However, he soon finds something even more engrossing than the war games he enjoys so much: pure mathematics. More specifically, a computer simulated mathematical world he can experience through a virtual reality set given to him by a stranger.

One intriguing idea that Quaternia raises is that mathematics would have some appeal to those who enjoy role playing games. Indeed, to those of us who appreciate it, mathematics can feel like a virtual reality with more beauty, excitement and power than can be achieved in any computer program. Shortly after his first visits to the stranger's virtual mathematical world, Ivan loses all interest in the games he played before and becomes completely obsessed with things like resolving the famous open question of the Riemann Hypothesis and also getting better acquainted with the beautiful girl he has met there.

Readers of Quaternia will encounter many pieces of real mathematics and mathematical history. In addition to the Riemann Hypothesis, there are brief but reasonably informative descriptions of Wiles' proof of Fermat's Last Theorem, Perelman's resolution of the Poincare Conjecture, and Goldbach's conjecture. Ivan “meets” virtual reconstructions of mathematicians such as Pythagoras and Bernhard Riemann himself. Some scenes taking place in the artificial world of Quaternia have the feeling of a children's fantasy novel but with a mathematical flavor: Ivan goes diving into a river made up of the digits of the transcendental number π and has conversations with the number i who talks about what it is like to be imaginary:

Although a visit to a fantastical world of talking numbers may have seemed original in The Phantom Tollbooth, by now such a mathematical “Wonderland” seems cliche and it makes it difficult to take those scenes seriously. Nevertheless, the plot does become rather serious. At the suggestion of the strange man who introduced him to Quaternia, Ivan runs away from home. It soon becomes clear to the reader that the more he visits Quaternia using the equipment in the stranger's house the less of him there is left in the real world. It is as if creating the virtual being “ivan” is achieved by emptying the brain in Ivan's physical body.

The stranger sometimes speaks eloquently about the importance of mathematics and how Ivan is better off in the virtual world. However, seeing this previously intelligent, even if misguided, young man reduced to a state in which he is barely able to form a coherent sentence, it is difficult even for a reader who loves mathematics to avoid hoping his parents can discover where he is hiding and save him from the man who is destroying his body. In this sense, even though some readers may gain some appreciation of mathematics from the brief non-fictional mathematical tidbits that the novel conveys, it is hardly a good advertisement for mathematics. The stranger, who we learn is a mathematician, seems to be selfish and villainous. And, if Quaternia is intended to be a metaphor for the world of mathematics, the novel seems to imply that those who do math research risk losing their minds, family, friends and health.

Petsinis, who is a lecturer in mathematics in Melbourne, Australia as well as a poet and the author of a mathematical novel about Galois, presumably had some idea of what he wanted to achieve with Quaternia, but it is not clear to me what that is. It is too serious and open ended to be light children's literature, but also too filled with fantastical cliches and exaggerated characters to be taken seriously as high literature. It spends too much time lecturing the reader about mathematics to appeal to those with no interest in math, but presents a frighteningly negative portrayal of the field that those who already love math may find offensive and worrisome. In the end, I am left with the impression that it is an allegory designed to attract the attention of those who might be interested in math, and then frighten them away from it. That, unfortunately, is not the sort of book I feel that I can strongly recommend to anyone.