Revisiting the Mathemagician

Or: Milo's Lesser-Known Second Trip Through the Phantom Tollbooth
(with apologies to Norton Juster)

By D. Aviva Rothschild


Everyone has wondered whether Milo's experiences in the Lands Beyond "took"; that is, whether he didn't become bored with his games and puzzles and whatnot a week after the Phantom Tollbooth vanished and he still had to rely on his own devices to amuse himself. The answer is that they took—all too well, they took. From a dull lump he became, literally overnight, a creative fanatic. He hungered for amusement. He used up his best puzzles and books in a month. He constantly built elaborate structures of Lincoln Logs or Lego, then tore them down to start anew. He wrote stories and painted pictures to illustrate them. As he grew older, he even painted his little electric automobile in psychedelic colors, emulating John Lennon (his god of creativity) and his Rolls-Royce.

His parents, while cautiously pleased that their son was finally in motion, grew concerned that the boy seemed a little unrealistic about his future. "I'm going to be an artist!" Milo would proclaim, scattering paint around a canvas. Or, fingers flying furiously on the typewriter: "I'm going to be a writer!" In school, his grades, briefly stellar in his first year after the Tollbooth, gradually dropped to their normal dull levels, with the exception of those for his art classes.

And, indeed, for a while it seemed as if Milo would make good on his predictions. He won prizes in high school for both paintings and stories, was editor of the school's literary journal, and even won 5th place in a national writing contest for kids, plus a $10 check. Although his grades weren't good enough to get him any scholarships, he impressed a local millionaire and art patron enough for the woman to sponsor him through school, with the conditions that he keep his grades reasonable and produce art regularly. These he did, and he enjoyed four years of peer acclaim and professorial respect.

But love at the college level does not always translate into love at the public level, and after graduation Milo spent a miserable nine months in a near-jobless state, barely able to afford painting materials, let alone the rent on his tiny studio apartment. He subsisted on his parents' largesse--doled out grudgingly--and the occasional sale on the sidewalk of his original works. It broke his heart to see a piece he'd worked on for a week going for $25, but what else could he do?

Finally, Milo got a job as house artist at a midsized publishing company. At first delighted to be working creatively, he soon discovered just how uncreative the work was. Between the demands of the head of marketing, whose ideas about how books should look were both unimaginative and law, and the crushing amount of work he had to do, Milo found himself doing the minimum amount on each cover. Many times he turned down reasonable requests from authors who had good ideas for covers, simply because they involved too much effort.

Somehow he stuck it out for five years, mostly because he was terrified of being unemployed again. But he paid a price. Too burned out at night to work on his own paintings, he stopped doing them. A novel he'd been noodling around with never got past chapter 10. And so, through an irony of circumstance, Milo went full circle and became the listless person he'd been as a kid.

Except that the death of his parents in a washing-machine accident woke him up at age 31. As the only child, he inherited every bit of their modest estate, which was enough for him to quit his job and take time off to think about the future. What he wanted, he realized, was to get out of the arts altogether, go back to school and take a degree in something useful. He fixed on computer science, a burgeoning industry that desperately needed people. Considering how many games needed fully painted art these days, his art degree might actually help open some doors for him.

So back to school he went. Because his math was all but nonexistent, he started at the bottom with a single class in algebra. When he did well in that, he took its successor, trigonometry. And hit a snag. The basics of trig seemed easy enough—angles, sine waves, stuff like that—but when he moved into the more complicated processes, his mind shut down and refused to comprehend the material. He read the book backwards and forward, studied with his classmates, and even hired a tutor once, but the material just wasn't getting through. He sank into depression, terrified that he wouldn't even make it to his first class that would actually deal with computers.

One day, while he lay on his bed with a wet cloth on his forehead, he heard a slight noise. He didn't think anything of it, but when he sat up later to go into the kitchen and make dinner, he beheld an unexpected but familiar sight: Sitting on the floor was a large, brown-paper-wrapped package with a fancy envelope on top.

Knowing what it was immediately, Milo leaped out of bed, headache forgotten, and seized the letter. It read:


The envelope contained two small copper coins. Milo put them in his pocket and proceeded to tear open the package. Just as he remembered it, there sat the mysterious little purple tollbooth, with its three signs: SLOW DOWN APPROACHING TOLLBOOTH, PLEASE HAVE YOUR FARE READY, and HAVE YOUR DESTINATION IN MIND. He placed them in order like Burma Shave signs before the tollbooth.

"Oh, boy!" he said. "I'll finally get to see Tock and the Humbug and King Azaz and Chroma and Rhyme and Reason again after all these years. Will they be surprised at how big I've gotten! I hope they haven't died or anything." But of course, he knew where he wanted to go first.

