Incandescence

When Tan analyzed a path on a curved surface, he broke it up into a multitude of tiny, straight line segments of equal length. These small straight lines acted as markers for the direction of the curve. The geometry of the surface could then be embodied in a simple mathematical rule that Tan called a "connection". The connection allowed you to take a direction at one point and shift it to another, nearby point, in a manner that respected the geometry of the surface. If a curve was a natural path, then when you broke it up into line segments and used the connection to shift them all one step forward, the shifted segments would coincide with the originals: shifting the first segment one step along the curve would give you the original direction of the second segment, and so on. If the curve was not a natural path, then the directions would fail to agree, and the resulting discrepancies would be a measure of how much the curve swerved unnecessarily, as opposed to merely following the geometry.

That the curves were broken into line segments of equal length was a crucial part of the recipe, because the analysis had to yield the same verdict if the surface was picked up and rotated, or if two people were viewing it from different angles. If you decreed that the curve should be broken up some other way, such as into segments that spanned equal horizontal distances, then different people would be left arguing over which direction was "horizontal". Nobody would argue as to whether two successive segments were of equal length. With the connection respecting this rule — preserving the lengths of the segments as it moved them from point to point — everything worked smoothly, and everyone agreed on which paths constituted natural motion and which did not.

What happened, though, when you considered the path of a tossed stone, moving forward in time as well as through space? Anyone could draw a picture in which some chosen direction represented time, and the path of a moving object slanted across the skin, but how could people ever agree on the correct scale for such a diagram? Whether one heartbeat, one shift, or one lifetime passed from the top of the skin to the bottom was a completely arbitrary choice.

Nevertheless, suppose you settled on a scale. What would happen if you divided the path of a stone into segments of equal length? To Roi, who tossed a stone forward across the Null Chamber at one span per heartbeat, the path she drew would slant across the skin. If Zak happened already to be moving at the same pace in the same direction, the stone would be motionless to him, so he would draw a line that stretched solely in the time direction. Suppose that after five heartbeats, the stone hit an obstacle. Zak's line would be "five heartbeats" long, whatever the scale of the picture made that. Roi's line, though, would have to be longer: it would stretch five heartbeats in the time direction, but it would also cross five spans of space. The accounts of the two experimenters had to be compatible somehow, but they couldn't expect to draw their separate diagrams and then measure the same path lengths.

What could they agree on? The simplest answer anyone in the team had been able to suggest was the time that had elapsed. If you marked off segments of the stone's path representing equal intervals of elapsed time, everyone would agree how many segments there were from start to finish. If you looked for a connection that respected this scheme by never changing the amount of time spanned by a segment, then everything would have a chance to work smoothly.

This was what the team had tried first. They had hunted for a geometry of space and time whose connection left intervals of time unchanged, and which obeyed Zak's principle.

In less than one shift, they had found one. In this geometry, everything was symmetrical about a special point, where the Hub could sit. The natural paths of the geometry included circular orbits around the Hub. The square of the period of each such orbit was proportional to the cube of its size. And the ratio between the garm-sard weight and the shomal-junub weight was precisely three. Close to the Hub, far from the Hub, always, everywhere, three.

It was the answer that Zak had guessed long ago, when he'd thought the Map of Weights might still hold true. It possessed an elegant simplicity, but it was impossible to reconcile with the measurements they had made. The current ratio of weights was two and a quarter; that had been confirmed a dozen times.

This failure had cast some doubt on the idea that natural motion could be described by the same kind of geometrical principles that applied to space alone. The team had considered looking for a completely new direction, but the consensus had been that they shouldn't give up on Tan's ideas so easily.