by Sonia Balcer © December, 1987
In honor of all good Teachers, especially in the area of Science
"Something strange is happening here," said Jeff as he stepped towards the window. The lighting in the room had changed in a subtle way, the way it does when you move a lamp or rearrange the furniture, yet nothing had really been moved. It was sort of a shift of ambiance, indefinable yet evident, as if Space itself was changing its shape, unfolding with a creeping motion very like that of a piece of cellophane crumpled and left to itself. A moment later, the process seemed to reverse, and everything returned to normal.
"A disturbance in the space-time manifold", said Professor Bundle as he looked at his wrist-worn Curvometer. The bright face of Greg, a sandy-haired little boy who was the youngest of the three children, became focused with curiosity. "What's a man-eefold?" The professor gently spread out his napkin on the table and smiled at the intent youngster, his own venturesome eyes sparkling brightly the way they always did when he was about to explain such abstract things as this, which were to him the most delightful of all. "It's sort of a galactic cloth, an underlying fabric for Space-Time that tells you what shape is had by Space itself. In two dimensions, you could think of it as a grid on a sheet of paper, laid out flat for what you'd know as 'flat space', or shaped into a round globe for space that is curved around like a sphere. For more than two dimensions, we can't visualize the Fabric, but nonetheless it has a shape." The professor picked up the napkin and supported it across his fingers. "And this shape can change whenever there is an intense gravitational disturbance somewhere in it-like the way this napkin droops and changes its shape when I put a marble on it. In other words, the Fabric can get ripples in it."
"So that's what just happened?" "Exactly," said the professor, with a sense of awe not unlike that of the little boy's, "we felt a gravitational ripple, caused by a giant star somewhere that has collapsed into what we call a 'Black Hole'-spewing out material from another universe." Greg listened wide-eyed and tried to imagine what such a thing would look like as it distorted the galaxies he had seen in the astronomy picture-books the Professor had shown him. He finally compared it with a volcano, although he knew it wasn't really the same kind of thing. This "eruption" was not within any towering, dark mountain, but in the Fabric of Space itself.
"We get a lot of these here", said QueFoure, a youngster who was a native inhabitant of Entrance Tower, and whose parents has arranged for him to be excused from Curriculum that day so that he could see Professor Bundle even if they couldn't be In-Tower themselves. "You get used to them. We call them Ripple Storms.".
"We don't have them at all on our world," said Kelly, the bright ten-year-old to whom the Professor had taught Tensorial Calculus and General Relativity, and in whom he saw such promise. "That must be because you come from only a four-dimensional universe-which includes the dimension of time," said QueFoure, who was just learning Manifold Geometry himself. "So you have only three space-dimensions. The other dimensions are folded up, including the direction that ripples." After a pause, he added thoughtfully, "I've always wondered what that would be like."
Kelly leaned towards him, her eyes brightening with imagination. "You should come visit us. You'd like Lone Pine a lot-it's a very pretty place, and you could see our 3D world for yourself." "But he couldn't come into our universe," said the older boy, Jeff, with a puzzled expression, "he'd be, well, sort of crushed, wouldn't he? I mean, how could his 4D body fit into our 3D world?"
From the window where he stood absorbed in the wonder of the metropolitan sights, the Professor smiled endearingly and listened for Kelly's answer. The child thoughtfully looked at her hands and moved them in the direction of the additional dimension. "The same way that we adapted to his world without being pulled apart-we unfolded in the 'fourth direction'. We're not used to moving that way, yet we adjusted alright because we didn't change-we are just in a different place. "So QueFoure would simply 'fold up' in the 'eckward-andward' direction. It's not at all like being crushed, because it's not his body folding up; it's Space itself folding up. It's a Professor what was that word?" "Compactification." Yes. A compactification of the space-time manifold," she said with the same fascinated, far-away look that her father, a mathematician and colleague of the Professor's had when he was talking about his work. QueFoure looked at them as if he were trying to imagine what it would be like. Kelly laughed, and said to Jeff, "It would feel kind of weird for QueFoure, not being able to move his body eckward or andward, but he would be unharmed."
