In the mid-nineteenth century, Henry Slade of Albion, Michigan was notorious as an accomplished spirit medium. His séances and slate writing were sufficiently impressive that European nobles invited him to their courts, and it was on his tour of Europe in the 1870s that he convinced a handful of notable German scientists of the reality of the spirit world. Dr. J. C. F. Zöllner of Leipzig published an account of Slade’s demonstrations in his 1878 book Transcendental Physics, arguing that the physical impossibilities must have been caused by spirit beings living in the fourth dimension.
The 1881 Atlantic Monthly‘s humorously scathing review of his work says:
One opens this work of Zöllner with great interest, in the expectation of something substantial and more edifying than the dreary accounts of table-tippings, and the insane conversations of great men who, entering into a Nirvana, have apparently forgotten all they learned in this world, and have nothing better to do than to move chamber furniture. Unfortunately, this hope is not realized.
Nevertheless, Zöllner’s book and the controversy it engendered led to a surge in popular interest in the fourth dimension that lasted well into the 20th century.
So, what exactly did Slade do? Along with a torrent of slate writing, apparitions, and mysterious noises, he really got Zöllner’s attention by causing knots to appear in a loop of cord, as shown below. Try this yourself, and you’ll find it can only be done by unsealing the ends. Though the actual cord hung below the table, Zöllner observed the wax seal for the entire séance and was convinced by Slade’s claim of spiritual intervention.
Zöllner designed several physical challenges for Slade, to test his fourth dimension hypothesis. To understand the challenges (and how Slade’s responses almost entirely fail to meet them), we need to first understand the fourth dimension.
Mathematically, “dimension” refers to the number of coordinates needed to describe a point, or equivalently the degrees of freedom of motion in a space. A line is one-dimensional because a point in the line needs only one coordinate for its description. You could say “dang, there’s a hundred people in front of me,” which describes very well your sorry position in line. As another example, the volume of sound is a one-dimensional concept. A particular volume needs only one number to describe it, possibly from the scientific decibel scale, or maybe on the stereo knob “turn it up to 10″ scale.
A two-dimensional space needs two numbers for each point. The flat, infinite plane from high-school geometry is the prime example, with each point given an x and a y coordinate. The surface of a sphere is also two-dimensional; for example, points on the Earth are described by longitude and latitude. Though we spend most of our days wandering the two-dimensional surface of the Earth, our space is in fact three-dimensional, which means we can move on three axes, North-South, East-West, and Up-Down. Describing points in space requires three coordinates: to spot an airplane, you need longitude and latitude, plus elevation.
Question: What dimension is “color space”? That is, how many coordinates does it take to describe a color? (This is especially interesting because there are many different ways to describe color, yet all have the same number of coordinates!)
The next step is the fourth dimension. Mathematically, it’s no problem to define four-dimensional space, or “hyperspace.” It’s just an abstract space that needs four coordinates to describe each of its points, which works very well for computations, but is not much help in visualization. Trying to think in four dimensions is a serious challenge, and requires a complicated collection of mental crutches to make any progress.
The most effective crutch is the analogy with one lower dimension, a trick perfected in the novel Flatland, written by the 19th century minister E. A. Abbott. The book is the story of A. Square, who lives in a two dimensional world. Mr. Square describes his world, with some not-so-subtle criticism of Victorian society, and then is visited by a sphere from the third dimension.
You can imagine a two dimensional being as an amoeba trapped in a microscope slide, or as an ink spot moving on a piece of paper. Often it’s easier to picture him as very flat and living on the surface of a table. Let’s use this analogy to explain Slade’s feats of four-dimensional dexterity. Consider a challenge for a two-dimensional spiritual medium. We present him with a rubber band and a penny, and challenge him to put the penny inside the rubber band. You too, can play this game, but as a two-dimensional being you’ll need to keep the penny and rubber band flat on a table at all times. Clearly, it can’t be done. However, using the third dimension you can pick part of the rubber band off the table, slide it over the penny, and set it back down. The two-dimensional being would see part of the rubber band mysteriously disappear, then reappear on the other side of the penny.
Question: How would a two-dimensional being know that the penny was actually inside the rubber band?
