analogous to moving ana, then moving across the location where the wall would be, and then coming back kata into the room’s space.

    Getting the food out of someone’s stomach is a little trickier, since you yourself won’t fit in the stomach. Let us imagine that the stomach in question belongs to your uncle Embry, snoozing off his Christmas banquet on his bed upstairs. The thing to do is to go forward in time to later that afternoon when he’s back out of bed. Take a little scoop and set it right on the bed where his stomach was. Now send the scoop back a few hours, to when he was there, and then bring it back to the empty bed. Do this a few times, and you can get all the chewed food out of Uncle’s stomach. Hide it under his pillow so he has a nice surprise that night!

ANSWER 10.2

If we assume that the time machine goes straight backward in time, then “right here one week ago” is wherever the time machine appears when I send it one week back. One would not expect the Earth to still be “right here” a week ago, so traveling backward in time might well land one in empty space. Science fiction writers usually deal with this problem by somehow setting their time machine to track along the Earth’s spacetime path.

    It is interesting to realize that even without the paradoxes, time machines are already ruled out by relativity’s basic assumption that there is no such thing as absolute rest or motion in space. Of course, if people somehow could build time machines, then we would probably just add some “except by using time machines” clause to the relativity principle. Or it might be that one would have to “aim” the time machine at some definite past object.

    Note that just as a time machine can be used to define “right here,” a machine that transmitted objects instantaneously could be used to define “right now.” One could, for instance, spew out a whole lot of synchronized clocks all over space at the same instant. This also, of course, would violate relativity.

ANSWER 10.3

The point is that “instantaneous” is a relative concept. Relative to Earth, B happens at the same time as A, so one can travel instantaneously from A to B. Relative to the distant galaxy that moves away from us, C is simultaneous with B, so one can travel instantaneously from B to C. Combining the two trips takes one from A to C, and thus into one’s own past.

ANSWER 10.4

Equip the rocket with a good robot brain so that after its hundreds-of-thousands-of-years-long journey, it can use the time machine to jump hundreds of thousands of years back in time and find the Earth. Once it finds the Earth again, it makes a small time jump to the right day (the day of launch) and lands.

ANSWER 10.5

The idea of a beacon that lasts all the way around time would lead to difficulties. If it drifted away from Earth never to return, then there would be, it seems, endlessly many of them out there, as a consequence of the one launch! This seems nonsensical. The situation is particularly vicious if we suppose that B₁ sends out a signal that can inhibit the launch of B₀. A yes-and-no paradox!

One becomes infinite?

    More thought is required. Let us look at the individual particles that make up the beacon. If the universe really repeats itself, then it must be that each particle returns — at the end of each cycle — to its starting position. The world lines of particles are thus like rubber bands looped around the spacetime cylinder.

    Now say that we are in such a circular-time universe, and say that we have built a durable radio beacon. The beacon has the form of a thick tangle of “bands” reaching around spacetime. Now, since we assembled the beacon — as opposed to just finding it — it must be that all of the beacon’s component particles are going to end up back on Earth: as ore, as glass fragments, and so on. Therefore, we can logically conclude that in a circular-time universe, any ship we build and launch into space must eventually crash back onto Earth … so that its particles can be assembled back into the ship to be launched! It is, in other words, impossible to build a truly indestructible object if time is circular! For anything you build must in time disintegrate into pieces so that you can “again” build it.

ANSWER 10.6

We have a yes-and-no paradox here. I dial “1” at 11:00 if and only if I don’t get a call at 10:00, but I get a call at 10:00 if and only if I dial “1” at 11:00. In other words, I get a call at 10:00 if and only if I don’t get a call at 10:00. This particular paradox was first raised in a paper by G. Benford, D. Book, and W. Newcomb, “The Tachyonic Antitelephone” (1970). Gregory Benford is, by the way, a science fiction writer, as well as being a physicist. The “tachyons” his paper refers to are hypothetical particles that, unlike ordinary mass particles, always go faster than light. Benford’s paper argues that since tachyons could be used to send messages into one’s own past, it must be, even in principle, impossible to detect them. If, then, tachyons are real, they fill a sort of undetectable ghost universe whose time direction is in a sense perpendicular to our own time direction.

