He indicated the bench where he had been sitting. “Please. Join me.”
I went up and sat with him on the bench. A breeze ruffled our hair and stirred the vines on the lattices. “This is beautiful,” I said.
He stared out at the gently rolling lawns. “Mara and I used to sit here together.”
“I’m sorry for your loss.” I thought I would have liked his wife. Although I didn’t agree with her choices regarding the High Mesh, she wasn’t the one who’d broken the law. As with Bessel and his love of knowledge, her interest in the Mesh was also for the scientific treasures they might uncover. And she had mentored Bessel, the only person who supported and encouraged him rather than using his genius.
Lukas spoke with pain. “The cosmos, fate, a deity in heaven, the pantheon of our ancestors, whatever you believe in—they have no sense of justice, that they should take someone so worthy of life.”
“I don’t know if this helps or not—but I found something you might like to have.” I reached into my pocket and took out the medallion I’d picked up in the cave under the starships. It showed an image of Mara and Lukas with their arms around each other, smiling in the sun. “The people working on the project left a box of keepsakes of their loved ones, each person picking something they valued over all others, to honor their hopes for the project. This is what your wife chose.”
Tears formed in his eyes. “I wondered where this had gone. I was so upset when I couldn’t find it.” He took it from my hand. “I’ve always loved this picture. That was such a wonderful day we spent together.” He looked up at me. “It means a lot to know she chose this medallion.”
I spoke gently. “She didn’t die in vain. What happened will help future scientists solve the secrets of our ancestors. The work she began may someday change human life.”
His voice caught. “Thank you, Major.”
I couldn’t tell him the rest, about the avarice and thirst for power that had led the House of Vibarr to fund the High Mesh. Their secret cabal had cost more, in human terms, than I knew how to quantify. However, I also knew one more fact: The Majdas would make sure the High Mesh paid dearly for his wife’s death.
The sunrise soothed the ragged edges of my thoughts. I walked across the rocky desert far from any settlement, alone under the brilliant sky, with its red and gold bands on the horizon and the arch of purple overhead. The last stars of the fading night were giving one final glimmer before dawn washed away their gleam. Stone-vines curled in silver lace across the land. I had no idea why Ruzik and Angel had sent word through the whisper mill for me to meet them here. They should be at home relaxing, recovering from their tykado exams yesterday while they waited to hear whether or not they had passed.
I saw no one. Wind stirred my hair, which I’d left loose around my shoulders. I stopped and listened to the vast silence of the land. Except the Vanished Sea never truly fell silent. Somewhere a flying lizard called out, its voice carrying across the desert like the spirit song of ancients who had lived and died here so long ago.
I realized I was no longer alone. In the distance, three animals were approaching, silhouetted against the sunrise. Large animals. They glinted in the predawn light, with iridescent sparks of green, gold, and blue glinting on their bodies.
Ruziks.
I went still, like a statue hiding in plain sight. The beasts galloped across the land, powerful and deliberate. I knew so little about these rulers of the desert. They stayed away from humans, and in return we left them alone. As they came nearer, they slowed to a walk. I saw them more clearly now, the giant heads, their large eyes, the huge tails. One walked on its massive back legs, with its front legs in the air, and the other two walked on all fours. Something didn’t look right—those two each had an extra ridge on their backs—no, not a ridge. Ho! Those were riders.
The animals continued their approach, looming larger and larger. I held still, waiting until they stopped in front of me. Ruzik was sitting on the one in the middle and Angel rode the animal to his right. The third came down on all fours, standing like a beast from a mythological tale.
“Eh, Bhaaj,” Ruzik said.
“Got a big lizard,” I said.
The animal without a rider came forward, watching me out of one giant eye, its head turned sideways. It lowered its snout and shoved my shoulder. I swayed back, then regained my balance. Its breath curled around me, smelling of musk and lemon.