Now, we have some of the tools to help us understand Slade’s challenges. In further sittings, Slade’s “spirits” caused rings of wood to disappear from a tabletop and reappear encircling the table’s leg, caused burns to appear on pig intestines held below the table, and caused snail shells to teleport from table to floor.
The rubber-band-and-penny thought experiment shows us exactly how Slade’s rings-around-the-table trick could work, if the medium had access to the fourth dimension. He simply lifts “up” the ring into the fourth dimension, and sets it back “down” around the table. But putting rings around a table is not what Zöllner had challenged Slade to do! In fact, Slade was to link the two wooden rings to each other. The rings were of different woods, each carved from a single piece. Two such linked rings are physically impossible to create, so their existence alone would provide excellent evidence for the fourth dimension. Linking the table, though impressive, is possible to fake.
So are Slade’s other feats. His initial feat, tying knots in a closed loop of rope, could also be done with four dimensions: move part of the rope out of our three-dimensional space, move it across the other part of rope, then bring it back to this world. But Zöllner was obviously suspicious of the rope trick, because his second challenge to Slade was to tie a knot in a closed loop cut from a pig’s bladder. Unlike the sealed loop of rope, which could be switched or tampered with, Slade had no way to create a knot in any continuous piece of pork. He had three choices: cut the loop and risk exposure, actually use the fourth dimension, or claim that the spirits weren’t in the mood. Not surprisingly, he chose the latter.
Slade’s final feat was to teleport some snail shells. Again, the fourth dimension is a good way to do this sort of thing. You move the shell into the fourth dimension, move it where you want it to go, then drop it back into our prosaic three-space. The two-dimensional analogy should help make this clear, as a third-dimensional being could lift an object out of the plane, move it, and set it back down. But again, this was not what Zöllner had asked for. In fact, Zöllner’s challenge to Slade was to take the snail shells, which had clockwise spirals, and turn them into snail shells with counterclockwise spirals.
Way back in 1827, the mathematician Möbius, of “Möbius strip” fame, realized that a trip through the fourth dimension could turn an object into its own mirror image. To understand, we return to the two-dimensional analogy. Take a symbol which looks wrong in a mirror, such as an N, and cut it out of a piece of paper. If you set it down on a table, you’ll find there’s no way to turn the N into the backwards N just by sliding the paper around the tabletop. But if you allow yourself a third dimension, you can simply lift up the N, flip it over, and place it back on the table. The four-dimensional version works the same way. You could use the fourth dimension, for example, to turn a right shoe into a left shoe.
Question: You could use the fourth dimension to turn a right glove into a left glove. But you can already do this by turning the glove inside out. What’s the difference?
In 1909, Scientific American held an essay contest to explain the fourth dimension, and many of the essays focused on mirror reversals. Isomeric chemicals such as dextrose and levulose (literally right- and left-handed sugars) were presented as evidence for the existence of the fourth dimension on the molecular scale, and one Zöllner enthusiast claimed that clockwise and counterclockwise snails are produced by a hyperspace reversal, right down to their “juices.”
H. G. Wells used the mirroring phenomenon in “The Plattner Story” of 1896, which is about a man who accidentally blasts himself a short distance into the fourth dimension. The man finds himself in a greenish world populated by spirits of departed humans, and can see faint images of the earthly realm overlaid on his new reality. After a week he manages to return home, but has become his own mirror image, as evidenced by photographs, his writing, and most impressively his heart, which now beats on the right side of his chest.
“The Plattner Story” was not the only appearance of the fourth dimension in literature of the period. It is the science behind The Time Machine, and also the home for the angel that falls to Earth in A Wonderful Visit, Wells’ first two novels. It is jokingly referred to in Oscar Wilde’s “Canterville Ghost” of 1887, about an English spirit who is snubbed by the new American owners of his ancestral manse. And Joseph Conrad’s The Inheritors of 1901 is about four-dimensional humans, devoid of conscience, who assume control of the earth.
Like many of the Victorians, I had my first exposure to the idea of the fourth dimension through science fiction, Madeleine L’Engle’s A Wrinkle In Time and its sequels. In these novels, Charles Wallace, Meg, and Calvin “tesser” between worlds, traveling through the fourth dimension. The word “tesser” means four, and shows up in the word “tesseract,” which is the four-dimensional analog of the cube.