There was a slight problem. Two, actually. The first was that he no longer owned the little electric car. He'd sold it for cash during his lean months. The second was that he was way taller than the booth now—he couldn't have fit into the car anyway. He solved both problems by getting on his hands and knees and crawling past the signs toward the little tollbooth. He fished out a coin and managed to flip it into the hopper. Then, thinking "Mathemagician," he crawled past the booth.


Suddenly he found himself crawling along a rocky dirt road. His head butted against a signpost, and he stood up. From the mountainous terrain, he knew he'd arrived at Digitopolis. Yet as he looked at the signpost, his elation dimmed. In the distance, in a straight line from the signpost, shimmered the Mathemagician's castle. The signpost had a single sign pointing in that direction. But the road forked, and both tines seemed to run in precisely straight lines—yet neither traveled to the castle. "Well, that's no problem—I just won't use the road," Milo said. He tried to circle around the sign—and found that he couldn't, as if a force field had been erected. In fact, while he could have gone back the way he arrived, the only ways he could go forward were the two wrong-pointing roads. which themselves had little signs: U and V.

Unhappily he started down U, the road that seemed closer to Digitopolis. He traveled on for quite a way, occasionally trying (and failing) to detour off it. Then it abruptly ended. But there was a console at the end of the road, with two buttons. One was marked "V," and the other merely had a big black dot on it: · . Of course, he pressed the dot button first, but nothing happened. Then, suspecting what would happen next, he pressed the V button.

Poof! When the smoke cleared, Milo found, to his great dismay, that he now stood at the start of the V road, his entire journey down U completely negated. And V looked even longer than U, and was even less likely to take him to Digitopolis. How was he to get there? He heaved a deep sigh and sat down on a rock to think.

What was the dot? Why hadn't it worked? This whole situation was naggingly familiar, somehow, not because he'd done this the last time he'd come, but because he remembered something like it from his class....

Milo snapped his fingers and stood up, proceeded down the V road. It was a long trek, but at last he came to the end of the road, where stood another console, with a dot button and a U button. Keeping his hand well away from the U button, he pressed the dot.

Poof! Milo reappeared at the signpost—but now both roads had vanished, and the right road--the right vector—to Digitopolis had finally appeared. Grumbling, "Well, this would discourage casual visits," for he was already pretty tired, he trudged down the road towards the castle.

The Mathemagician awaited him at the end of the road. He hadn't changed a bit: robe covered in mathematical equations, pointy cap, long flowing hair and beard (but no moustache), giant pencil-staff in hand. "MILO!" he roared, rushing to embrace his old friend—and proving to be quite a bit shorter than him now, which was kind of embarrassing. "BY THE FOUR MILLION, FIVE HUNDED AND EIGHTY THOUSAND, FIVE HUNDRED AND SEVENTY-THREE THREADS IN MY ROBE, I'M SO GLAD YOU CAME BACK!"

"So am I," replied Milo, after they finished hugging, "but couldn't you have made the road here a little less, well, long?"

The Mathemagician shrugged. "It keeps the traveling salesmen away. So!" He tapped Milo on the shoulder with his staff. "Having trouble with trigonometry, eh? You seemed to have figured out the vector-roads well enough."

Milo nodded. "It does seem easier when there's a practical application that I can use with it. The trouble is, the things I'm working on now don't seem very practical."

"NOT PRACTICAL!" The Mathemagician's beard quivered in indignation. "Come with me, my boy, and I'll show you ‘not practical.’ Not practical indeed! Hmph!"

He led Milo into the castle and to a large room that looked like a rabbitry, with hutches lining the walls and straw on the floor. "By the way," he said, jabbing Milo in the ribs playfully, "I want to warn you about breaking the law of sines or the law of cosines while you're here. You can get into a great deal of trouble if you do."

Milo wondered how one could possibly break those laws when they were more formulas than laws, but he forgot all about that when he saw what lived in the large room. Or, rather, what he thought he saw, out of the corner of his eye. Little straight-line figures darted about the room, running from shadow to shadow. When Milo tried to look straight at one of the figures, it became invisible to him.

"This is where we breed imaginary numbers," the Mathemagician said proudly.

"You breed them?" said Milo. "I thought you mined numbers."

"Only real numbers, my boy—only real numbers have the solidity to be mined. Imaginary numbers, on the other hand, are born of human ingenuity and perversity, for where else in nature could you get the square root of negative 1?" The Mathemagician stretched out his arm, and Milo had the impression that several i's ran up to sit on the magician's shoulder.

A door creaked, and the Mathemagician said in delight, "Good! You can see how we train complex numbers." Through the door came a miner with a wheelbarrow full of sparkling mined real numbers. Milo saw a 4, a pi/2, and even a -Ö 17. The Mathemagician whistled, and the real numbers jumped out of the wheelbarrow and lined up like little soldiers. One or two quivered.