“Eh,” I told it. I laid my hand against its head. The ruzik whistled, a sound much lower than the cries of the smaller lizards, almost a rumble. I stood quietly, letting it smell me. I should have been afraid, yet I felt only curiosity. No, curiosity wasn’t the right word. I had trouble defining the emotion, as if it belonged to someone else. The ruzik was deciding if it wanted to accept me.
The animal pushed my shoulder again, harder this time. I stepped back and waited. It turned its head to look at me with its other eye. With another whistle, it straightened up to its normal height, then higher still, onto its back legs, towering, its giant claws glittering in the dawn. If it decided to attack, I wouldn’t have a chance. It could kill me with one swipe of those claws. No matter. Nothing could have frightened me away from this glorious moment. I’d never known an animal with such majesty.
The ruzik came down again, less than a meter away. It slowly lowered its bulk to the ground, folding its front and back legs under its body. Then it waited.
“You honor me.” I stepped forward and set my palm on its back. A bony ridge ran down its neck and another crossed its lower back like a natural saddle. I understood the invitation, the rarity of what it offered. For a moment, I stood with my hand resting on its scales. Then I climbed onto the animal and settled into the indentation in its lower back. The ruzik stood, and the ground seemed to drop away beneath us. My breath caught. I looked at Ruzik, the human one, and he grinned. Beyond him, Angel lifted her chin, her gaze radiant.
“Good way to celebrate new belts,” I said, giving them the good news. They’d passed their exams and were now first-degree black belts. And celebrate was just the right word. Their achievement deserved three syllables.
They nodded, accepting the congratulations Undercity-style, with no words or excessive emotion, but I saw the gleam of satisfaction and yah, relief, in their gazes. Ruzik laid his hand against the shoulder ridge of his mount, and the animal stepped around until it faced the sunrise. I put my hand on the ridge of the ruzik I rode, but nothing happened. So I tried pressing. It responded then, turning to join Ruzik’s mount. Angel bought hers around and together we walked toward the sunrise. The wind blew across my face and swirled my hair around my shoulders. It took me a moment to realize I’d been holding my breath. I inhaled again.
“Press inward,” Ruzik said. “With knees.”
I tried his suggestion and the ruzik walked faster. Another press, and it speeded up again. I took a deep breath, filling my lungs with the clear desert air—and pressed hard.
We took off then, thundering across the desert. I tilted my face up to the red sky, glorying in the ride, one that no one had taken in thousands of years. Freedom coursed through us, the freedom to run beneath the sky of a world that had become more ours than Earth would ever be.
We rode across the desert in joy.
APPENDIX
Mathematical Methods and the
“What if?” of Science Fiction
by Catherine Asaro
In The Vanished Seas, the idea of Kyle space depends on a mathematical concept called Hilbert space. I’ll describe here what that means and how I played with those ideas for the story.
To start, we’ll define a “vector space.” Consider the three-dimensional world we live in. If you want to tell someone your exact location, you can do it by using three numbers. Imagine you are sitting in a room. You can describe your position by saying how far you are from some point in the room. We will choose the corner where two walls and the ceiling meet (assuming they are all perpendicular). We call that corner
the origin. You say the top of your head is two feet from the wall in front of you, three feet from the wall at your right, and seven feet from the ceiling. Those three distances define your coordinates relative to the corner. Specifically, your coordinates are (2, 3, 7). You’ve described your location in “coordinate vector space,” which is just the three-dimensional universe where we live.
It’s possible to specify the coordinates with arrows. One arrow starts at the origin (the corner described above) and points along the seam where the wall in front of you meets the ceiling. It extends for two feet and ends at the point we’ll call X. The next arrow starts at X and comes straight out from the wall along the ceiling, parallel to the wall on your right, for three feet. The point where it ends is Y. The third arrow goes straight down from the point Y, perpendicular to the ceiling, for seven feet. It should touch your head. That is point Z. So in this case, X=2, Y=3, and Z=7. The arrows are called vectors.
