Visions of the Future (8 page)

“I calculate a fifty-three percent probability that you will survive ejection.”

Well, that was better than zero. Even if he didn’t make it, at least the Glint’s mind would survive. The computer was better shielded even than the reactor.

Kelric touched the console. “Jessa and her team will have you fixed up in no time.”

“I don’t think that will be possible,” The Glint sounded subdued.

“Why not?”

“When we reinverted, I created a cybershell for your mind. It damaged my systems.”

“A what shell?”

“Cybershell. I ran your brain as a subprocess of my own. Your mind wouldn’t have survived reinversion otherwise,”

Kelric whistled. “That’s impossible.”

“Not completely. However, it did leave me unprotected during reinversion. It corrupted my systems. By the time we crash, my functions and memory will be degraded past recovery.”

No, Kelric thought. “You killed yourself so I could live.”

“I’m only a computer.”

“A computer, yes.” He spoke quietly. “But ‘only’? I would never use that word for you.”

“Captain, thank you.” Then it said, “We’re disintegrating.”

“I won’t forget what you did for me,” Kelric said.

“Take care.” With just the barest pause, the Glint added, “Let yourself heal, Kelric.”

The plane ejected him.

Kelric went out the top, shooting upwards as the Glint fell away from him in pieces. Windblast buffeted him so roughly that he almost blacked out.

He began to fall. Tucking his chin to his chest, he held his legs together and crossed his arms while he tumbled through the air. Silence surrounded him and clouds covered the landscape. He tried to look at the altimeter on his arm, but the numbers blurred. It took his groggy brain a moment to comprehend that he had used up the air in his emergency tank. He clawed at his helmet, his fingers scraping across the face plate. Darkness closed around him, warm, inviting…

A blast of cold air slapped Kelric awake. His helmet had opened and his suit timer was going off, triggering the release of a parachute. It jerked him so hard, it felt as if his arms would rip off his shoulders.

Mercifully, the buffeting soon eased. His survival kit deployed, its life raft dangling from his suit like seaweed waving in an ocean of air. Clouds closed around him and he fell through a wet mist that ate away at his sense of up and down, right and left.

Gradually Kelric realized the world wasn’t silent. A rumble throbbed below him. As he fell through the fog, the growl swelled into the roar of waves hitting land. Even when he sealed his helmet, he heard the thundering voice.

He had no warning before he hit water. He plunged into it, his limbs tangling in the parachute’s suspension lines. As he struggled to free himself, he plowed into sand. With a huge kick he shot out of the water, breathing the few moments of borrowed air in his suit.

Kelric pulled free of the parachute and grabbed the life raft. He had run out of air, but when he opened his face plate, a wave smashed into him. Another wave came, another, and another. The breakers tore away the life raft and rolled him over and over, his lungs straining while he struggled not to gulp in water. If he didn’t get air soon, he would pass out.

His feet scraped the bottom for only an instant, but it was enough. He shoved against the sand and shot upwards, clearing the breakers long enough to gasp in a breath. Then he was back in the water, fighting the waves. He touched sand again, again, and again, and then he was stumbling up a sandy slope, waves crashing around him in frothy turbulence.

Kelric staggered onto the beach, wading out of the mist into watery sunlight. Ahead of him, a hill slanted up to a road—where a hovervan with flashing red lights was braking to a stop. As people jumped out of the vehicle, engines rumbled overhead. He looked up to see a flyer circling, its military insignia gleaming in the sunlight.

A woman in a blue jumpsuit ran toward him, her shoulder-length hair glinting like copper. As Kelric sunk to his knees in the sand, people surrounded him. The woman knelt in front of him, tears on her face. “You crazy man.”

Kelric barely managed to croak out an answer. “Zaub? How did your hair grow so fast?”

“Six months you’ve been gone.” Her voice shook. “Six months we’ve been thinking your hide was finished.”

“I came back to get the money you owed me.”

She pulled him into her arms. “Welcome home.”

Kelric hugged her back, unable to respond as silent tears ran down his face.

The broadcasts that aired following his return made him out to be a bigger-than-life hero. Over and over they showed the scene of his parents embracing him, the son they thought they had lost, his breathtakingly beautiful mother with tears streaming down her golden face. Space Command took advantage of the good public relations and paraded him around in his uniform, keeping quiet about that fact that they also took him off flight status and sent him to a therapist. Kelric went where they told him to go, stood where they told him to stand, and endeavored not to look like an idiot.

All the reports went on with great enthusiasm about the dramatic moment when he wept on the beach for the joy of seeing home. Kelric let them say what they wanted. He knew the truth, deep inside where suppressed grief had once crippled his heart.

Those healing tears had been for Cory.

Notes

The first time I heard about Riemann surfaces, I fell in love with the subject. It was during a course in applied math for physics majors that I took as an undergraduate. I was intimidated by the course but it also looked intriguing, so I gave it a try.

I adored that class.

To this day, applied math remains my favorite subject. Give me an equation to solve and I’m happy. This essay describes some of the ways I’ve incorporated math into my stories. I’ve started out with a few equations for those who enjoy them, but it isn’t necessary to understand those to follow the rest of the essay. I’ve also included analogies and pictures I hope will elucidate the beautiful concepts behind the mathematics.