"I see. It kind of makes sense," said Jeff, whose interest was much less in mathematics and book-work than it was in his friend Kelly, and who had come along for this afternoon jaunt in the Starship with the Professor because he liked all the strange sensations of space-time travel and would much sooner be enjoying these than doing his homework. His younger brother Greg, on the other hand, liked mathematics better than almost anything else (except for his puppy, Bunsen Burner) and at age eight was quite a part of the family. Jeff and Greg's parents were happy about this too because, as High-Energy-Particle Physicists, their work often took them out of town or even out of the country for weeks at a time, and they were glad to have a stable place for their sons to stay, where they didn't have to move constantly and lose friends or school progress.
And it certainly was an interesting household for the boys to be in, for the Professor didn't stop at working his ideas on paper- he tried them out whenever he could, including his Theory of Folding Manifolds that led, "inevitably" (as his wife would always say with fondness about his inventions) to the building of his Spacebender Starship. Mrs. Bundle turned out to be a splendid partner in this endeavor, since her years of experimenting with chemistry and art while the children were growing up had developed in her a remarkable ability to mentally picture his invention concepts. She was always making sketches for him and thinking up strange new materials for constructing his prototype devices. Furthermore, her artistic inclinations carried over into a skill for crafting things with her hands-everything from unusual toys for the neighborhood children and special layered cards for friends, to decorative tiles in the shower, creative stone arrangements in the garden, and, during long evenings in the laboratory, with her gentle but quietly vibrant husband, strange machines made of exotic materials. The children never knew what new invention would emerge from this strange Lab which had come to engender wild fantasies of time travel and was, in a hushed and enchanted sort of way, the talk of the neighborhood.
So, it wasn't long at all after they had built their SpaceBender StarShip and convinced everyone that it worked reliably that the Professor began taking the children along with him for short trips. For this was indeed his greatest discovery, and he wanted to enjoy it and share it. He had been recognized the world over, and had been bestowed with numerous honors and awards, but nothing could compare with the simple joy of watching the children's faces as he set his manifold-curvature coordinates and guided the ship through infinite halls of indescribable color, twisting the fabric of space itself into pulsating bundles of glorious sights. "Gravitational Optics", as he called it, a far-reaching result of the General Relativity connection between the gravity arising from matter and the changes in motion and position arising from a bending of the space-time Fabric. Indeed, his was no ordinary space-ship or time-machine. This was a Manifold-Curving Starship, which could take them anywhere by simply bending space itself. Moreover, by using a suitable "gravitational lens" such as a collapsed star, it could funnel them into different universes altogether-ones with completely different space-time manifolds, even different dimensions-universes parallel but unreachable by any amount of "travel" in the ordinary sense. Mrs. Bundle had been his first companion on these early voyages, and over the years, she had created many fabulous 3-dimensional "space-scapes" based upon the sights they had discovered on their unique "vacations" together.
The professor smiled as he watched the children survey their four-dimensional surroundings. What a way to spend an afternoon. And the sensation of being in 4D space! How could one describe it? It was like waking up, like all the room becoming more open, and you suddenly realizing that you had been standing in the same position all your life and that now you were able to move in a direction you hadn't known before. And the people of Corridor World were so hospitable. They seemed to pick up on his language right away, see the pattern all at once without having to learn it piece by piece. During his first visit, when he almost didn't make it across the Barrier between the universes and got brutally ill with Time-Dilation Sickness, they took care of him, and invited him back. They even helped him map out a better route through the manifold. It was an ideal place for a little field trip-a perfect setting for the children to learn about dimensions, space-curvature, manifolds, and the like. For here they were: totally outside their own universe, yet sitting and talking with QueFoure as if they were at home, and drinking some absolutely delicious four-dimensional hot chocolate. Yes, this was his idea of a nice field trip. "We could certainly make science more interesting for children," he mused to himself.
