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Spherical Harmonic (2001)
Catherine Asaro
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As a child, Dyhianna Selei created a transformation, just a mathematical construct, mapping the real world into an abstract space of "thoughts" (whatever that means) spanned by an infinite set of spherical harmonic eigenfunctions. As an adult, she finds herself transported into that space which she thought of as a theoretical construct and finds that it is inhabited by monsters, people, danger and intrigue.

Asaro, who has degrees in physics and chemistry, includes discussion of the spherical harmonic functions (both as part of the story and in non-fictional appendices). So, let me say a little bit about what these functions are.

As a build up, let us consider a continuous function f(x) which is defined on the interval [0,2π] and has the value 0 at each endpoint. An example of such a function is f(x)=sin(nx) where n is a positive integer. If you know some calculus then you can easily verify that this function also has the property that it satisfies the differential equation f''+nf=0. A less obvious fact is that whatever function f(x) you think of that has the property in the first sentence, it can be made as a sum (perhaps an infinite sum) of multiples of functions of the form sin(nx) over all positive integers n. (This is the fact underlying Fourier analysis, breaking up a sound into the different frequencies that make it up.) In this sense, they form a basis for the space of such functions.

Spherical harmonic functions are similar functions but in a higher dimensional space. These are functions defined in 3-dimensional space that similarly satisfy a differential equation depending on a discrete parameter and together span a space of functions. They are of importance not only in mathematics itself (analysis), but in quantum physics, chemistry (think about the "electron orbitals" you learned about in school!) and even computer graphics.

Asaro does an okay job of describing them in the book (going so far as to try to explain what "orthonormal" means). Her analogy to the common mathematical procedure of applying a transformation between vector spaces, letting the result evolve, and then transforming back to the original space was interesting, as was the idea that certain "higher order terms" are getting lost in the transformation, resulting in her not being entirely solid.

But, overall, this novel reminds me of Asaro's The Spacetime Pool in that the math is basically just a vague explanation for how the protagonist gets to the other universe where the story is just a non-mathematical fantasy/sf tale of romance and war.

Note: This book is part of the author's "Skolian" series. See also Primary Inversion where she includes a bit about combining complex numbers with general relativity to break the "speed limit".

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Works Similar to Spherical Harmonic
According to my `secret formula', the following works of mathematical fiction are similar to this one:
  1. The Spacetime Pool by Catherine Asaro
  2. Primary Inversion by Catherine Asaro
  3. Mathematica by John Russell Fearn
  4. The God Patent by Ransom Stephens
  5. Catch the Lightning [Lightning Strikes Vols. I-II] by Catherine Asaro
  6. The Labyrinth Key by Howard V. Hendrix
  7. Singleton by Greg Egan
  8. Schild's Ladder by Greg Egan
  9. Light by M. John Harrison
  10. Six Thought Experiments Concerning the Nature of Computation by Rudy Rucker
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GenreScience Fiction,
MotifFemale Mathematicians,
TopicAnalysis/Calculus/Differential, Mathematical Physics,

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Exciting News: The 1,600th entry was recently added to this database of mathematical fiction! Also, for those of you interested in non-fictional math books let me (shamelessly) plug the recent release of the second edition of my soliton theory textbook.

(Maintained by Alex Kasman, College of Charleston)