"You train them?" said Milo.

"Of course, of course, you don't think real numbers would naturally submit to being saddled with an imaginary number, do you?"

A clap of the Mathemagician's hands seemed to produce an equal scurry and lineup among the imaginary numbers, although Milo wasn't too sure about that. The two lines faced one another, and the Mathemagician stamped his staff on the ground. The real numbers rushed towards the imaginary ones and clambered atop them like so many riders on horses. The effect was astonishing. Many of the imaginary numbers started to buck and shake, trying to throw off their real components, and some succeeded. A few ran off or leaped into the hutches. Only a few submitted to the indignity with equanimity.

"It takes some time to get the imaginary numbers used to their new partners," said the Mathemagician. He slapped his stomach, and the three or four tame complex numbers came meekly to his side, the real numbers leading the imaginary ones on either minus-sign or plus-sign leashes. "But when they're tamed, we can train them to do all sorts of tricks. They can plot themselves on axes, turn themselves into trigonometric forms, and even divide themselves into nth roots (with the help of my assistant DeMoivre). Now!" The Mathemagician clapped Milo on the shoulder. "Would you like to see our scalar garden? It's really very charming to see the scalars sunning themselves on rocks."

"Thanks, maybe later," Milo replied. "But what I still don't understand is what all of these things are good for. They certainly seem like fun, but I really need some practical uses for them—"

He stopped and shrank back as the Mathemagician swelled up in indignation. The magician lowered his staff's eraser towards Milo's head....


…. and rapped it smartly. "I ought to rub some sense into that imaginationless noggin of yours, boy! Or maybe I should just erase it altogether and give you a more interesting one. A parabola would work. At least you'd always be smiling! Except you'd constantly be hitting your curves on the ceiling, since they're infinite." He moved his staff away, much to Milo's relief, and began drawing on the floor. "Not practical indeed! Hmph! Well, I have a thing or two to show you!" He stepped away from his drawing and waved his hands. "Don't just stand there, get on!"

Milo peered uncertainly at the Mathemagician's drawing, which looked like one of those concrete skateboard parks: a wide U deep in the ground. "What's this?"

"A cycloid, of course! The fastest way to get down there—the curve of quickest descent, if you will. Follow me!" The wizard sat down on the edge of the cycloid and shoved off. He slid down and vanished in a blink at the bottom. Milo didn't at all care for this form of travel, but he had no choice, and awkwardly he sat down and


slid down the almost frictionless surface, directly into a hole at the bottom, like a pinball. Immediately he changed direction and found himself shooting through a curving tunnel, going so fast that he slid along the side of the tunnel rather than the bottom. For an eternity of seconds he whizzed around in what appeared to be a circle, but turned out to be a spiral when he shot out the end and got a good glimpse of the tunnel from mid-air.

But his journey wasn't done yet. Somehow, his momentum had vanished, and he now tumbled lazily in the center of what seemed to be a huge transparent sphere. Every so often he floated through a blackish patch of air, and when he did, a set of coordinates would briefly light up: (3, -45º), (5.4, pi/6), and the like. Also floating around like strange fish were various ellipses, hyperbolas, parabolas, conics, and even long, complex equations. The sphere, he soon discovered, was revolving on a central X-Y axis—he couldn't quite figure out how the sphere revolved around both the X- and Y-axes simultaneously, but it was—and as he floated near the origin, he saw the Mathemagician standing on 0,0, waving.


"Yes, actually," Milo called back.


Milo made some swimming motions, then flapped his arms, but could not get any closer. "How?"


24y=X^2 was the one drifting by, so Milo grabbed it and said politely, "Would you take me to your vertex?"

Instantly, the equation swooped down to 0,0. Milo jumped off, and the equation floated serenely off.

"Come," said the Mathemagician, walking along the X-axis as if it were a tightrope, using his pencil-wand as a balance. Milo followed carefully. Once he had to duck as a conic, covered in two layers of fur, swooped over his head.

"A double-napped right circular cone," the Mathemagician said without looking back. "And there's a cardioid. We grow them here for our heart patients. Oh, watch out for that hyperbola!" Milo dodged just in time. "If you get caught in its asymptotes, you'll get sucked into the void."

After walking for what seemed like forever, the two abruptly emerged into a garden in which all the flowers' petals were actually graphed onto their own tiny axes. "This is the rose garden," said the wizard. "I never promised it to you—" he snickered "—but notice how the flowers are rooted in equations!" He pulled a pretty four-petaled rose up to display how its roots were tangled around (7 sin 2t cos t, 7 sin 2t sin t).

"Very nice," said Milo, who was wondering what such flowers were fertilized with.