Question: There are two points on the segment, four segments on the square, and six squares on the cube. How many cubes should be on the hypercube?
The tesseract, or “hypercube,” is the most accessible four-dimensional object, so it’s worth trying to understand. We work by inductive reasoning, starting with a point, and dragging it to trace out a segment. Then, drag the segment to trace a square, and drag the square to trace a cube. The next step is to drag the cube in a fourth direction, perpendicular to all edges of the cube, resulting in a tesseract or “hypercube.” The last step, as usual, is difficult to imagine because it requires the fourth dimension. We get the flavor with some drawings:
Using perspective, we can draw a cube a little differently. Doing a similar projection to the hypercube leads to the three-dimensional picture below. Your mind reconstructs the picture of a cube into a mental image of “cube” quite easily. Do the same with the hypercube, and you should have a pretty good three-dimensional image of a cube inside another, with corners connected by lines. However, this is only a picture of the hypercube, projected into our space using perspective. The smaller cube in the middle is smaller because it’s further away, in that fourth direction. To get an even better feel for the hypercube, play with this moving stereographic image.
Perspective images seem natural to us in part because we’re used to looking at them, especially as photographs, and in part because our eye functions in a similar manner. But in fact, perspective results in tremendous distortion of images. Close objects are shown grotesquely large while distant objects become tiny. At the start of the 20th century, a group of painters led by Picasso and Braque led a crusade against traditional perspective. They argued not only that perspective destroys proportion, but that in fact we don’t see like a camera — we see with two eyes, and our eyes move to understand a scene.
Although many other factors were involved, one of the instrumental ideas in the development of Cubism was that the fourth dimension could provide a viewpoint from which to observe the undistorted forms of objects. To understand how this might be true, imagine a two-dimensional creature looking at a square. Because the creature lies in the same plane as the square, it can see only one or two edges of the square at most, and seen corner-on, the angle measure would be difficult to determine. It would have to infer the shape to be a square. In fact, in Abbot’s Flatland, class distinctions among the 2-D beings were based on measures of angles, and a man with irregular angles could disguise his lower class status by concealing one side of his body. In our three-dimensional world, you can look at a cube from the side, but only know it is a cube when you turn it in your hands or walk around it. To overcome this, the Cubists attempted to portray all sides of an object at once, as if viewed from the fourth dimension.
Here are two fine examples of this technique, one by Picasso, who never explicitly acknowledged the influence of the fourth dimension, and one by Jean Metzinger, who clearly stated it as his goal.
P. Picasso, Portrait of Ambrose Vollard (1910)
© 2002 Estate of Pablo Picasso / Artists Rights Society (ARS), New York
J. Metzinger, Le Gouter/Teatime (1911)
© 2002 Artists Rights Society (ARS), New York / ADAGP, Paris
In both, you can see the similarity between the faceted figures and the angled planes of the hypercube, and the teacup in “Le Gouter” is a perfect demonstration of multiple viewpoints combined to give a full impression of an object. As another example of four-dimensional cubism, look at Marcel Duchamp’s “Nude Descending A Staircase, No. 2.” There’s a somewhat robotic figure shown in various stages of descent, as if we’re seeing multiple exposures. In this picture, Duchamp (who was the greatest advocate of the fourth dimension in the art world) considers the fourth dimension as time.
M. Duchamp, Nude Descending a Staircase, No. 2 (1912)
© 2002 Artists Rights Society (ARS), New York / ADAGP, Paris / Estate of Marcel Duchamp
Interestingly, this idea was one of the triumphs of Einstein’s theory of relativity, but the relativity papers were published in 1916, four years after “Nude Descending”! Duchamp, though a brilliant artist, wasn’t anticipating modern physics. He was simply following the lead of scientists who, from the mid 1800s, used time as another mental crutch towards understanding hyperspace.
The time crutch works as follows: Take your four dimensional object and cut it into a succession of three dimensional slices. Duchamp explains,
The shadow cast by a four-dimensional figure on our space is a three-dimensional shadow . . . by analogy with the method by which architects depict the plan of each story of a house, a four-dimensional figure can be represented (in each one of its stories) by three-dimensional sections. These different sections will be bound to one another by the fourth dimension.
