Tom stood perfectly still as the cool evening wind rustled his shirt and raised tiny bumps on his bare arms. He counted deep, methodical breaths from one to ten, over and over, until he was satisfied his heart rate was lowered acceptably. A faint stirring of electricity in his fingertips informed him that he had probably been breathing a bit too deeply, but that would pass.

Beyond an intimidating set of black iron bars stood an austere Victorian cottage, its grayish paint peeling to reveal dry, crumbling wood. A wooden fence extended from the gate and ran along both sides of the property. Every rotting post was laden with a chaotic tangle of vines and ivy. An ancient live oak sprawled its gnarled limbs overhead, threatening to collapse onto the heads of unwary intruders.

It was time. Tom could no longer convince himself that this pause at the threshold was just to avoid being out of breath when Madam Aaron answered the door. The simple fact was that he was afraid of the old woman. Or at least, he decided, of the situation.

He'd never been to a tutor before, and he was confident he could get through seventh-grade geometry well enough on his own, but his mother had insisted he take Madam Aaron up on her generous offer. Free instruction from such a distinguished personage was an honor that could not be refused.

Tom gathered his courage. He gave the gate a cautious push, and with a long, distressing creak, it swung open.

In response, a number of black birds sprang from their various hiding places and scattered in all directions across the twilit sky. Tom gazed up at the noisy creatures—crows or ravens, he guessed—trying to remember what such a group might be called. Not a "flock," he'd learned in school, something more ominous. A meanness? A conspiracy?

Murder. The word tore into the boy's mind, freezing him with terror. Still staring at the sky, stricken with a paralyzing dread, he felt a hot breath touch his exposed neck. His heart raced, providing a jolt of adrenaline just powerful enough to restore mobility.

He lowered his gaze to find an icy pair of eyes floating in the darkness, mere inches from his face. Two rings of blue in a field of black. Inside his head, Tom began to scream. He imagined falling backward, scrambling to his feet, and sprinting all the way home. But in reality, he couldn't be the first to break this silent stare. The eyes had him.

"Don't mind the hound, son. She's not half so mean as she looks." It was Madam Aaron, calling from her front porch. Before Tom's eyes, the blackness took shape and became nothing more than a dog—albeit a very large dog.

"Her name's Ella," she continued, ambling casually toward them. "She's part Newfie and part husky... and all sissy. See the cornfield back there?" She gestured behind the house. "Ol' Ella won't go near it. I don't even need a fence back there to keep her in. The corn does just fine."

Tom gulped and found his voice: "Miss... Dr. Aaron!"

"Madam Aaron will do," laughed the old scholar. "I'm no doctor."

#

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."

Tom looked around the room for a clock. He had the sudden feeling that it was time to leave. As if guessing Tom's thoughts, Madam Aaron said, "Don't worry about the time, son. It's Friday night. I told your parents we'd stay at it till we got it, and we ain't quite got it yet." Then, without another word, she stood up and left the room.

Tom considered making a run for the door. His mother would be upset with him for cutting the session so rudely short, but there were worse things than upsetting one's mother. The only serious barrier was that dog, Ella.

Before Tom could come to a decision one way or another, Madam Aaron returned. She carried a lighted candle, placed it on the end table, and looked at Tom hungrily. "Bring your feet up onto the sofa, hug your arms around your knees, and fix your eyes on the flame." The boy complied, uncomfortable but curious. Madam Aaron then cut the lights, and only a single dancing flame remained in the room.

"Okay, here's the tricky bit," she began. "The fire you see is neither inside you nor outside you, but right where you are, on the same three-dimensional plane. As it burns, a part of it travels inward on the W axis—that's the destructive force—and a part of it radiates outward. That's the energy, which is creative. I want you to look out and see that energy coming off the flame."

Despite his trepidation, Tom was excited at the idea of learning this strange old woman's mystical secrets. He imagined Madam Aaron as a powerful witch, and himself as her protégé wizard-in-training.

After a moment, Tom realized his mind had been wandering and he hadn't responded to Madam Aaron's request. In another moment, he realized that the small flame now filled the room. It was surely a trick of his eyes, but he couldn't bring himself to test the theory by looking away. It was simply too beautiful.