Ships that pass in the night.

ANSWER 11.1

No. There are probably some sections of y our life that do not involve any thoughts at all about the other sections. In “A New Refutation of Time,” Borges argues that any state of mind that recurs in your life is actually the same event.

ANSWER 11.2

Dunne’s idea is that at any instant we have a fixed future along a familiar time axis called T₁; but that as a second, higher sort of time, T₂, lapses, our future changes. Our actual time motion is a combination of T₁ motion, which we might think of as moving into the future, and T2 motion, which we might call moving into alternate worlds. When I see something unpleasant ahead in this Monday’s T₁ future, I am able to move in the T₂ direction into an alternate spacetime with a different T₁ future.

Saved by precognition!

    Dunne tended to think in terms of infinite regresses. His notion of time leads to a regress, for if the dreaming mind can see ahead into T₁ future, it is not unreasonable to suppose that it can also see into T₂ future. And if you can see T₂ future, you can change it, which means that you are really moving along, let us say, T₃ time to a whole different T₁-T₂ plane! And, of course, there is no stop-ping at T₃: the regress goes on forever.

    The reason Dunne was willing to embrace such a crazy system is that he was so dissatisfied with a simple Minkowski diagram’s inability to depict our feeling that time really passes. As we discuss in chapter 11, any attempt to “animate” a Minkowski diagram by thinking of a “focus of consciousness” moving up along one’s world line leads to the Dunne-style regress.

“You’re going to do it till you get it right!”

    In his 1885 story “An Unfinished Communication,” Charles Howard Hinton describes a somewhat different sort of two-dimensional time. Hinton’s concept is that one’s life is repeated over and over, but with slight changes possible each time. After doing y our life often enough, you finally get it right.

ANSWER 11.3

No, this is just another example of a region of fact-space having a vague position relative to one axis (your opinion), but a sharp posi-tion relative to another axis (the other person’s opinion). If you are close enough to a person to be in some sense part of him (or her), then your opinion of him will evolve continuously with him. But if you are separated for a while and then brought back together, the other person’s many possible states will seem to collapse down to one or two definite facts. In quantum mechanics this abrupt change is called “the collapse of the wave function.” It is worth noting that, in certain relaxed states, you are yourself spread out in fact-space, even relative to your own axes. That is, if you are not presently asking yourself whether you are happy, then there is, at present, no definite answer to the question. Sharp or hazy, reality is just what it seems.

Bibliography

This is a free online webpage edition, Copyright (C) Rudy Rucker 2016.
Purchase paperback or ebook editions from Dover Publications or Amazon.

Abbott, Edwin Abbott. Flatland: A Romance of Many Dimensions. 1884. Reprint. New York: Barnes & Noble, 1983.

Borges, Jorge Luis. “A New Refutation of Time.” In Labyrinths: Selected Stories and Other Writings. New York: New Directions, 1962.

———. Borges: A Reader. New York: E. P. Dutton, 1981.

Bork, Alfred. “The Fourth Dimension in Nineteenth-Century Physics.” Isis 181 (1964).

Bragdon, Claude. More Lives Than One. 1938. Reprint. New York: Alfred Knopf, 1971.

———. A Primer of Higher Space. 1913. Reprint. Tucson: Omen Press, 1972.

Breuer, Miles J. “The Appendix and the Spectacles.” 1928. Re-printed in The Mathematical Magpie, edited by Clifton Fadiman. New York: Simon & Schuster, 1962.

Burger, Dionys. Sphereland. New York: Apollo Editions, 1965.

Calder-Marshall, Arthur. The Sage of Sex: A Life of Havelock Ellis. New York: G. P. Putnam’s Sons, 1959.

Campbell, Lewis, and William Garnett. The Life of James Clerk Maxwell. London: Macmillan, 1884.

Carroll, Lewis. Through the Looking-Glass. 1872. Reprint. New York: Random House, 1946.

Castaneda, Carlos. A Separate Reality. New York: Simon & Schuster, 1971.