We can define the location of any object by figuring out the numbers (X, Y, Z) that indicate the location of the object relative to some origin. Once we know X, Y, and Z, we know the exact location. To make the process more general, define three arrows (x, y, z), each with a length of one. Then any point P can be described by P=Xx + Yy + Zz. The position of your head would be P = 2x + 3y + 7z.
The arrows (x, y, z) are mutually perpendicular vectors with unit length. They are called orthogonal coordinate vectors, and we say x, y, and z span coordinate space. The word “span” means that any location can be described using just these three unit vectors, multiplying each by appropriate numbers and adding them together.
Mathematically, other “vector spaces” exist. Some have an infinite number of dimensions! I’ll describe here the type I play with in this story. As it turns out, the solutions to certain differential equations act in a manner analogous to the unit vectors described above. If we draw the solutions to such an equation, we get a family of curves, all of which have similar properties. To see what this means, consider the vibration of a string on a musical instrument. It is fixed at its endpoints and goes up and down in a wave with periodic peaks and valleys. These are called sinusoidal waves and give the solutions of the equation that describes the motion of such a string. The more peaks and valleys in the wave, the higher the pitch; that is, it has a higher frequency. Only certain frequencies work, however—those where an exact number of peaks and valleys fit along the length of the string. We say the frequencies are quantized. An infinite number of possible waves still exist, going to higher frequencies as we cram in more peaks and valleys. The functions that describe these waves are called eigenfunctions.
Many differential equations follow this behavior. Each has a family of eigenfunctions as solutions. We call the eigenfunctions “vectors” because they behave in a manner analogous to the x, y, and z unit vectors above. Just as the location of any point P in three-dimensional space can be described by the sum P=Xx + Yy + Zz, so any function P(x) can be described as a sum constructed from a family of eigenfunctions. We add them up with each vector weighted according to its contribution to the total result. However, we have an infinite set of eigenfunctions! So they span a space with an infinite number of dimensions. Universes defined by such vectors are called Hilbert spaces.
Think of it as preparing a dish for a meal. We add ingredients together, varying the amount depending on the recipe. If you have an infinite number of ingredients, you can make any dish in the universe. For a particular recipe, you use only a few ingredients; the contribution of the rest is small (say a pinch of salt) or zero. So it is with eigenfunctions. The number needed to create the wavefunction is infinite, but the main contributions usually come from a small number of eigenfunctions, with the others adding corrections or having no effect.
A well-known family of eigenfunctions are the curves sin(nπx/L) and cos(nπx/L), which can describe functions that repeat like a wave. The value of 2L is the length of one cycle or period of the wave, and π = 3.14159 . . . is the number pi. The number n takes the values n = 1, 2, 3 . . . on to infinity. These are seen in an area of physics called Fourier analysis. We write a function P(x) as a sum of those “vectors.” For example, a triangular wave looks like a triangle repeated over and over. It is used in many disciplines, such as electronics, and can be written as
The sum goes on forever, but the contributions of the waves for large frequencies (large n) are tiny. The constants (8/π2), – (8/9π2), (8/25π2) and so on specify the contribution of each sine wave to P(x).
For the spatial vectors, the values of X, Y, and Z depend on the origin. In the example we started out with, if we define the origin as a different corner in the room, then X, Y, and Z would be different from (2, 3, 7). The choice of origin should be sensible. If I wanted to specify my position in a room in New York, it wouldn’t make sense for me to choose the origin at a bar in Toledo. Likewise, the description of P(x) using eigenfunctions depends on which family of functions we use. For example, the radial position of an electron in the hydrogen atom uses Laguerre polynomials, whereas Bessel functions are better suited to describe the radial position of a free particle in spherical or cylindrical coordinates.
