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.
* * * *
Bryan Clair is a professor of mathematics at Saint Louis University. His previous publications in Strange Horizons can be found in our Archive.
Further Reading:
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:
[Back to Table of Contents]
Interview: Brad Strickland
By James Palmer
9/23/02
Brad Strickland has been a professional w
riter since his first sale to Ellery Queen's Mystery Magazine in 1966. His first SF novel, To Stand Beneath the Sun, was published in 1985. Since then he has written or co-written fifty novels, including five Star Trek young adult works written with his wife Barbara and several Are You Afraid of the Dark?, The Mystery Files of Shelby Woo, and Wishbone adventures done alone, in collaboration with Barbara, and in collaboration with fellow Georgia writer Thomas E. Fuller. In 1992, Brad began a collaboration with the late John Bellairs, finishing The Ghost in the Mirror and The Vengeance of the Witchfinder from uncompleted manuscripts and then going on to continue the YA gothic mystery series begun by Bellairs. Currently he is collaborating with Fuller on Pirate Hunter, a YA adventure series they created. It will be published by Simon and Schuster beginning this fall. In the daytime, Brad is Associate Professor of English at Gainesville College in Oakwood, Georgia, where he teaches American and British Lit, English Composition, and the occasional SF and Fantasy writing class.
This interview was conducted via e-mail.
James Palmer: What prompted you to become a writer?
Brad Strickland: I come from a family of storytellers. As a child, I listened to my grandparents, parents, aunts, and uncles tell all sorts of stories around the fireplace of my grandfather's farm at night. And I always loved to read, so storytelling came as a natural step for me. Despite being a teacher who stands in front of classes all day and an occasional actor, I'm actually pretty shy about talking to groups, so writing the stories rather than reciting them also seemed a natural step.
James: Who are your biggest influences as a writer?
Brad: I have a good many. For the characters, I love the works of Charles Dickens—flamboyant, memorable people who stick in your mind long after you forget the details of the plots. Ray Bradbury's wistful, gentle, but surprisingly muscular and emotional style was an eye-opener for me as I was beginning to write. And just lately, the Napoleonic-era historical novels of Patrick O'Brian have been a composition course on how to re-create the feel and style of a bygone era.
James: How did you get started writing young adult novels?
Brad: It began when The Byron Preiss publishing company put together a series of books for younger readers, based on fantasy characters or creatures. I did Dragon's Plunder, a comic fantasy involving some inept pirates and a studious dragon. Then a little later, the fantasist John Bellairs died suddenly of a heart attack. John is well-known as the writer of a cult favorite, The Face In The Frost, but he had also done more than a dozen YA gothic thrillers. He had left two books partially finished, and his son, Frank, wanted to have them completed (Frank was not himself a writer). He knew some of my work and suggested me to Richard Curtis, who was John's agent as well as mine. Richard asked if I would complete The Ghost In The Mirror.
I wasn't sure I could do it. After four or five long phone conversations with Toby Sherry, John's wonderful editor at Dial Books for Young Readers, and after meeting the publisher face to face, I got up the nerve to finish that book (it needed a better climax and resolution and some earlier chapters to “plant” one or two plot developments). I also completed The Vengeance Of The Witch-Finder. Those did well, leading me to go on to write two more books from the briefest of descriptions left by John ("Johnny and Fergie fight a voodoo priest whose magic threatens to kill Johnny's family") and then to extend the series with original novels written in the style of John. Those did well enough that, somewhat to my surprise, other YA publishers began to offer me projects.
James: Was it hard to follow in his footsteps?
Brad: It still is. I get mail from fans who berate me bitterly for this or that in the books. However, I also get much more mail from fans who like my books, so it more than balances out. I'm always keenly aware that I have a responsibility to do the best I can to live up to John's books, though I'm also aware that I am not John Bellairs and will never be. The best I can do is to keep the characters true to their backgrounds and to write the stories that seem to me to show off their personalities best.
James: How are YA novels and adult novels different?
Brad: YA novels are a bit shorter and more economical. Of course, there aren't graphic scenes of violence or erotica in YA fiction, but then I didn't have a plethora of these in my other fiction, either, so I hardly noticed the difference. I don't attempt to simplify the style or anything as I write for a younger audience; it's mainly a matter of thinking back to when I was younger and to the kinds of things I liked to read then.