They went through a door that bore the word "Jail" and into a room where a very peculiar (relative to everything else) sight greeted Milo. He saw a proud-looking coordinate system in a military uniform endlessly making ovals around two points in the air. Next to this was a young-looking coordinate system standing dejectedly in a curve inside a parabola that had been hammered into the wall.

"That's Major and Minor Axis," the Mathemagician explained. "Both of them are in focus traps for the time being, until they shape up and agree to have their equations assume standard form, rather than any old list of numbers and letters."

"What's a focus trap?" asked Milo.

"Each one depends on the kind of conic section. A parabola has one point of focus, from which each pair of points on the parabola is equidistant, and Minor Axis is caught in there—he won't be able to move until we move the parabola. The ellipse, on the other hand, has two foci, and the prisoner must always remain at a distance no greater and no smaller than the sum of the distances of the foci from a given point. Normally the size of the ellipse determines where the foci appear, but we just reversed the phenomena—put up a couple of foci, and Major Axis can't go anywhere except that narrow path of points. A good trap if you really want to annoy your prisoner." The wizard chuckled. Major Axis looked annoyed.

The Mathemagician continued: "I'd introduce you to the Directrix of the jail, but we haven't formally assigned her yet. She—"

He was interrupted by a series of soft gasps and thuds from a door marked "Eccentricity."

"What's in there?" asked Milo.

The wizard looked at the floor, kicked some invisible rock. "That's our eccentricity room—where we keep the degenerate conics."

"Really?" Milo couldn't resist—he opened the door and peered in—then quickly shut it, blushing from head to toe. "How could they?"

"They can't be satisfied with a normal plane pass-through. No, these wonderful creatures absolutely insist on having a plane pass through their vertex." The Mathemagician shook his head sorrowfully. "What can you do? You can't outlaw them—no point in passing a law that won't get obeyed. All we can do is hide them and hope an equation isn't tempted to drop a y here or an x there and tempt yet another a conic to go degenerate."


Milo didn't know what to say. He chose his words careful: "Well, they certainly are useful here. But, uh, I'm not sure, uh, that, uh, that I'll be able to, uh, find a parabola to ride on back home."

The Mathemagician swelled up in annoyance, then shrank back to a teeth-gritted, determined man. "Useful in your world, eh? Well, with ellipses you can help destroy kidney stones by placing a lithotripter at one of an ellipse's foci near a kidney. Hyperbolas and parabolas have impressive reflective properties and are used in cameras, glasses, and telescopes." He drew a diagram in the air that showed how hyperbolic and parabolic lenses reflected light. "Vectors, of course, are useful for everything from airplane flights to determining the effects of gravity. Polar coordinates can help you navigate your world-sphere (the Lands Beyond are, of course, flat). Do you want to know whether your star batter is going to hit a home run if he hits the ball at a certain angle with a certain amount of power? Use parametric equations."

Milo's face lit up. "All that?"

Pleased, the wizard said, "Oh, yes! And, of course, all these things build on themselves and on math to come, so that much, much more can be determined. My goodness, Milo, keep going, and soon you'll be building bridges and determining the position of stars in space!"

"Wow! I never realized how useful this stuff was. I can't wait to go home and try it! After I visit the rest of my friends here, of course."

The Mathemagician smiled sadly. "I'm afraid that your time is just about up here."

Milo's face fell.

"I'm sorry, but once you actually learn something here, it's back home to the real world with you. Those are the rules. However," the Mathemagician added, "if you have any problems with any of those other disciplines—and frankly, how anyone can use letters without going insane continues to amaze me—perhaps the Tollbooth will come back to you again."

Although Milo was annoyed that the reward for having things cleared up in his head was no reward at all, he accepted his fate—after all, he hadn't expected to ever return to the Lands Beyond, and being here for these few hours was much better than nothing. Still, he heaved a deep sigh before hugging the old wizard until the man's bones creaked and he protested happily, "Don't break my spine! Otherwise I'll have to transplant a y-axis in to replace it. You can stay a bit longer if you'd like dinner. Remember subtraction stew? We have a new recipe that adds a pinch of exponentiality to it. Now, if you eat three bowls, you'll be nine times hungrier than you started, rather than just three."

Milo was tempted—for the company, not for the food—but now that he'd accepted leaving, he wanted to go before his heart broke. So he politely declined and headed back down the vector-road. Soon he'd passed the original signs—and once he did, the vector-roads returned to normal—and not long after that, he paid his money at the tollbooth and passed through to his home.

His trig textbook lay on his desk. Riffling through it, he recognized many of the things that the Mathemagician had shown him. Now they didn't seem so mysterious and abstract. He smiled back at the Tollbooth, which seemed to smile back as it faded from view.

Copyright 2000, Aviva Rothschild

About the Author

See the "About the Editor" page. This was one of three stories I wrote for a trigonometry class in 1995.

Tell Aviva Rothschild what you thought of her story!

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