Now imagine the slices played back as a movie, using the flow of time to “bind them to one another.” The classic example of this, used in Abbott’s Flatland, is to imagine a ball passing upwards through a plane. A being in the plane would first see a tiny dot, the top “slice” of the ball. As the ball moves up, the two-D observer sees the dot grow into a larger and larger circle. When the ball is halfway through the plane, the circle will be as large as possible, and then the observer will see it shrink to a point and disappear.
Just as we can’t play kickball with a frisbee, a four-dimensional athlete would need a “hypersphere” in lieu of a ball. And if she kicked it through your room, you’d first see a pea-sized object, which would quickly grow to a melon, hover, shrink back to a pea, and disappear.
Capturing this sort of movie was the goal of the Italian artist, Boccioni, who brags:
It seems clear to me that this succession is not to be found in repetition of legs, arms, and faces, as many people have stupidly believed, but is achieved through the intuitive search for the unique form which gives continuity in space. . . . If with artistic intuition it is ever possible to approach the concept of the fourth dimension, it is we Futurists who are getting there first.
U. Boccioni, Unique Form of Continuity in Space (1913)
Ironically, as Einstein’s theory of relativity was accepted in the early 1920s, its elegant definition of four-dimensional spacetime killed the romance between the public and the fourth dimension of space. Now that physicists were treating plain old time as a fourth dimension, speculations about mysterious “other” directions seemed ludicrous, and the fourth dimension disappeared from art and literature.
The surrealist art movement was one of the few reappearances of hyperspace. The spiritual associations and irrationality of the traditional fourth dimension must have appealed to Salvador Dali, who used many images and allusions to the fourth dimension, for example in the “Crucifixion (Corpus Hypercubicus)” and “In Search Of The Fourth Dimension.”
S. Dali, Crucifixion (Corpus Hypercubicus) (1954)
© 2002 Salvador Dali, Gala-Salvador Dali Foundation / Artists Rights Society (ARS), New York
The fourth dimension in this crucifixion is the cubical “cross.” We know from grade school that a flat paper cross can be folded into a cube, so we should be able to fold a three-dimensional collection of cubes into a tesseract. It seems that it would be impossible to “fold” two stuck-together cubes, but it’s not, and if you can visualize this maneuver you’re well on your way to higher dimensions. At the very least, take some solace from the plight of a two-dimensional being faced with two squares attached along an edge. He would assure you that folding along the edge is an absurd idea — the two squares would surely rip apart.
If you’d like more help folding your hypercube, turn to Robert Heinlein. Heinlein’s short story “And He Built A Crooked House” is the tale of an ambitious architect who designs a house in the form of an unfolded tesseract, only to have it collapse in a California earthquake and fold into the fourth dimension.
Even relativity theory didn’t answer the big question: does a fourth dimension of space exist? Physics says time is a fourth dimension, and modern string theories suggest a whole bunch of dimensions on the sub-atomic scale. But none of this precludes another direction, perpendicular to space, in which we could move if we only knew how. We are like the men in Plato’s Republic, chained in a cave and illuminated from behind. Their entire world consists of their own shadows, thrown on the cave’s wall. Shadows are all they have ever seen, shadows are all they know, and shadows are their reality. To tell these men that they are solid beings living in space is impossible, and it could be that way with us and the fourth dimension. If it’s there, it’s in a direction for which we have no conception and no way to look.
E.A. Abbott’s Flatland: A Romance In Many Dimensions (1884) is available online. It’s still the definitive discussion of the fourth dimension using the one-dimension down analogy.
Rudy Rucker’s The Fourth Dimension (1984) is fabulous and readable.
Answers to Questions:
1. Color space is three dimensional. Colors are described by Red-Green-Blue coordinates, as shown below, or often by Hue-Saturation-Value coordinates. CMYK seems like it has four coordinates, but that’s just to save ink by not mixing Cyan, Magenta, and Yellow to make Black.
2. The 2D being couldn’t see the penny from any side: it’s surrounded by the rubber band. It would have to push on the rubber band and infer that the penny was inside.
3. If you turn a right glove into a left glove using the fourth dimension, it will really be a left glove. It won’t be inside out.
4. There are 8. Here they are: