Geometry might well kill them in the end, but only a rigorous understanding of its principles could make their situation intelligible, let alone survivable.

That quote comes from Part 4 of this novel, but it encapsulates what’s intriguing and (at least potentially) frustrating about the story—it’s about spacetime geometry.

I’ve written about Greg Egan before, when I reviewed his Orthogonal trilogy. Egan has always written about big ideas, and pushed farther into mathematical physics than most of his contemporaries. In the Orthogonal series, his novels were set in a universe in which the time dimension has exactly the same geometrical properties as the spatial dimensions. This is in contrast to our own universe, in which the time axis of spacetime works differently from the space axes, creating a non-Euclidean, hyperbolic geometry which underlies the counter-intuitive physics embodied in special relativity.

In a way, Dichronauts represents a companion volume to the Orthogonal novels—having asked what the world would look like if time worked the same way as space, Egan now flips the problem over and considers a universe in which one of the spatial dimensions is timelike. His new universe’s spacetime therefore has two axes with timelike geometric properties—the time axis itself, and one of the space dimensions. Hence the title of the novel, which isn’t actually explained in the book—fashioned after the pattern of aeronauts and astronauts, its Greek roots give it the meaning “sailors in two times”.

In our universe, the hyperbolic relationships appear when a space coordinate is plotted against the time coordinate—so they show up when position changes over time; if an object has velocity, in other words. An example of that hyperbolic relationship is that, no matter how hard you accelerate, you can never exceed the speed of light, only approach it asymptotically.

In the Dichronauts universe, the same relationships also appear if you plot a normal space coordinate against the timelike space coordinate. That is, when you change position along one space axis relative to position along the other axis—with rotation, in other words. So in the Dichronauts universe, it is impossible to rotate an object from north to east—no matter how hard you try, you can only approach northeast asymptotically, and never rotate any farther. And the spacetime distortions caused by the hyperbolic coordinate system (which in our universe show up as changes in length and clock rates when travelling close to light speed) appear as changes in the shape of rotated objects—as they approach a 45º rotation angle, they grow asymptotically towards infinite length and zero thickness.

Egan talks the reader through some of this material in an Afterword, which can be read with advantage before starting the story, because it contains no real spoilers. And (as with Orthogonal) there is a great deal more background information, including mathematical detail, on Egan’s website. There, he also explains how light cannot travel within a cone surrounding the timelike space axis.

It turns out that self-gravitating objects in this sort of spacetime collapse to form hyperboloids rather than spheres, with the symmetry axis of the hyperboloid aligned along the timelike space axis. So Egan’s alien protagonists inhabit a region near the equator of a huge hyperboloid world orbited by a tiny hyperboloid sun—it’s shown in the cover illustration at the head of this post.

Egan’s aliens live in a world where they can’t see to the north or south; where they can’t turn around but have to walk forwards or backwards in the east-west direction, or “sidle” to the north or south; and where a fall to the north or south can trap them into a runaway lengthening of their bodies as they topple towards a 45º angle with the ground. The restrictions and opportunities afforded by such an environment are worked out in loving detail—doors can only face west or east, for instance, and must pivot upwards, keeping their plane of rotation entirely within the spacelike axes of the world.

Again as in Orthogonal, Egan gives his aliens one truly alien characteristic, and otherwise portrays them as essentially amiable and thoughtful humans. It’s a plan that previously worked well for Hal Clement—trying to tell an engaging story set in a totally alien environment is hard enough, without stirring in an alien culture and alien thought processes, too.

Egan’s alien protagonists are bipartite beings—a large, roughly humanoid creature with eyes, orientated in the east-west direction, called a Walker; and a small, blind, intelligent, commensal organism called a Sider, that is threaded through the Walker’s skull in the north-south direction, and which “sees” in those lightless directions using echolocation. The two share sensory information and thoughts through a nerve linkage. The Walker and Sider who are Egan’s composite point-of-view character bicker cheerfully and engagingly throughout the novel, like a long-married couple.

The main plot driver is the Migration—because of the changing position of their sun, the planet’s inhabitants are forced to move their towns and farms endlessly southwards.* Egan’s story follow the Surveyors, who search ahead in order to plan the migration route. This lets him gradually expand the picture of his strange world and its inhabitants. And when the Surveyors encounter an apparently impassable barrier, the story takes an unexpected twist.

I enjoyed this one very much—in large part because the characters and problems become very engaging as the story progresses, but also because I just liked messing around with the maths. I do think Egan skipped rather lightly over some problems with the physical environment he builds—zeroes and infinities are never too far away. For instance, two objects that are aligned northeast-southwest or southeast-northwest in his world will have a separation of precisely zero, no matter how far they are separated along the north-south and east-west axes. But they will also have zero thickness measured at those 45º angles, no matter how wide they are north-south and east-west, so they shouldn’t collide—the world just seems to go a little indeterminate at those special limiting angles. And it’s not clear what actually happens to a vertical object that falls to the south or north. It gets longer as it topples, certainly, but it shouldn’t be able to get closer to the ground than a 45º tilt. Egan refers to this situation a couple of times but doesn’t get into detail. I think what he envisages happening is that the endlessly lengthening and thinning object breaks up into sections under the differential torque of gravity (like a toppling factory chimney), and then the broken sections fall vertically to the ground with minimal farther rotation.

But these tilted segments should then start to undergo their own asymptotic lengthening …

And I do think there may be a problem with this novel if you’re not a special-relativity junkie, like me. While the odd spacetime of Orthogonal was only an occasional intrusion in the narrative, which could be skimmed over, the counterintuitive spacetime distortion in Dichronauts is front-and-centre, influencing plot and the characters’ behaviour on every page. It may simply be too weird an environment for a reader who doesn’t enjoy playing with maths a little.

So the question is: when I described those exotic spacetime axes, did you perk up and want more detail? Maybe feel the need for a graph? In that case, take a look at Egan’s website, and then go and buy the book.

* Remarkably, and I’m sure coincidentally, Egan’s is not the first novel to describe a society obliged to migrate continuously over the surface of a planet with hyperbolic geometry. Christopher Priest’s 1974 novel Inverted World did the same thing, expanding on a 1973 short story with the same title. But Priest’s planet was a different shape from Egan’s (a pseudosphere), and Egan’s makes actual mathematical sense, whereas Priest’s probably falls into the category of “a cool idea I’d rather not have to justify”. (The ending of Priest’s novel was deeply unsatisfactory for those of us who’d been on the edges of our seats waiting for an explanation.)