Spherical Harmonic

by Catherine Asaro

  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 dur 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 Hubert 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 Hubert 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 systern; the three vectors we talked about earlier define a Cartesian coordinate system.

  Spherical harmonics depend only on the two angles; to add the third dimension, we need to multiply them by some function that has radial dependence. This is what Kelric meant in chapter XXXII when he told Dehya she was missing the “radial extent” from her model of psiberspace.

  Thought Space

  Now comes a central extrapolation in the book: the fictional Selei functions span a Hilbert space, but instead of its functions depending on any of the usual coordinates, they depend on thought.

  Well, what does that mean? To start with, we need to decide how to define a thought. We can’t really isolate a single thought completely; they blend into one another and we can 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, whether it is a single thought about going to the store, or a mingled thought about fixing the car and feeding the cat. More complicated thoughts could be broken into individual components, such as the ideas that go through your mind when you read an essay about the science in science fiction.

  You could think of your brain, at any moment, as being in a “slice” of a thought. Sum up the slices over the time it takes to complete the idea, from an approximate start to finish, and you have the whole thought.

  The size and structure of our brains are reasonably fixed, but the brain isn’t identical from moment to moment. As we think, neurons fire and other chemical reactions take place. At each instant, everything in the brain from its structure to its chemical processes can be described by what we call a quantum wavefunction. We don’t need to understand details about the wavefunction, only that it exists and that physicists know (in theory) how to calculate it. Then a thought could be described as the sum over the wavefunctions for all the slices of the thought, from start to finish. Voila! We have a mathematical description of a thought.

  In theory, we know how to calculate a wavefunction for the entire brain; in practice, it would require computer memory and speed far beyond our current technology. But given the theory, we can jump to a science fictional “What if?;” suppose we let the thoughts act as coordinates and have our eigenfunctions depend on them? These are the Selei eigenfunctions in the story. They span psiberspace.

  Transformations

  In physics we often use transforms. These are mathematical procedures that transform a function depending on one type of coordinate into a function that depends on a different type of coordinate. It is like a caterpillar turning into a butterfly; the cocoon acts as the transform, causing the change. Just as the appearance and other characteristics of the caterpillar determine those of the butterfly, so do the characteristics of the original function determine those of the transformed function.

  A big difference exists between the caterpillars and functions, though: you can do an inverse transform on the transformed function and go back to the original type. Imagine if the butterfly could return to th
e cocoon and come out as a caterpillar again. The new caterpillar wouldn’t be exactly like the original because the butterfly wouldn’t be exactly the same when it transformed back as when it emerged from its cocoon.

  Fourier transforms are used extensively in science, math, and engineering. They take a function that depends on time and change it into one that depends on energy. In the time “universe,” the function has a fixed energy and varies with time. If the time function is transformed into the energy “universe,” the new function has a fixed time and varies with energy.

  Now consider our wavefunction for a thought. It depends on many things, mainly the position of all the particles in the brain (spatial coordinates) and how long it takes to have the thought (time coordinates). In this book, the fictional Selei transform takes a wavefunction for a fixed thought, which varies with space-time coordinates, and transforms it into a wavefunction for fixed space-time coordinates that varies with thoughts. However, just as a thought isn’t precisely defined in this universe, so the spacetime coordinates aren’t precisely defined in the “thought” universe. Some degree of uncertainty exists in both.

  In the book, I called the thought universe by several names: Selei space, because the character Dyhianna (Dehya) Selei derived much of the theory to describe it; psiberspace, because it’s fun; and Kyle space, after a character who helped the Ruby Dynasty in the early days, before the Skolian Imperialate existed.

  The fictional part of all this is the transform that can convert a spacetime wavefunction describing a thought into a universe defined by thoughts. However, if we assume that “psiberspace” behaves like a Hubert space, then we can apply all the rules of mathematical physics to its behavior, keeping in mind that we are playing a math game now.

  The (real) wavefunction for a thought has a shape, probably curved around the brain. If we imagine the wavefunction as a hill, its peak would probably center on our brain in space for however long it takes to think the thought. In thought space, the hill would correspond to a specific thought, and as you move down its slopes you would sample related thoughts. Say the central peak is your thought of your next-door neighbor. As you move down the slope, your thoughts might turn to other people with a close association to your neighbor. Far down the slope, the thoughts become less distinct and refer to subjects with only a distant relation to your neighbor.

  In quantum mechanics, wavefunctions can be built out of eigenfunctions. We add up the eigenfunctions, weighting each by a value that determines how much it contributes to the overall shape of the wavefunction. You can think of it as making a dish for a meal. We add ingredients together, varying the amounts depending on what type of dish we’re making.

  Now imagine you have an infinite number of ingredients. With that many, you can make any dish in the universe. For many dishes, the amount you add of some ingredients will be zero or very tiny (say a pinch of salt). So it is with eigenfunctions. To build a wavefunction, we might add in major amounts of some eigenfunctions and lesser amounts of others. The number of eigenfunctions needed to make the wave-function may be infinite, but for higher order terms we are probably adding only pinches.

  A wavefunction can be made from any set of eigenfunctions. Two well-known sets are Bessel and Laguerre functions. We could build a wavefunction out of either type, but the amount we use of each eigenfunction would differ from set to set, because the shape of the eigenfunctions vary from set to set. Usually an optimum set of eigenfunctions exists for making a particular a wavefunction. It’s roughly equivalent to finding a recipe that requires the fewest possible ingredients.

