Part 2

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Once inside Madam Aaron's home, conversation quickly shifted to the business at hand: geometry. Tom had expected a barrage of rules, theorems, and formulas, but the woman never even pulled out a pencil and paper. She simply conversed with the boy as if geometry were a part of life, as mundane and ever-present as the weather.

They soon made their way to the "sitting room," as Madam Aaron called it. Indeed, that did seem to be the sole purpose of the room. Its windowless walls were plastered with maps and charts, covering such varied topics as the planets and constellations, phrenology, and music theory. Other than these, the room offered few distractions. There was only a long, gray sofa, a small end table, and a chartreuse armchair that put Tom in mind of a throne. Tom knew without asking that the armchair was not for him and so seated himself on the sofa.

An ornate rug covered most of the floor. Its intricate forms in various shades of purple and blue gave the impression of writing, but not of a sort familiar to Tom.

The pair sipped a bitter herb tea cut with milk and heavily sweetened. Tom wasn't much of a tea drinker, but Madam Aaron had insisted: "Sugar's just the thing for studying. And besides, the tea will help you concentrate. I don't want to waste my time on a sleepy, distracted pupil."

After an hour or so, Tom felt he had a much better understanding of Euclidean space and the Cartesian coordinate system. He might even have been able to make up a rule or theorem of his own if he gave it some thought. The more they talked, the more it began to sound like science fiction. He had never known the real world of math and science was so absurd, and so little understood.

"There really are other dimensions?" Tom asked excitedly. "How many?"

"Something on the order of nine or ten, as far as we can tell," replied Madam Aaron. "Not counting time, of course. That's another matter altogether." Her brow furrowed as she seemed to recall some ancient, unsettled dispute. "But really, what do we know? Not long ago, we thought there were only three total! Well, most people did anyway. Some of us knew better." She winked impishly.

"I wonder what they're like," Tom mused. "Do you think people will be able to visit the other dimensions?"

Madam Aaron lightly closed her eyes as if she were embarrassed for the poor boy. "You use the word dimension as if it referred to some other world, disconnected from what you see all around us right now. A dimension is nothing more than a way to measure the size and location of things—to describe them in space." She said all this with her eyes closed, as if she couldn't face the child again until he grasped this basic concept.

After a moment, the old scholar opened her eyes, took a deep breath, and continued: "In the three dimensions of space that you're already familiar with, you can picture the universe as a large box. You can measure your height inside the box, you can move up and down, and—in theory at least—you can measure the height of the box. That's one dimension. Do you remember what we call it?"

"Up and down is... Z," Tom answered. He pictured the Z axis, an imaginary line starting infinitely low underground and running infinitely high, out into space.

"Good," Madam Aaron continued. "Imagine standing inside this universe box and facing north. If you move forward or backward, that's in the Y dimension. You also have some thickness from front to back on Y, just as you have height on Z."

This seemed an odd description of some very obvious facts, but Tom made no comment. He had faith that his tutor would eventually come to the point.

"You can also move from side to side, of course. That would be east and west in dimension X—or put another way, on the X axis. At any given time, your location—and in fact, your size and shape—can be described in terms of X, Y, and Z. That's just in three dimensions, mind you, not in reality."

At this, Tom had to object: "But reality is three-dimensional! Our reality, at least. I mean, that's obvious, right?"

Madam Aaron seemed to expect such a comment. She responded with the sort of patient determination one might expect from a dog trainer: "What's obvious is rarely true. Let's talk about a two-dimensional reality for a moment. It would be like a sheet of paper, infinitely large and infinitesimally thin. Really, it should have no thickness at all, but even a sheet of paper has to have some thickness, or it simply wouldn't exist. So it's not truly two-dimensional, though it seems to be. It has thickness in all dimensions or none at all."

Tom nodded absently. So even a flat sheet of paper was three-dimensional because the space it occupied was three-dimensional.

"As an exercise," she continued, "imagine yourself as an ant crawling around on that sheet of paper, unable to look up or down, just crawling along. You might think yourself as flat as the paper, never realizing that you have height—or that there even is such a thing as height. For you and your ant friends, the paper is a whole universe, apparently two-dimensional. If you jumped off the paper, the other ants would think you simply vanished."

Tom contemplated this. He was beginning to see where his tutor was headed. "So if there's a fourth dimension..."

"There is a fourth dimension. We call it W." Madam Aaron gave Tom a harsh, penetrating look. For an instant, her eyes were as icy cold as Ella's. They seemed greedy, pitiless, and unfathomably old.

"I know this is hard to picture," she continued, "so I'm going to attach some words to dimension W that might help. Before learning how to move on the W axis, you need to learn how to see. Just like an ant has to learn how to look up and down on Z, you have to learn how to look in and out on W." An unnatural grin spread across her face. "I can help you with that."

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