Clifford, W. K. Mathematical Papers. London: Macmillan, 1882.

Conklin, Groff, ed. Science Fiction Adventures in Dimension. New York: Vanguard Press, 1953.

Davis, Andrew Jackson. The Magic Staff. New York: 1876.

Dewdney, A. K. Two Dimensional Science and Technology. Ontario: [No imprint,] 1980.

———. The Planiverse. New York: Poseidon Press, 1984.

Dolbear, A. E. Matter, Ether and Motion. Boston: Lee & Shepard, 1892.

Dunne, J. W. An Experiment with Time. 1927. Reprint. London: Faber & Faber, 1960.

Durrell, Fletcher. “The Fourth Dimension: An Efficiency Picture.” In Mathematical Adventures, by Fletcher Durrell. Boston: Bruce Humphries, 1938.

Eddington, Arthur. Space, Time and Gravitation. 1920. Reprint. New York: Harper & Row, 1959.

Edwards, Paul. “Panpsychism.” In The Encyclopedia of Philosophy. New York: Macmillan, 1967.

Einstein, Albert. “Ether and Relativity.” In Sidelights on Relativity, by Albert Einstein. New York: E. P. Dutton, 1920.

Ernst, Bruno. The Magic Mirror of M. C. Escher. New York: Random House, 1976.

Faser, J. T., F. C. Haber, and G. H. Muller, eds. The Study of Time. Berlin: Springer-Verlag, 1972.

Gardner, Martin. Relativity for the Million. New York: Macmillan, 1962.

———. The Ambidextrous Universe. New York: Basic Books, 1964.

———. “The Church of the Fourth Dimension.” In The Unexpected Hanging, by Martin Gardner. New York: Simon & Schuster, 1969.

———. “The Hypercube.” In Mathematical Carnival, by Martin Gardner. New York: Alfred A. Knopf, 1975.

———. “Parapsychology and Quantum Mechanics.” In Science and the Paranormal, by G. Abell and B. Singer. New York: Charles Scribner’s Sons, 1981.

Gillespie, Daniel. A Quantum Mechanics Primer. New York: John Wiley, 1970.

Gödel, Kurt. “A Remark on the Relationship Between Relativity Theory and Idealistic Philosophy.” In Albert Einstein: Philosopher Scientist, edited by Paul Schilpp. New York: Harper &. Row, 1959.

Golas, Thaddeus. The Lazy Man’s Guide to Enlightenment. Palo Alto: The Seed Center, 1972.

Gribben, John. Timewarps. New York: Dell, 1979.

Gustaffson, Lars. The Death of a Beekeeper. New York: New Directions, 1978.

Greenberg, Marvin. Euclidean and Non-Euclidean Geometrics. San Francisco: W. H. Freeman, 1974.

Hawking, S., and G. Ellis. The Large Scale Structure of Space-Time. Cambridge: Cambridge University Press, 1973.

Heinlein, Robert. “And He Built a Crooked House.” 1940. Reprinted in Fantasia Mathematica, edited by Clifton Fadiman. New York: Simon & Schuster, 1958.

———. Starman Jones. 1953. Reprint. New York: Ballantine Books, 1978.

Henderson, Linda Dalrymple. The Fourth Dimension and Non-Euclidean Geometry in Modern Art. Princeton: Princeton University Press, 1983.

Hilbert, D., and S. Cohn-Vossen. Geometry and the Imagination. 1938. Reprint. New York: Chelsea, 1952.

Hinton, Charles Howard. Selected Writings of C. H. Hinton, edited by R. Rucker. New York: Dover, 1980.

Houdini, Harry. A Magician Among the Spirits. New York: Harper, 1924.

Huxley, Aldous. The Perennial Philosophy. New York: Harper & Row, 1944.

I Ching. Princeton: Princeton University Press Bollingen Series, 1950.

Jung, C. G. Synchronicity. Princeton: Princeton University Press Bollingen Series, 1973.

Kant, Immanuel. Kant’s Inaugural Dissertation and Early Writings on Space. Chicago: Open Court, 1929.