My introduction to Riemann sheets came about in that long-ago math class when we delved into the subject of complex analysis, or the math of complex numbers, a subject seen by students of all ages, from preteens first learning about imaginary numbers to doctoral candidates studying theoretical physics. So what is a complex number? We can call it z, where

z = x + iy.

Here x and y are real numbers, that is, numbers such as 42, 3.64, 84/7 or π. However, i is a different beast altogether; it’s an imaginary number, specifically the square root of −1:

i = √−1 .

So z has a “real part” equal to x and an “imaginary part” equal to y. We can plot a complex number on what is called the z-plane. It looks the same as the x-y coordinate plane, except the x axis corresponds to the real part of the complex number and the y axis corresponds to the imaginary part. On such a graph, the coordinates (x, y) = z give the complex number.

Figure 1: The tip of the arrow gives the complex number z = x + iy, where x is the real part and y is the imaginary part.

We can also represent z by
polar coordinates.
Its position is again given by two numbers, but in this case the two numbers are r and θ, as shown in figure 1. The line drawn from the origin to z has length r and the angle it makes with the x axis is θ. Polar coordinates and x-y coordinates are related; x = r cos θ, and y = r sin θ. So

z = r (cos θ + i sin θ) = rei
θ
.

The term
ei
θ
= cos θ + i sin θ
is called the complex
exponential function
, which often shows up in physics classes, where it has bedeviled many generations of incipient young scientists. The value of r doesn’t really matter in these discussions, so to make life easier, I’ll set r = 1. The angle θ can still vary. In fact, if we let it increase from 0 to 2π, the point z will move in a circle around the x-y plane and come back to where it started.

Imagine that the line from the origin to z is the hour hand on a clock, with z at its tip. When the hand moves once around the clock face, that’s analogous to θ moving through a total angle of 2π. After twelve hours, midnight until noon, the hand is back where it started. Go around a second time and the hour goes from noon to midnight. The hours in the first go-around have the same numbers as in the second one, but refer to different times. Go around a third time, though, and we’re back to the morning hours.

The convention in math, however, is that θ = 0 when z is on the positive x-axis, which corresponds to an hour hand at 3 o’clock. When z is on the positive y-axis (θ = π/2), that corresponds to the hour hand at 12, which means θ = π/2 at midnight or noon. Also, in math the convention is that θ increases as z goes counter-clockwise around the plane. If z goes clockwise, θ
decreases
from 0 to −π when the time goes from 3 am to 3 pm (or 3 pm to 3 am). The discussion is essentially the same, however, regardless of whether or not we have a minus in front of the angles, and we can just as easily go from 3 am to 3 pm as from midnight to noon. The important part of the analogy is that going once around the clock face corresponds to z moving through a total angle of 2π. So in that sense, z is analogous to a clock.

Something amazing occurs when we take √z, the square root of z. Let’s check it out. We write

√z = z
1/2
= ei
θ/2
= cos(θ/2) + i sin(θ/2).

What happens now if the “hour hand” moves clockwise? When θ goes all the way around the clock, √z only makes it halfway around because it depends on θ/2. To see what that means, we’ll look at θ = −π, which corresponds to 9 o’clock. For that angle,

√z = cos(−π/2) + i sin(−π/2) = −i.

If we go around the clock once (say 9 am to 9 pm), the angle θ changes by −2π, which means θ = −3π (since we started at θ = −π). That’s also 9 o’clock. So if √z were well-behaved, it would have the same value at −3π as it did at −π. However, instead we get

√z = cos(−3π/2) + i sin(−3π/2) = i.

The square root has different values for −π and −3π even though −π and −3π are in exactly the same place, the same “hour.” So is √z = i or −i at 9 o’clock? It’s ambiguous. That’s why double-valued functions aren’t allowed; z must be unique at every point to be a valid function.

You might wonder what happens if we go around a third time. Is √z triple-valued? Quadruple valued? Where does it stop? As it turns out, the third go-around gives √z = −i again and a fourth gives √z = i. So √z alternates between only two values. Unfortunately, even two is too many. In our universe, the math that describes physics requires single-valued quantities. They give unambiguous results; otherwise, we wouldn’t know which number to use. But terms like √z come up all the time in the equations of physics. So we seem to be stuck.

The solution to this conundrum is an elegant idea developed by the mathematician Bernhard Riemann in the nineteenth century. Instead of one x-y plane, he suggested stacking two of them together. The top plane, or “sheet,” is where z has its first value, and the bottom is for its second value. To go from one sheet to another, we slit them from the origin out to infinity. That slit is called a
branch cut.
If we connect the sheets at their branch cuts, we can go around the top sheet once and then slip through the cut to the bottom sheet for the second time around. Then back to the top sheet. That allows √z to have one value on the top and a different one on the bottom.

Voila! The function is no longer double-valued. The ambiguity goes away as long as we know which sheet we’re on. It’s like stacking two clocks. The hour hand goes around from 3 am to 3 pm on the top clock, then slides through the branch cut and goes around the second clock from 3 pm to 3 am. Then back to the top clock.