No reporters managed to shove into the elevator with us, but when we reached the roof, many more were there, shouting. With our bodyguards clearing a path, we ran through the crowd to the racer. Even after we boarded the ship and were closing the hatch, the reporters continued to call out questions. As we strapped into our seats, the comms in the cockpit lit up like a festival tree. The racer engines rumbled and the people outside scattered. Then the ship leap off the roof and soared into the sky, headed for orbit.
Messages flooded in as we traveled to
Roca's Pride.
Ragnar and Naaj hated my idea, an odd alliance given Ragnar's antipathy toward the Houses and Naaj's arrogance toward anyone not of them. Chad Barzun pledged support. And so, of all people, did Vazar Majda.
Then the pilot said, "Pharaoh Dyhianna, I'm receiving a message through Secondary Jinn Opsister." She paused. "It comes from Barcala Tikal."
I sat up straighter. "What does he say?"
Her voice quieted. "Three words. 'Yes, I agree.' "
Relief flooded over me and I sat back in my seat, my breath coming out in a long exhalation. For the first time since I had awoken on Opalite, I felt hope for the future.
* * *
The darkness in my quarters gave a much-needed respite. The day had seemed endless, while Barcala Tikal and his Assembly councilors met with me and my people. We had debated for hours, trying to hammer out a compromise. It was going to take time, and Kelric and I were in for some rip-roaring fights— but saints almighty, it looked like it would work.
Footsteps entered the room. "Dehya?"
I turned onto my side. "My greetings, Dryni."
Eldrin sat on the bed, a silhouette in the dark. "A long day."
"Truly." I tugged playfully on his sleeve. "Come, Husband."
He gave a low laugh. "You come, Wife." Then he slid under the covers and pulled me into his arms.
I touched the bare skin of his neck. "When did the doctors operate?" No one had told me they had finished mapping the intruding web within his body and could remove the restraints.
"Today. We didn't want to distract you." He settled me in his arms. "It is true that Jaibriol the Third has responded to your speech?"
I nodded, my head on his shoulder. "He will meet with us."
"Gods," he murmured. "It's incredible."
"I hope so. But we've so far to go. It may take decades. I ran models on it."
"I didn't think you could predict that far ahead."
"I can't tell much, only that the next fifty years will be rocky." I thought back to the sessions today. "But we have the separation of powers formalized now."
"It was good what you did."
I winced. "Not everyone agrees. And I wish people would stop acting so shocked."
Amusement lightened his voice. "The Ruby Dynasty isn't known for a willingness to compromise."
"I suppose." That intransigence had started wars in ancient times. "Many Aristos don't want peace either. My models predict the Aristo line of Iquar will resist any treaties between Skolia and the Traders."
"Iquar? That's the line of the late Empress."
"Yes." ISC had been responsible for her death. Their vengeance would culminate in fifty years, though I had no idea how. Nor did I know yet how Seth and his vanished foster children came into it, but they too were in the models. I would discover why and how even if it took me the next five decades to solve the mystery.
Eldrin was playing with a tendril of my hair, twining it around his four fingers. "Do you remember those black shoes the computer imaged for you when the PR people were deciding what you should wear for the holocast we made at Earth?"
"The high heels?"
"Yes." He spoke in a low voice near my ear. "You should get those. Just to wear for me."
I laughed softly. "All right." I had asked my personal EI why Eldrin liked those shoes, given that they made me taller and he preferred small women. The EI gave me a dreary discourse about how raised heels changed human female posture in a way that inspired human males to think of reproduction. I supposed it made sense, but at a gut level I still couldn't fathom why stiletto heels would make Eldrin want to make babies.
"Eldrin?"
He kissed the ridges in my ear. "Hmmmm…?"
"My models predict something else."
He sighed. "Dehya, you need to stop thinking about math and concentrate on your husband."
"I am."
He raised his head, looking a bit alarmed. "You are?"
"Dryni… what if we had another child? A son?"
He laid his forehead against mine. "Dehya, love, we don't dare."
"Maybe we could find a way."
"I wish it could be true. But you know the risks."
"Medicine is always advancing."
His voice softened. "We can always hope." His thoughts carried a bittersweet ache, remembering both joy and anguish with Taquinil.
I touched his cheek. "If we do ever have another son, I would like to name him Althor. For your brother."
"Yes. I would like that." He pulled me closer.
So we loved each other, at the dawn of this new era we had entered. The future still held many unknowns. It wouldn't be easy to solve the problems that plagued humanity. But hope had come.
Perhaps we could find a way, after six millennia, to reunite the sundered children of Earth.
Author's Note: Science in Science Fiction
I
THOUGHT HARMONICS
Spheres of Art
Spherical harmonics are among the most beautiful functions in physics. In this essay I will describe what they are and try to give a feel for how and why they inspired this book.
Spherical harmonics can be used to form spheres, rings, teardrops, and other rounded shapes. They appear in many areas of physics, including quantum theory, electromagnetism, and optics. When I do physics, I tend to associate mathematical terms with colors, textures, and other traits. I envision spherical harmonics in shimmering hues: rose, lavender, blue, silver. My doctoral thesis brims with them, a pleasant set of equations to work out given the lovely images they create in my mind.
