Incandescence

The other symmetry they were committed to retaining was the assumption that the geometry around the Hub was unchanging over time. Though the Splinter and other objects might be nudged into different orbits, it was the fact that they had shifted in space that altered the geometry they experienced; the geometry itself was not melting beneath them.

"The question then," Tan said, "is how are the two symmetries related? In our last calculation we assumed that the symmetries of space always acted in a direction perpendicular to the time symmetry. But do we have any evidence for that?"

Roi hoped Gul was filling their children's minds with ideas that would prepare them for questions like this. She had been raised with an understanding of three perpendicular directions in space — garm/sard, rarb/sharq, shomal/junub — and if you added time as a fourth, it seemed obvious that it ought to be measured perpendicular to all three. Certainly, any clock you carried with you would measure time that way, and even in the abstract world of Tan's geometry, at any given time and place you could simply pick four perpendicular directions.

However, the directions of symmetry weren't a matter of choice or convenience; they were properties of the geometry itself. And while the framework for the calculations would become more complicated if the two symmetries were allowed the freedom to slant against each other, it would be even worse if they could not rely on a measure of time in which the geometry was unchanging.

Roi said, "What would count as evidence?"

Tan couldn't answer that immediately. He took a sheet of skin and started doodling. "Throw out one dimension of space, the one that takes us out of the plane of the Incandescence, and use that instead to picture time." He drew a point for the Hub, then sketched a circle around it for their old, un-Jolted orbit. "The symmetry in time takes this circle into another one in the future, tracing out a cylinder." He sketched in the cylinder, drawing lines rising straight up from the circle to indicate the direction in which it could be pushed without its geometry changing.

Roi said, "And if the time symmetry isn't perpendicular to the rotational symmetry?" She scratched a second diagram beside the first, in which the lines that carried the circle forward in time wound around the cylinder in helices. "But wouldn't we always be able to straighten out these lines?" she said. "The geometry doesn't change, whether you move around the cylinder as you travel along its length, or just slide straight up and down. It's all the same."

Tan thought for a moment. "With one cylinder you could always do that, but don't forget the rest of the geometry." He drew in a second, larger orbit on Roi's diagram, then sketched in helices with a different, steeper pitch. "Suppose the time symmetry makes a different angle with the rotational symmetry at different distances from the Hub. We're free to combine this whole motion with any fixed amount of rotation around the Hub, but we're not free to rotate around the Hub by different angles at different distances, and that's what we'd need in order to straighten everything out."

Roi said, "So there'd be a kind of unavoidable twist in the geometry?" She pondered this. "Then wouldn't motion around the Hub in the direction of the twist be different from motion in other directions?"

"That sounds plausible," Tan said.

"When we throw a stone out of the plane of the Incandescence," Roi said, "it completes the orbit in much less time than it takes to fall down and rise up again. It's almost as if it's being swept around the axis of symmetry, forcing it to go around faster than the other cycle it's completing, the shomal-junub cycle."

Tan said, "I think you've just answered the question. There might turn out to be some other explanation, but for now we definitely can't assume that the symmetries are perpendicular."

That would make the calculations harder, but at least they were doing it for a reason. Roi felt buoyed; the idea that they could anticipate a feature of the geometry that might allow it to conform to the new observations was encouraging. Most of what she'd seen in the void remained utterly mysterious to her, but they were moving in the right direction.

"There's one more thing we need to decide before we call in the calculating team," Tan said. "How are we going to measure distances from the Hub now?"

In the previous calculation, they'd described each point's relationship to the Hub by the size of the sphere on which it lay. You didn't need to worry about the actual, messy curved geometry all the way from the point to the Hub itself; instead you imagined rotating the point around the Hub in all possible directions, sweeping out a sphere whose surface area would increase the further the point was from the Hub.