Kaufmann, William J. Relativity and Cosmology. New York: Harper & Row, 1973.

Lewis, C. S. The Lion, the Witch and the Wardrobe. 1960. Reprint. New York: Collier Books, 1978.

Mach, Ernst. The Science of Mechanics. Chicago: Open Court, 1893.

Manen, Johan von. Some Occult Experiences. Chicago: Theosophical Publishing House, 1913.

Manning, Henry. Geometry of Four Dimensions. 1914. Reprint. New York: Dover, 1956.

Maxwell, James Clerk. The Scientific Papers of James Clerk Maxwell. 1980. Reprint. New York: Dover, 1963.

Minkowski, Hermann. “Space and Time.” 1908. Reprinted in The Principle of Relativity, edited by A. Sommerfeld. New York: Dover, 1952.

Misner, C, K. Thorne, and J. Wheeler. Gravitation. San Francisco: W. H. Freeman, 1973.

Nabokov, Vladimir. Look at the Harlequins. New York: McGrawHill, 1974.

Neumann, John von. Mathematical Foundations of Quantum Mechanics. Princeton: Princeton University Press, 1955.

Nicholls, Peter, ed. The Science Fiction Encyclopedia. Garden City, N.Y.: Doubleday, 1979.

Ouspensky, P. D. Tertium Organum. 1912. Reprint. New York: Random House, 1970.

———. “The Fourth Dimension.” In A New Model of the Universe. 1931. Reprint. New York: Random House, 1971.

Pagels, Heinz. The Cosmic Code. New York: Simon & Schuster, 1982.

Pearson, Karl. The Grammar of Science. London: Walter Scott, 1892.

Peebles, P. J. E. Physical Cosmology. Princeton: Princeton University Press, 1971.

Plato. “The Republic.” In The Dialogues of Plato, translated by B. Jowett. New York: Random House, 1937.

Reichenbach, Hans. The Philosophy of Space and Time. 1927. Reprint. New York: Dover, 1958.

Rucker, Rudy. Geometry, Relativity and the Fourth Dimension. New York: Dover, 1977.

———. Spacetime Donuts. New York: Ace, 1981.

———. Infinity and the Mind. Boston: Birkhauser, 1982.

———. The Fifty-Seventh Franz Kafka. New York: Ace, 1983.

———. The Sex Sphere. New York: Ace, 1983.

Schoefield, A. T. Another World; or, The Fourth Dimension. London: Swann Sonnenschein, 1888.

Schubert, Hermann. “The Fourth Dimension.” In Mathematical Essays and Recreations, by Hermann Schubert. Chicago: Open Court, 1898.

Stewart, Balfour, and Peter Guthrie Tait. The Unseen Universe. London: Macmillan, 1875.

Swenson, Loyd. The Ethereal Aether. Austin: University of Texas Press, 1972.

Taylor, Edwin, and John Wheeler. Spacetime Physics. San Francisco: W. H. Freeman, 1963.

Thorne, Kip. “The Search for Black Holes.” In Cosmology + 1, edited by D. Gingerich. San Francisco: W. H. Freeman, 1977.

Wells, H. G. “The Time Machine.” 1895. Reprinted in Seven Science Fiction Novels of H. G. Wells. New York: Dover, 1955.

Wheeler, John. “Frontiers of Time.” In Problems in the Foundations of Physics, edited by N. di Franca and B. van Fraassen. Amsterdam: North-Holland, 1980.

Willink, Arthur. The World of the Unseen; An Essay on the Rela-tion of Higher Space to Things Eternal. New York: Macmillan, 1893.

Wolf, Fred. Taking the Quantum Leap. San Francisco: Harper & Row, 1981.

Wolfe, Tom. The Electric Kool-Aid Acid Test. 1968. Reprint. New York: Bantam, 1969.

Zöllner, J. C. F. Transcendental Physics. Boston: Beacon of Light Publishing, 1901.