  In Spherical Harmonic, Dehya becomes a thought wave-function when she goes in psiberspace. As she reverse transforms back to our universe, she fragments into the individual eigenfunctions used to make her in the thought universe. Those eigenfunctions are the spherical harmonics, chosen on the soundly scientific principle that the author thinks they are gorgeous. Given that Selei eigenfunctions have thoughts as coordinates, they would be wickedly messy conglomerations of math. Remarkably, in psiberspace they reduce to a beautiful simplicity, because I’m the author, so I get to make up the rules.

  II

  WORLD BUILDING

  Star Makers

  One aspect of science fiction I’m often asked to talk about at workshops is world building. Over the years I’ve accumulated many folders full of notes and equations about the various worlds in my books. I’ve also put each system in a spreadsheet that calculates many properties of the worlds and their stars. In the three-body system of the moon Opalite, the gas giant Slowcoal, and their parent star, for example, I worked out the size of each, their orbital motions, their relative distances, temperatures and luminosities, densities, gravity, the albedo of Opalite, how close Opalite can get to Slowcoal without being torn apart, the angular extent of Slowcoal in the sky as seen from Opalite, and many other details. It is fun to create such systems, like solving a puzzle.

  Since so many readers have asked about the world building, I decided to write these essays at the end of the books. For those interested in creating their own systems, I recommend the Writer’s Digest series edited by Ben Bova, including Alien and Alien Societies, by Stanley Schmidt, and World-Building, by Stephen L. Gillett.

  All in a Word

  In this essay I’ve picked an area I’ve been asked about on convention panels and in interviews: planning languages.

  To develop languages for the Ruby Dynasty universe, I first needed its history. The background: an unknown race took humans from Earth about a thousand years ago, from Mesoamerica, North Africa, and India. The aliens not only moved humans in space, they also shifted them in time, dumping them on another planet about six thousand years in our past. Then they vanished, stranding the confused humans.

  Earth’s lost children eventually developed star travel and built the Ruby Empire. Unfortunately, it collapsed after a few centuries, leaving its colonies isolated for five millennia, until humanity made its way back to the stars and built the Skolian Imperialate. For me as a writer, this offers great possibilities. I have a slew of colonies that evolve for five thousand years on their own, so I can experiment with many ideas for their cultures as long as they remain true to the roots of the situation.

  Modern Iotic, Dehya’s first tongue, is spoken by only a few people in modern Skolia: the Ruby Dynasty, people who interact regularly with them, descendants of the ancient noble Houses, and scholars. Modern Iotic descends from Classical Iotic, a language no one speaks any more. All Skolian languages, including Classical Iotic, derive from the languages humans brought from Earth from Mesoamerica, India, and North Africa.

  The resource I most often use is The Great Tzotzil Dictionary of Santa Domingo Zinacantan, compiled by Robert M. Laughlin with John B. Haviland, Smithsonian Institution Press. It describes the language of the Tzotzil Maya. For the Shay language in the book Spherical Harmonic, for example, I use variations of Tzotzil words. The language schemes in The Last Hawk and The Quantum Rose are even more detailed, so I will also talk about those.

  Hawks

  The Last Hawk takes place on the world Coba, one of the lost Ruby colonies. The Coban settlers included descendants from all the areas on Earth where humans were taken from. The name Coba comes from Mesoamerica, but other words in the book have North African or Indian roots. Languages evolve, of course, so the words shouldn’t be identical to those on Earth now. So I used Ahl Majeb River (Al Maghrib/Morocco) and Raajastan Cliffs (Rajasthan, India). Mesoamerican influence shows in the Teotec Mountains, Jatee River, and Olamec Desert.

  When writing science fiction, you get to pick what language the story is told in. For The Last Hawk I decided on the modern language of the Coban people. If you imagine that an SF writer translates the language of the characters into that of the reader, then words in the modern Coban language would read as English, like Forest of the Mists or Lake of Tears.

  A language scheme needs a consistent basis. However, if it derives from too narrow a base, th
e words will all sound the same and the design may lack depth. When I started The Last Hawk, I had Dahl, Khal, Kehsa, Vahl, Kiesa…(yawn). As I developed the culture and its languages, a richer tapestry took form.

  In real life, we use words from many languages, such as Sierra Nevada or Lafayette in the United States; in The Last Hawk, I had words for ancient customs derive from their older languages, such as the word Calanya for the elite dice players. Just as we combine English and other languages for some names, like New Mexico, so Cobans have names like Little Jatee River. Different geographic areas on Coba have different schemes: Estates in the western desert draw more on languages of North Africa and India, those in the upper mountains of the northwest use modern Coban, those in central areas mix Mesoamerican influences with modern Coban, and so on.

  Roses

  In Spherical Harmonic and The Quantum Rose I draw primarily on Mayan. Four main languages appear in The Quantum Rose: Bridge, spoken on the world Balumil; classical Iotic; modern Iotic; and Iotaca, an altered version of classical Iotic that evolved on Balumil during its isolation.

  To complicate matters, many names in The Quantum Rose refer to physics, because the story is an analogy to quantum scattering theory. Mayan languages from one thousand years ago had no words for many of those concepts. In such cases, an author has two options: have the people in the story create new words or have them give new meanings to words already in their language. That also left me with two constraints; I wanted to distinguish those two cases, and I also didn’t want nuances of the quantum analogy to be lost in the linguistics.