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. 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. In Madeleine L'Engle's "A Wrinkle In Time," 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.

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.

J. Metzinger, Le Gouter/Teatime (1911)

P. Picasso, Portrait of Ambrose Vollard (1910)

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)

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.

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."

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.

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.