Index

Abbott, Edwin Abbott, 11, 56, 94, 96, 120

    and Flatland, 11-13, 40

    quoted, 28, 77

Absolute. See One, the

A Cube, 41-43, 44-45, 46, 47, 48, 97

Aether, 71-79

    and Einstein’s theory, 75-77

    and fourth dimension, 87-88

    and Michelson-Morley experiment, 75

    vortex rings in, 83-87

Ana (hyperspace region), 26, 28, 30

Analogy, reasoning by, 8-9, 37

“And He Built a Crooked House” (Heinlein), 34

Andrus, Jerry, 49

Another World (Schofield), 56

Anthony, Piers, 121

A Sphere, 16, 17-18, 39-40

A Square, 10, 11-12, 39-41

    and block universe, 140-143

    dream of, 202, 203

    in Globland, 114-117

    and left-right, 41-43, 97

    lifted into space, 21-22

    and space bump, 77-79

    and Sphereland, 99, 101

    Sphere stabbed by, 19, 21

Astria, 22, 24

Augustine, Saint, quoted, 145

Bell, John S., 187

Berkeley, George, 193

    quoted, 194

Black hole (door into Nothing), 92, 114, 120, 121

Black Hole, The (Disney movie), 120

Block universe, 135, 136, 139-140

    and absolute “Now,” 148-149, 154-155

    of Flatland, 137-138, 140-143

    of Lineland, 139

    See also Spacetime

Boole, Mary (wife of Charles H. Hinton), 64, 66, 68

Borges, Jorge Luis, 159, 193

    quoted, 140, 197

Breuer, Miles J., quoted, 32, 33

Burger, Dionys, 23-24

    quoted, 102, 103

Burgess, Gelett, quoted, 67

Cantor, Georg, 201

Carroll, Lewis, 113

    quoted, 115

Castaneda, Carlos, quoted, 148

CAT scan, 3-D model by, 44

Causal loops, closed, 173-176

Cause and effect

    as principle of universe, 179-181

    and synchronicity, 185, 186-187

Cave, Plato’s allegory of, 8-10

Circular time, 160-161, 163

Clifford, William K., 82, 83, 89

    quoted, 94

Coincidence, as synchronicity, 181-191

Contradictions of time travel, 166-174

Coriolis force, 126, 128-129

Cosmology, 91

Crookes, William, 53

Cube, A. See A Cube

Dali, Salvador, 34

Davis, Andrew Jackson, quoted, 30

Death, and spacetime, 134, 144, 147

Degrees of freedom, 5-6

Durrell, Fletcher, quoted, 22, 23

Eckhart, Meister, 136

Einstein, Albert

    and absolute “Now,” 149

    and aether, 72, 75-77

    determinism of, 186, 188

    relativity principle of, 150-151

    and space curvature, 79, 98

    and speed of light, 155

Einstein-Podolsky-Rosen paradox, 186

Einstein-Rosen bridge, 113, 118-121, 124. See also Hyperspace tunnel

Episode of Flatland: Or How a Plane Folk Discovered the Third Dimension, An (Hinton), 22-23

Escher, Maurits, 108

E.T. (Spielberg movie), 183-184

Etch-A-Sketch, 4, 58

Eternal return, 159-161

Ether. See Aether

“Ether and Relativity” (Einstein), 76

Fact-space, 195-199, 202

Fechner, Gustav, 53

    quoted, 13

Feynman, Richard, 176

Flatland, 10, 12-14

    as analogy, 37

    A Square lifted out of, 21-22

    block universe of, 137-138, 140-143

    discussion of higher dimensions in, 40

    false individuation in, 57-58

    and Globland, 113-118

    and Lineland, 14-16, 18

    motion through, 26, 35

    as pseudosphere, 104-105, 111-112

    rotation through, 42-43

    sphere in, 16, 17-18, 19

    as Sphereland, 93-97 (see also Sphereland)

    stretching of, 77-78, 119

    thicknessin, 19-21, 97

Flatland (Abbott), 11-13, 20, 21, 40

Fourth dimension, 3

    and aether, 87-88

    experiments on, 53-55

    hands “turned over” in, 38-39

    human beings in, 58-59

    and human thought, 67

    hypersphere in, 29-31, 93-101, 125

    left-right reversal in, 49-50

    as modern idea, 38

    movement in, 16-17, 18-19, 28, 35-37

    and mysticism, 59-64

    and Necker cube, 46-49

    as reality levels or colors, 6, 7

    rotation in, 43

    as space dimension, 7-8

    spirits in, 52-54, 56

    3-D perception of, 29

    time as, 7, 138-139

    universe in, 123-129

    very small particles in, 129

“Fourth Dimension, The” (Ouspensky), 59

Free will, 147-148

Fresnel, Augustin, 72

FTL (faster-than-light) travel, 155-159, 165-166, 169, 171, 173. See also Time travel

Gardner, Martin, quoted, 59

Geller, Uri, 55

Geometrodynamics, 82

Gerrold, David, quoted, 174, 175

Ghosts. See Spirits

Globland, 113-118

God

    design argument for, 185-186

    in ultimate dimension, 56, 200-201

    as watchmaker, 179, 180

    See also One, the

Gödel, Kurt, 145

    quoted, 145

Golas, Thaddeus, quoted, 199

Grammar of Science (Pearson), 87

Gravity

    in general theory of relativity, 72, 79-82, 90

    higher-dimensional analogue of, 126-127

Gustafson, Lars, quoted, 47

Hand, right or left, 38-39

Heaven, as higher-dimension world, 26, 120, 123

Heinlein, Robert A., 113, 121

    quoted, 34, 128, 129

Heisenberg, Werner, 202

Hell, as higher-dimension world, 26, 120, 123

Helmholtz, Hermann von, 84

Higher dimensions, nonrigid

    definition of, 138-139. See also Fourth dimension; Infinite-dimensional space

Hilbert, David, 201

Hilbert space, 57, 89, 202

Hinton, Charles H., 22-23, 26, 200

    life of, 64-68

    mysticism of, 60-64

    quoted, 72, 93

    on right-left confusion, 50

Hinton cubes, 64-65, 67

Homaloidal space, 101, 102, 112

Human existence, as four-dimensional, 58-59

Huxley, Aldous, quoted, 135

Hylozoism, 199

Hyperbolic space, 102, 109-110

Hyperbrain, 50

Hypercube, 29, 31-37

    Hinton’s visualization of, 65, 67 and Necker cube, 47

Hyperspace

    fourth dimension as, 3, 25-26, 124-129 (see also Fourth dimension)

    left-right reversal in, 49-50

Hyperspace tunnel, 113, 118-121

Hypersphere, 29-31

    in four-dimensional hyperspace, 124-129

    space as, 93-101, 102, 162-163

    through three-dimensional space, 18-19

Hypersurface, space as, 31, 93-101, 125

“Hypotheses Which Lie at the Foundations of Geometry” (Riemann), 93

I Ching, 181-182

Immaterialism

    Berkeley’s theory of, 193

    paradoxes of, 198

Individuals, in fact-space, 198-199

Individuation, and fourth dimension, 57-58

Infinite-dimensional (∞-D) space, 3, 57, 62, 89, 139, 199-202

Infinity and the Mind (Rucker), 145

Jung, C. G., 181

Kant, Immanuel, 38

    quoted, 39

    on right and left hands, 38-39

Kata (hyperspace region), 26, 28, 30

Knotted string, as fourth-dimension proof, 53-55

“Lands” of various dimensions. See Flatland, Globland; Lineland; Spaceland; Sphereland; Thought-land