I've also always liked the name, in part because of the musical associations it evokes. Music has played a big part in my life; I've studied both ballet and piano, and to a lesser extent voice. The word harmonic simultaneously makes me think of melodic harmonies and pleasing equations.
The spark of creativity I feel when choreographing a dance is similar to what I feel when solving an equation, and the meditative quality of ballet class reminds me of working on a satisfying derivation. I first combined the two in a ballet called
Spherical Harmonic
that I choreographed for several students in a dance program I directed while in grad school at Harvard. Spherical harmonics often describe waves, so the ballet evoked wave motions as the dancers wove in and out of spherical patterns. The colors of their skirts, leotards, and tights matched the colors I see for spherical harmonics. I set the ballet to a Gymnopedie by Eric Satie, which has a delicate beauty that fits the way I imagine the functions.
That ballet became the seeds of this story.
Spanning Space
When I was choreographing the dance, I couldn't resist playing with some science fiction "What if?" questions. I wondered what a universe spanned by spherical harmonics would be like if you could actually visit there.
What do I mean by
spanned
? Let's talk about coordinates. Imagine you are sitting in a room. 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: (2, 3, 7). They specify your location. We need three to specify your position completely because we live in three dimensions.
If we imagine arrows pointing from the two walls and the ceiling to your location, we have one arrow that is two feet long, one three feet long, and one seven feet long. We call them
vectors,
and they define three directions. When we say those vectors
span
our three-dimensional universe, it means we can specify the location of
any
point in the universe by using those three vectors, if we make them long or short enough. For example, if you stand up, the top of your head moves closer to the ceiling, until it is only, say, four feet away. The same three directions that specified your position before do now as well, but the vector that points from the ceiling to your head shortens to four feet, making your coordinates (2,3,4).
Now let's look at some really odd vectors. The solutions to certain equations are functions that act in an analogous manner to vectors. However, they span a universe with infinite dimensions! To specify a point requires an infinite number of them. The functions don't point anywhere; they are shapes or curves. But mathematically they can be treated like vectors. Such universes are called Hilbert spaces, and the infinite set of functions that span them are called eigenfunctions.
Many eigenfunctions are named after the people who figured them out, like Bessel or Legendre functions. In the book
Spherical Harmonic,
the Selei eigenfunctions refer to the fictional functions discovered by Dehya Selei, the main character.
In our real universe, the three vectors we use to specify our position are mutually perpendicular, that is, they intersect at right angles. We say they are orthogonal. Eigenfunctions are also orthogonal, but here the meaning is 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 parts of the two functions overlap; rather, when certain math operations are perpetrated on them, they cancel each other out.
On the other hand, if we have two copies of the same eigenfunction, they will overlap. We can scale that overlap so it equals one; then we say our functions are
normalized
as well as orthogonal; in other words, they are
orthonormal.
Now we have a wonderfully arcane description of our new universe; it is a Hilbert space spanned by an infinite set of orthonormal eigenfunctions.
Spherical harmonics form such a set.
Figuring a Good Angle
To understand what these eigenfunctions mean, recall how we specified our position in a room. Our location depends on our distance from the walls and ceiling, so we say it varies with distance. Other quantities can depend on other coordinates. For example, if you keep track of your temperature during the day, you would say your temperature
varies with
or
depends on
time. If I keep track of how my weight varies during a diet, I can say my weight depends on what I eat. I could also say it depends on time, since it changes during the diet (hopefully!). Likewise, a function must depend on something, such as time, distance, energy, momentum, or any other physical quantity.
Spherical harmonics depend on angles. Imagine you stick a fork in the center of a pie (so much for the diet). Next you lay a knife flat on the pie with the end of its handle against the fork. Then you rotate the knife around so its end stays against the fork and its tip moves in a circle. If you rotate the knife one quarter of the way around the pie, we say it makes an angle of 90 degrees with its original position. Halfway around is an angle of 180 degrees and the full circle is 360 degrees.
Of course if we just push our knife around on top the pie, we will never cut out a tasty piece. Suppose we stick the knife in the pie right next to the fork. When the two are straight up and down together, the angle between them is zero. As we bring the knife down, the angle it makes with the fork increases. When the knife reaches the pie, it makes an angle of 90 degrees with the fork. If we cut through the pie (and the pie dish) and continue the knife down until it is opposite the fork, the angle between them is 180 degrees. If we bring the knife all the way around, cutting through the other side of the pie (and the beleaguered dish), the knife has gone through 360 degrees.
Now we've described two angles; one is measured around the surface of the pie and the other is measured on a circle that cuts through the pie. The knife in both cases makes a circle, either on the pie or cutting through it. The two circles are perpendicular to each other, just as were the directions that gave our location from the walls and ceiling.
But wait! We've only defined two angles. Space has three dimensions, so we need a third coordinate. That's easy; we let the length of the knife vary— it can be as short or as long as we want. This is called the radial coordinate; together with the two angles, it can completely specify any point in space. These three coordinates define a
spherical
coordinate system; the three vectors we talked about earlier define a
Cartesian
coordinate system.
