Lloyd Rose (pen name for Sarah Tonyn) has a “Doctor Who” book called “The Algebra of Ice”. It describes the attempted invasion of our universe by mathematical beings from another dimension. These aliens have evolved into near non-entropic mathematical entities existing as mathematical equations which capture their basic essence. They subsist close to Absolute Zero temperature, using very little energy in their interactions-through-transformations but now crave the cornucopia of energy available in ours. They’re trying to build a computational bridge over to our side but do not have sufficient power to do it on their own.
The catch is that the aliens are only able to operate in a rational number system (no irrationals, imaginaries, etc. The author does not state this anywhere but it is implicit in the entire plot). This is a needless bind into which the author gets herself as the novel’s central concept could have easily worked without it and in a far richer, consistent fashion. Leads to silly plot ideas like the doctor drawing simple circles at the sites of invasion to thwart the aliens (pg 36: “the circle contains irrationality and straight-sided figures don’t”, a patently absurd statement.). The doctor comes across as a particularly inept savior of the universe. Riemann hypothesis and prime numbers make their obligatory appearances (the computational bridge is numeric, not geometric), with a mention of fractal graphs/curves. The final battle between the doctor and the aliens is a very interesting and tongue-in-cheek description of how one might fight a
mathematical entity (“his numbers shifted and regrouped, snatching at the being’s equations. No. No. Ah! He trapped the string of numbers in parentheses and drew it close”). I wish the novel had gone slightly deeper in its description of the alternate universe.
|(quoted from Doctor Who: The Algebra of Ice)|
`Doesn't make much sense, does it?' They sat by the fire. The Doctor gestured to Lethbridge-Stewart's glass, but he shook his head. `But, if these beings are, for example, pure equations, then no doubt they could manage it. You know mathematical graphs quite often have more than three dimensions. Fractals can be graphed in one and one half dimensions. It's more flexible in there than you'd think. Of course, it's also completely rigid.'
`Of course,' said the Brigadier unperturbedly. Over the years, he'd discovered that it was perfectly possible to work with the Doctor without understanding even half o what he said. `But, if they are equations or some such, how would they function in this world?'
`It's a problem, isn't it?' said the Doctor unhappily. `It's very, very hard for any being with a body to conceive of their reality; I'm guessing more than I like.'
`Are they likely to break through?'
The doctor went quiet for a few minutes, leaning back with his hands clasped, watching the flames. `I wouldn't think any time soon. The computations from our end are extraordinarily difficult. PEople can take decades solving mathematical problems. Some were posted centuries ago and remain unsolved today. Even on Gallifrey, there are things that, so far, are beyond us. To use an Earth phrase, they exist only in the mind of God.'
Perhaps it is not fair of me to use different standards for different books, but I can't help thinking ``For a Doctor Who book, this is not bad mathematical fiction!'' (In other words, it may not be the greatest science fiction novel ever, but what do you expect for a novel based on a BBC series known for being cheesy?) [For those who care, let me add that this novel involves the seventh Doctor and his popular side-kick, Ace.]
I think it is worth mentioning that the two mathematician characters in the book represent some of the worst stereotypes: one is an anti-social nerd who has to take anti-psychotic medicine to avoid hallucinations and the other, while not totally unethical, is working for `the bad guys' and is prone to saying things like
|(quoted from Doctor Who: The Algebra of Ice)|
I am a genius! All I want is to do my work! It is not my fault that I can't without ugly things happening. I don't want them to happen. I only want to do my work and it's not possible without...
to justify the torture of his colleague.
In addition to the mathematical topics mentioned by Vijay Fafat above (e.g. entropy, Riemann Hypothesis, rationality, etc.) an interesting theme of the book is the connection between music and math. In particular, a musical piece created out of primes in the same way that Pythagorean harmonics are created from multiplicative inverses of sequential integers is literally `key' to the plot.
I dislike the Doctor's suggestion at one point that humanity is about to enter a mathematical dark age following the death of Paul Erdős, ``the last great abstract thinker''. Not intending to insult Erdős, who did lots of really interesting things in combinatorics, but this really elevates him beyond what is reasonable while simultaneously undervaluing the talents of every other living mathematician.
Finally, let me also mention that this book mistakenly remarks that Wiles' proof of Fermat's Last Theorem was not entirely valid because it involved the use of computers. A similar remark appeared in The Last Theorem, and I am supposing that my having seen it twice indicates that this is a common misconception. Probably, the confusion is caused by the fact that the Four-Color Theorem was proved using a computer to check a large number of cases, and some people do argue that this is not a valid proof. However, this was not an issue with Wiles' proof of Fermat's Last Theorem. That proof involved lots of complicated mathematics, but it really is a traditional mathematical proof in which every claim is supported by a reasoned argument without resorting to ``computer experiments'' as evidence.