Laplace, Pierre-Simon, quoted, 177

Left-right reversal, 38-39, 49-50

Lewis, C. S., 113

quoted, 116

Light, speed of, 121, 151, 155-159

Lineland, 14-16, 18

    as analogy, 37

    A Square’s dream of, 42

    block universe of, 139

    fact-space of, 195-198

    in spacetime diagrams, 169

Logical positivism, and other universes, 123-124

Lurie, Alison, 182

Mach, Ernst, and Mach’s principle, 127

Macroscope (Anthony), 121

Manen, Johan von, quoted, 20, 21

“Many Dimensions” (Hinton), 60

Matter

    Berkeley’s denial of, 193

    in general theory of relativity, 79-82

    geometrodynamic view of, 82-83

    and perceptions, 194

    as space bump, 79-82, 89, 119

    as vortex rings, 83-87

Maxwell, James Clerk, quoted, 71, 79, 86-87

Michelson-Morley experiment (1887), 75

Minkowski, Hermann, 143

Minkowski diagrams, 143-145, 149-151, 161, 169

Möbius, August Ferdinand, 45, 53

More, Henry, 52-53

Morosoff, N. A., quoted, 26, 27

Mysticism

    and fourth-dimensionality, 59-64

    and spacetime, 136, 145

Nabokov, Vladimir, quoted, 49

Necker cube, 45-49

Neutron star, 119

New Era of Thought, A (Hinton), 66-67

“New Refutation of Time, A” (Borges), 193

Newton, Isaac, 71-72

One, the, 61-62

    and fourth dimension, 59-61

    as reality, 61, 202

    See also God

“On the Space Theory of Matter” (Clifford), 82

“On the Various Standpoints with Respect to Actual Infinity” (Cantor), 201

Ouspensky, P. D., 59

    quoted, 4, 37, 59

Panpsychism, 199

Parabox, 49

Paradoxes in time travel, 167-174

Park, David, 143

Particles

    in closed causal loops, 174-176

    fourth-dimensional motion of, 129

Pearson, Karl, 87-88

Peebles, P. J. E., quoted, 91

Perception

    of body rotated in 4-D space, 44

    of four-dimensional objects in three dimensions, 29

    of three-dimensional objects by hyperbeings, 26-27, 28-29, 64

    of three-dimensional objects in two dimensions, 13

Perceptions, as reality, 192-194

Plato, cave allegory by, 8-10

Poltergeist (Spielberg movie), 56

Pseudosphere, 102-112

Psychokinesis, 188-191

Quantum mechanics

    closed causal loops in, 174-176

    and Hilbert space, 202

    on nature of light, 73

    and objective-subjective interaction, 183, 186, 193

    and parallel universes, 123

    and synchronicity, 186-188

    and time travel, 165-166

    and unpredictability, 180

Quasars, 125

Reality

    dimensions of, 194-195

    as fact-space, 195-199, 202

    I as, 192

    infinite-dimensional space as, 199-202

    as One, 202

    perceptions as, 192-194

    See also Spacetime; Universe; World

Reality levels, 6

Reichenbach, Hans, quoted, 6

Relativity

    of left-and-right, 49-50

    principle of, 150

    of simultaneity, 153-155, 157-159

Relativity theory

    general, 72, 79, 98

    special, 75-77, 121, 149, 155

Reversal of left and right, 38-39, 49-50

Riemann, Bernhard, 93, 110

Right-left reversal, 38-39, 49-50

Rucker, Rudy, quoted, 17, 44, 118

Schofield, A. T., 56

Schrödinger, Erwin, 202

Schubert, Herman, quoted, 56

Schwarzschild wormholes, 113. See also Hyperspace tunnel

Sensations, as reality, 192-194. See also Perception

Separate Reality, A (Castaneda), 148

Sight. See Perception

Simultaneity, relativity of, 153-155, 157-159

Slade, Henry, 53-55, 86

Small particles. See Particles

Soul, as aetherial vortex ring, 85-86

Space

    aether in or as, 71-79, 83-89

    curvature of, 79-83, 90-91, 102, 112, 121-122

    fact-space, 195-199, 202

    fourth dimension in, 7-8 (see also Fourth dimension)

    in general theory of relativity, 79-82, 98

    gravitational distortion of, 79-82, 90, 102, 119, 122

    Hilbert, 57, 89, 202

    and Hinton on world unity, 60-61, 62-64

    homaloidal, 101, 102, 112

    hyperbolic, 102, 109-110

    as hyperspherical, 92-101, 102, 125, 162-163

    infinite-dimensional, 3, 57, 62, 89, 139, 199-202

    vs. spacetime, 135, 137

    three-dimensional, 4-6 (see also Three-dimensional space)

