As you probably know, according to the theory of relativity, the crew of a space ship that traveled quickly away from Earth and back would return to find that many years had passed here at home for each year they experienced on the ship. (This theoretical prediction, by the way, has been experimentally verified. So, as strange as it may seem, it really seems to be true.)

The very basis of this series of novels is that with the change of just a single sign (replacing the minus sign in one component of the metric tensor of space-time with a plus sign) the situation reverses. In particular, in such a universe the space travelers would experience many years for each year passed on their home world. This means that many generations of scientists, engineers and mathematicians can do research on a ship seeking a way to avoid an imminent disaster on their home planet.

But, of course, that is not the only thing that would be different, and Egan does a spectacular job of considering other ways in which that hypothetical universe would necessarily be different than our own. Probably the most dramatic is that the flow of time can not only be slowed by a change of inertial frame but actually reversed. As a consequence, when the Peerless is journeying home, the crew must deal with the seeming paradox of going backwards in time. The same ideas that made such a great joke in the Red Dwarf novel Backwards are treated with dead seriousness here. (Well, I guess it is a bit lighthearted when they land on a new planet to discover their own footprints are already there, and that the footprints disappear as they walk over them.) A major plot point is the question of whether they should seek advice from their future selves, a disagreement that literally leads to a civil war on the ship.

Mathematically, there are two things of interest going on here. One is the development of their analogue of General Relativity and (most mathematically) the discovery that their universe must have the topology of a torus. (One of the principle characters is a physicist working on this topic, and her work is clearly presented as mathematical. Essentially, she is trying to prove a theorem about the possible topologies the universe could have and to find a way to rule out the possibility that it is a 4-sphere.) The other mathematically interesting topic concerns "the arrows of time" themselves. In "our universe", this is still a contentious topic (more so than relativity or quantum theory, I would say): why is it that time seems to have a preferred direction, a direction in which entropy increases? This book seems to take a position on the question and suggests it is just a consequence of the fact that the universe happens to have had a very low entropy state in the distant past. Mathematics (in the form of probability and statistics) then become an important part of such arguments since the "second law of thermodynamics" then stops being a fundamental law of physics and starts looking like an unlikely coincidence.

The ending of a work of fiction is very important to me. No matter how good the rest of it is, I have trouble liking a work of fiction if I am not happy with the ending. Of course, I won't give any details about the way this trilogy ends because I don't want to spoil it for you, but I will say that the ending was perfect for me. If you have knowledge of and interest in mathematical physics, I strongly encourage you to give Egan's "Orthogonal Trilogy" a try.