Kelly stood up. Another ripple storm was beginning, but this one seemed stronger, almost violent, "Look!" The lighting not only changed, but it seemed to bend around cor-ners that weren't really there. And the walls of the room were no longer "hypersquare"-the shapes of things seemed to change. From far away there was the sound of an alarm, and announcements in a language foreign to them, that sounded like instructions.
"I've got to get you to your spaceship immediately," said QueFoure. "What's wrong?" asked Jeff as they got up and began making their way quickly down the hall. "We have to evacuate. This is more than a ripple storm; it's a Fold-Wave!" "You mean like a Tidal Wave back home?" asked Greg. The professor smiled gently with professorial pride and affectionately mussed the boy's hair in between strides. "Yes, it's kind of like that, only more drastic, because it's not a physical object that's making the wave, but space itself wrinkling up, the Fabric folding over under some tremendous disturbance, probably a Macro Black-Hole, swallowing up an entire galaxy and tunneling it from one universe to another with a huge, explosive force."
"But I thought it wouldn't hurt QueFoure to be folded up," said Jeff. "In your Starship it wouldn't hurt me," said QueFoure, opening the doors to the Space Bay, "because your Starship does it smoothly, keeping the time-coordinate in tact. A Fold-Wave just rips you apart, sending different parts of you into different times, and such a 'dissynchronization', even over very short periods of time, would have drastic effects. I mean, can you imagine what it would be like, for part of your body to be in the future and part in the past? It's as bad as being torn apart, because those parts would be disconnected and out of reach with the rest of you. That's why we have to go into Time-Shelters and wait it out, usually for a few hours."
"But what will happen to Entrance Tower?" asked Kelly as the Professor warmed up the Starship. "Oh, it's built to take the beating-it's reinforced with Time-Fibers," QueFoure said, as an alarm sounded nearby. "You'd better hurry-just two and a half minutes before the Fold-Wave hits." Just then another pre-shock swept over them. "What's keeping the Professor?" asked QueFoure, his face beginning to look distorted and his breathing becoming courser. "Just plotting in a course," said the Professor. "It's tricky navigating around a fold-wave. There's no room for error, and I don't want us to end up spread out all over the entrance subspace. I'm looking for a suitable trajectory. And QueFoure, don't you think you should get to your Time-Shelter?" "Well, I've been thinking" said QueFoure awkwardly " about visiting us?" said Kelly, her expression eager. "Oh please do!" said Greg, bouncing perceptibly with excitement. "I'd really like to," said QueFoure. "It sure beats waiting for hours in that Shelter!" added Jeff, giving voice to QueFoure's unspoken thought.
"You're welcome to stay with us; I'll transmit a message to your parents." said the Professor quickly but happily. "Close the door. I think we're ready." "And not a moment too soon!" said Jeff, who had been keeping track of the time and noticed that they had only a half of a minute left. The children settled back into their seats. The professor engaged the Spacebender sequence, and they slipped right out from under the wave into the wide-open space beyond the Tower and, with the Professor's skillful maneuvering, they traversed the hyperspatial strata and emerged in their own universe, right onto the launch platform in the Professor's backyard. "I'm ready for dinner!" said Jeff as they walked into the house where Mrs. Bundle was waiting-dinner in the oven and freshly-made hot-chocolate on the stove.
The pair of words, "eckward" and "andward" are credited to C.S. Lewis in his partial manuscript of the science fiction novel, The Dark Tower. He had invented these words for essentially the same purpose as in the Professor Bundle story: to signify a fourth distinct space-time dimension, which means that it had to be independent of the ordinary three directions, "up-down", "left-right", and "inwards-outwards" (or "forwards-backwards"), and of the single ordinary time-direction "back to the past-forward (ahead) to the future." It seems fitting to follow his philological lead in adopting these words which are not yet in any dictionary.
How to construct a cube of "n" dimensions from "squares" of "one less than n" dimensions:
by Sonia Balcer © December, 1987
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