    two-dimensional, 4, 6, 10

Space bubble, 121-122

Spaceland, 13, 40

Spacetime, 135-139

    acausal patterns in, 179

    bumps in, 79 (see also Space, gravitational distortion of)

    and eternal return, 159-161

    models of, 161-163

    and particle relationships, 176

    vs. passage of time, 139-149

    and relativity of simultaneity, 151-155

    and space, 135, 137

    synchronicity in, 181-191

    and time as fourth dimension, 7, 138-139

    world lines in, 149-150

    See also Block universe; Reality; Universe; World

Space tunnel, 113, 117-118

Speed of light, 121, 151, 155-159

Sphere, A. See A Sphere

Sphereland, 93-97, 99, 125, 126

Sphereland (Burger), 23-24

Spielberg, Steven, 56, 183-184

Spirits, in fourth dimension, 52-54, 56

Spiritualism, 51-52, 58

Spissitude, 53

Square, A. See A Square

Stars, black holes generated from, 119-120

Stewart, Balfour, 85-86

    quoted, 85

Superspace, 60

Surfaces, of constant curvature, 101-102, 112

Synchronicity, 181-191

    and EPR paradox, 186-188

    and psychic phenomena, 188-191

Tait, Peter Guthrie, 85-86

    quoted, 85

Telepathy, 188-191

Tertium Organum (Ouspensky), 37

Tesseract. See Hypercube

Thomson, William, 83-84

Thorne, Kip S., quoted, 120

Thoughtland, 40

Three-dimensional space, 4-6

    as degrees of freedom, 6

    4-D view of, 26-27, 28-29, 64

    hyperbeing’s movement through, 26, 35-36

    hypersphere in, 18-19

    two-dimensional perception of, 13-14

Time, 133-135

    circular, 160-161, 163

    as fourth dimension, 7, 138-139

    freedom from, 159, 173

    moving backward in, 176-179

    passage of, 139-143, 155

    See also Spacetime

Time travel, 165-166

    contradictions in, 166-174

    and speed of light, 157-159

Transcendental Physics (Zöllner), 53, 54

Two-dimensional space

    and Etch-A-Sketch, 4

    and Flatland, 10 (see also Flatland)

    and motion on earth surface, 6

Universe

    acausal patterns in, 179

    block, 135, 136, 139-140 (see also Block universe)

    cause and effect in, 179-181

    rotation of, 127-129

    shape of, 91-92 (see also Space)

    synchronicity in, 181-191

    See also Reality; Spacetime; World

Universes

    parallel, 3, 118-119, 123, 172-173

    plurality of, 123-129

Unseen Universe, The (Stewart and Tait), 85-86

Vision. See Perception

Vortex ring, 83-87

Waltershausen, Sartorius von, quoted, 14

War Between the Tates, The (Lurie), 182

Wells, H. G., quoted, 45, 171

“What Is the Fourth Dimension?” (Hinton), 65

Wheeler, John A., 82, 193

    quoted, 82, 196, 197

“Why Space Has Four Dimensions” (Fechner), 53

Willink, Arthur, 200-201

Wolfe, Tom, quoted, 152

World

    dimensions of, 194-195

    as fact-space pattern, 195-199, 202

    infinite-dimensional space of, 199-202

    parallel, 172-173

    See also Reality; Spacetime; Universe

World lines, 149-150, 156, 157, 169

World state, 197-198

World of the Unseen, The (Willink), 200-201

Worlds of various dimensions. See Flatland; Globland; Lineland; Spaceland; Sphereland; Thoughtland

Wormholes through space, 113, 117-118 (see also Hyperspace tunnel)

Wundt, Wilhelm, quoted, 52

Yes-and-no paradoxes in time travel, 167-174

Young, Thomas, 72, 73

Zeno’s paradoxes, 106

Zöllner, Johann Carl Friederich, 53-55, 56, 86

www.doverpublications.com

 

This is a free online webpage edition for browsing.
Purchase paperback or ebook editions from
Dover Publicationsor Amazon or B&N