"It's... everywhere," he finally managed to say.

"Then you're doing it right," said the old woman. "Can you see me?"

"Can I..." Tom turned toward the chair. The room was still dark, though now it was also pulsing with the radiance of a fire that burned not hot, but sweet. The chair itself was hardly visible, but Madam Aaron was glowing much brighter than the candle flame. She faced Tom and stood up, looking no older than a teenager.

Something was wrong here, really wrong. Tom suddenly had no interest in learning magic, becoming a witch's apprentice, selling his soul to the devil, or whatever this was. He sprang from the sofa in an awkward burst, his foot catching on the rug. This sent the end table tumbling, candle and all. Tom didn't wait to see his tutor's reaction. He was out the front door before he regained his usual sense of sight. No more dazzling candy flames—only the night, now dusty and gray.

Tom bolted straight for the gate, but halfway to it, a pair of icy blue eyes flashed open in the darkness. Ella sat serenely in his path, as if she'd been expecting him. Tom skidded to a halt and then stood there a moment, unsure of what to do next. As if to make his mind up for him, the massive hound delivered a hateful barrage of slobbery thunder.

Tom stumbled backward in terror, scrambling physically and mentally for a way out. The corn! If he could just make it to the cornfield in time, Ella wouldn't follow. At least that's what Madam Aaron had said.

Racing past the side of the cottage—now lit up like a jack-o-lantern—Tom pushed through the uneven stalks of corn. He ran for a bit in one direction and then shifted course rapidly in a clumsy attempt to throw Ella off his trail. He repeated this technique until he was painfully out of breath—and, it occurred to him, hopelessly lost.

All the energy now drained from his body, Tom fell to his knees and balled up for protection. His last hope was that Ella would find him so pitiful she'd simply leave him alone. But Ella never came. Tom lay curled up in the soil, gasping for breath, dripping muddy sweat and tears. Gradually, his heart rate slowed and he regained his composure. He hadn't been followed. He'd made it. Now all that remained was to find a way out of the cornfield.

Tom tried to get a sense of direction by remembering where the moon had been in the sky as he walked to Madam Aaron's house. The left, he thought. Almost certainly the left. But even with that knowledge, he didn't like the idea of keeping the moon to his right and heading right back into the jaws of Ella.

Then Tom had an idea. It scared him a little—quite a bit, actually—but it was also such an exciting possibility that he couldn't resist giving it a shot: He would try looking out for the way home.

As he'd done before on the old woman's sofa, Tom sat down where he was and hugged his arms around his knees. He fixed his eyes on the moon and allowed his gaze to seep through it... beyond it... beyond the three-dimensional paper universe—out, out into four-dimensional space. And suddenly, the light of the moon was everywhere at once, in all its honeyed technicolor splendor.

Tom felt himself drift up toward that gorgeous moon. Just a bit at first, and then fully out of his body (at least, out of the dull, ashen form he'd always considered his body). Now he was fire—all radiant flame, fierce and beautiful. And he wasn't the only one. A ways off in the corn stood another form like himself, glowing as brightly as the distant cottage.

"Who's there?" Tom asked. His voice escaped more as light than sound, but he was sure the blazing figure understood him. He focused his discorporeal eyes on the thing's face. It was young, feminine, and all too familiar: Madam Aaron.

As soon as this recognition had come, the figured sped toward Tom with all the fury of spreading wildfire. But it wasn't after his true glowing self, only the discarded husk on the ground below. It sank into that lifeless body, wriggled into every extremity, and held on tight.

#

The boy's body shivered violently as he awoke. Nausea always accompanied this part of the transition, this disorienting renewal of self. The night was dark again. Dark, but not drab. The color was always there, whether one chose to see it or not.

Brimming with eons of confidence, he rose to his feet and whistled for Ella. His faithful companion made her way through the corn, panting gleefully as she approached. Together, they made their way out of the cornfield—taking care to avoid the section that had caught fire—and started down the street to Tom's house.

"Won't mother be relieved to see us alive and well after this evening's tragedy?" the boy asked Ella with a wink. "Poor old Madam Aaron."