The three unit vectors x, y, and z are mutually perpendicular; that is, they intersect at right angles. We say they are orthogonal. Eigenfunctions are also orthogonal, but here the meaning becomes more complicated. Roughly speaking, two functions are orthogonal if they have no overlap. This is a simplification because it doesn’t actually mean that no part of the functions overlap; rather, it means that when certain operations are performed on the functions (integration over their inner product), the result is zero. If we have two copies of the same eigenfunction, they have one hundred percent overlap. If we scale that overlap so it equals one, the functions are normalized. It’s the same as taking the length of the x, y, and z vectors to equal one. We call such a family of functions an orthonormal set of eigenfunctions.
Quantum Dreams
Quantum physics predicts that any object can be described as a wavefunction. For example, electrons sometimes act like particles and sometimes like a wave. Both descriptions are valid! If we shoot tiny particles through a double slit, they diffract as if they were a wave. Any “solid” object can also be defined in that manner, including humans. You might say, “If we’re all waves, why can’t we see the wave properties of our bodies? People don’t diffract when they go through a double doorway.” The answer is that the wavelength for a macroscopic object like a human is so tiny that we can’t (yet) see it even with our most advanced technology. Quantum theory may seem nonintuitive, but the model gives results that match what scientists observe in the real word. In fact, it’s one of the most successful scientific models ever developed. It allows us to describe what’s happening at the atomic level, yet on larger scales it agrees with classical mechanics, that is, with what we see in our world.
For particles such as electrons, we can use wavefunctions to investigate their energy, position, momentum, and other properties. Wavefunctions that describe the spatial behavior of a system depend on position variables. Some variables are part of what we call a conjugate pair. A result of quantum theory called the Heisenberg uncertainty principle specifies that we can never know the value of both conjugate variables exactly; the more certain we are of the value of one variable, the more uncertain the value of the other. For example, the better we know the position of an electron, the more uncertain its momentum. Time and energy are also conjugates. We can give a highly accurate number for the lowest energy of the electron in a hydrogen atom, but we can’t say when that measurement was made; the number is always the same regardless of when we determine its value.
We find the eigenfunctions that describe quantum behavior by solving the Schrödinger equation. Its solution produces families of orthonormal eigenfunctions. To model the system, we take sums of those eigenfunctions just like we needed sums of sine waves to describe a triangular wave. In theory, the spatial part of the wavefunction that desc
ribes the electron in a hydrogen atom exists everywhere in space. It is centered around the nucleus, but its tails extend to infinity, becoming negligible at large distances. We can use the wavefunction to find average values for properties of the electron, such as its position and momentum.
Quantum theory can describe any collection of particles. The more particles are involved, the more complicated the problem. For example, a student can learn to calculate the wavefunction for the hydrogen electron with only pencil and paper, needing no computer. For a system even as simple as three particles, it is no longer possible to solve it by hand. A collection of just twenty particles requires intensive computer work. In theory, the behavior of any system can be determined by solving the Schrödinger equation as long as we include a term in the equation for every particle in the system. We don’t have the computer resources yet to do any system of substantial size, but our ability to solve such equations has grown over the years. Someday we should be able to solve even human-sized systems.
In theory we could calculate the wavefunction that describes a human brain during a thought. It would most likely be localized around the brain, with tails that trail off to infinity, rapidly becoming tiny. What does that mean? We can’t really isolate thoughts; they blend into one another and we can also think about more than one thing at once. One way to specify a thought might be according to how long it takes to complete an idea. It could be a single idea, going to the store perhaps, or a mingled thought about fixing the car and feeding the cat. More complicated thoughts could be broken into individual components. We could describe what goes in the brain at a specific moment as a “slice” of a thought. Although the structure of a brain is pretty much fixed, it isn’t identical from moment to moment. As we think, neurons fire and other chemical reactions take place. At each instant, the collection of all those particles can be described by a quantum wavefunction. A thought could be described as the sum over the wavefunctions for all the slices involved with a particular idea. Voila! We have a mathematical description of a thought.
The Vanished Seas (Major Bhaajan series Book 3) Page 35