The first concern, always, is to tell the best story possible. If the editor wants to change some of the vocabulary, or more often to explain it, then that comes later. I never really worry about it. When I was writing for the Wishbone series, the editors would frequently want to throw in a passage of explanation when they thought the vocabulary might be difficult. I would write, “Sam was wearing a poodle skirt and saddle oxfords as her fifties costume.” The editors would change that to “Sam was wearing a poodle skirt, which was a gray flannel skirt with an appliqué of a poodle on it, and saddle oxfords, which were white shoes with a black leather instep, as her fifties costume.” Lots of times I'd change it back or revise it to be somewhat less clunky.
James: Are YA novels more difficult to write than adult novels?
Brad: Yes and no. No because I don't really worry about it being a YA book while writing it, and yes because I have to be very careful about my research, about the clarity of the style, and so on. Kids are very sharp, and they'll eagerly point out any errors of research you might make. With adult readers, the main things you have to worry about are cars and guns. If you make a mistake about a car make or model, you'll hear from readers; if you make a mistake about the caliber of a weapon, you'll hear from armed readers.
James: How did you get started writing the Star Trek, Nickelodeon, and Wishbone books?
Brad: Star Trek came first. About the second or third year of Star Trek: The Next Generation, the producers felt the ideas were getting stale, and they asked the members of the Science Fiction Writers of America (now the Science Fiction and Fantasy Writers of America, but still “SFWA") to pitch story ideas to the writing staff. My wife Barbara is a huge Trek fan, and on a long car trip, we crafted the outline for a Next Gen episode to be called “Happy Birthday Data,” all about Data wanting a birthday, and interviewing other crew members about their favorite birthdays. Each cast member got a little star turn in a flash-back scene about the character's birthday. We passed that on to Paramount, who liked the idea but not for a script, and they sent it to Pocket Books. The editors there then got in touch with us and said, “We love this as a book idea! Would you please write it for us? Except take out the birthday stuff and Data, and instead concentrate on....” It became a whole different story, but it got us the first of five Star Trek assignments.
Because of that, the people at Pocket Books asked us to write for the Nickelodeon series. And because the editor of Wishbone books at Lyrick Press had been a coeditor on the Trek books, I had the opportunity to launch all three of the Wishbone series: the Adventures of Wishbone, the Wishbone Mysteries, and the Wishbone: The Early Years books about Wishbone as a pup.
James: How difficult is it to play in someone else's universe? Does it limit you creatively?
Brad: It varies enormously. The Bellairs books are fun because each of them hangs from some odd little fact—the Voynich Manuscript, say—or some strange historical event—the attempt by Count Cagliostro to live forever. I have a great deal of freedom within these books, limited only by remaining as true as I can to the conception of the characters John created. The Star Trek books, of course, take place within a well-defined universe. Fortunately, my wife is a great fan of the series, and she knows it so well that Paramount had only two changes in all the five books: we had invented a computer component, and the next season of Deep Space Nine was going to include something called “isolinear chips,” so they told us to change our term to theirs. And once they asked that a character not carry a
phaser, since he rarely if ever did on the show.
Sometimes, though, the publishers and/or the creators ask for changes that are more fundamental or even nix plot ideas altogether because they don't see how they will fit into the world that has already been established by the series. The creator of Are You Afraid Of The Dark? didn't really mind us using one of his characters, but insisted on re-dressing him, because “He likes earth tones."
James: You have collaborated on many books, both with your wife and Thomas E. Fuller. Is that an easier writing process? Is it more fun?
Brad: I enjoy collaborating. Writing is lonely, and you never know if what you are doing is much good. Having another mind and set of eyes to go over everything is reassuring! Barbara and I usually collaborate by talking through the plot idea. I then do an outline; she revises it; we talk through the changes and get a final version; I write a first draft; she revises it; then we again go to a final version for submission. That takes more time than a solo effort, but it's more enjoyable.
When Tom and I collaborate, we actually spend a great deal of time, a larger percentage of our time, on the outline; then we create a very detailed outline (about one page of outline to ten pages of book) and divide the labor. He'll do chapters 1, 3, 5, and so on, while I am doing 2, 4, and 6. We exchange chapters by email, revise each other's chapters, and put them together into the manuscript; then we go through and do a detailed revision and polish, meeting in person and turning page by page through the book as we fine-tune everything. It goes faster than a solo effort, and by the time we finish, it's awfully hard to tell who wrote what.
To be more specific, the first decision we make is who the protagonist will be and what problem he/she has. We build the conflict from there, and the plot from the conflict. Then we proceed as above.
James: You also write and perform radio plays. Does working in an auditory medium help you when you sit down to write a novel?
Strange Horizons, September 2002 Page 5