The Clockwork Rocket

She sketched a few examples to make the calculations tangible. The north and south hemispheres were identical, so she only bothered with half the sphere, drawing the waves along the various circles of latitude where they were strongest.

Around any circle of latitude, though—however large or small—each particular harmonic would execute the same number of cycles, making each one as distinctive as the harmonics of a plucked string. So if you were given the values on any circle of latitude of any wave obeying the equation, you could separate out the harmonics and determine their respective strengths—which would give you the entire, global solution. What’s more, the choice of where the “pole” lay in this scenario was completely arbitrary; in principle, you could perform the same analysis anywhere at all.

In practice, though? If the waves were already packed six dozen gross to the scant as they marched around the cosmic equator, how could you ever hope to observe their proportionately finer undulations around a circle that was just a stride or two across? And to make the problem even more acute, the higher the harmonic the faster its strength dropped as you approached the poles, so the wavefronts associated with it would be preposterously weak, as well as unmeasurably close.

Tullia’s objection, then, was purely philosophical: the idea that the entire cosmic history of light was writ small in every last corner of the world was simply too shocking to countenance—however useless it would be for aspiring fortune-tellers. Yalda was prepared to set her unease aside and see where the rest of the theory took her, but other physicists might easily view this as a flaw as fatal as Giorgio’s original complaint. What was the use in being half right, if she wasn’t even half-believed? She needed other scientists to pursue these ideas; imprisoned or free, she couldn’t begin to explore all of their ramifications herself.

Yalda slumped forward and rested her head against the aching circle of her arms. She wanted to drag the weary muscles in these limbs all the way back into her chest and replace them with rested flesh, but with no instinct or experience to guide her, she couldn’t find a safe, painless sequence of movements that would fulfill this wish. For all the many postures she’d tried since childhood, she had never before suffered a change in the topology of her skin.

She butted the gap between her arms with the top of her head, allowing the limbs to relax for a while without touching. The sense of respite was glorious, but she knew it would only be a lapse or two before her arms started to slip together.

The loose skin in the gap was puckered into folds that brushed against the top of her skull. Yalda played with the wrinkles, sliding them back and forth to massage the top of her head. To her amusement, she realized that they’d naturally arranged themselves into a set of evenly spaced “waves”, a few dozen oscillations circumnavigating the sleeve of skin. She was practically a spherical harmonic come to life—except that she was no longer even roughly spherical. She was more like a torus, now.

A torus, instead of a sphere.

What would that change?

A torus would still prevent the exponential blow-up—you couldn’t wrap an exponential growth curve around a torus, any more than you could around a sphere—but its fundamental solutions would be different.

Yalda lifted her head and stared into the darkness. A torus didn’t even need to be curved; mathematically, you could slice it open and lay it flat, turning it into a rectangle or a square. You simply had to guarantee that the wave at each edge of the square agreed with its value on the opposite edge, so the whole thing could be put back together smoothly.

The fundamental solutions would be waves that executed whole numbers of cycles as you moved across the square in either direction, bringing the wave back to its original value. The sum of the squares of those two whole numbers would have to equal a constant—fixing the relationship between the size of the cosmos and the wavefronts’ separation.

She quickly sketched some examples, picking a constant small enough to be manageable, but large enough to be broken down into a sum of squares in a few different ways.

For the kind of waves she could draw on her body, completing a few dozen oscillations at most, there would only be a handful of solutions—which was tantamount to saying that light could only travel at a handful of different velocities, equal to the ratios between its frequencies in space and in time. But in the real, cosmic, four-dimensional case, the sum of squares would be so vast that it could be written in more ways than there were grains of sand in a prison cell, and the ratios would be so numerous and closely spaced that you’d never know they were not continuous.

For each choice of the number of waves spanning the square, you could also choose to have the wave in each direction either start at zero on the edge of the square, or start at a peak. With that additional flexibility, a completely general solution—whatever its complexities and quirks—could always be written as a sum of the fundamental solutions, multiplied by various factors.

What data would you need, in order to find those factors and reconstruct the whole wave—the whole history of light for a toroidal cosmos? Unlike the spherical harmonics, the imprint of these fundamental solutions didn’t get funnelled down toward any poles. To measure their contributions you’d need to know what the wave was doing along one entire edge of the square—and not just its value, but also its rate of change in the orthogonal direction, in order to learn about the waves that had a value of zero along the chosen edge.

These requirements were almost exactly the same as those for the physicists’ beloved plucked string: you set the initial shape of the string, and its initial motion, and the equation told you what followed. The only difference was that this equation allowed waves of any speed, so you needed to gather the same information from far away—potentially, across the whole width of the cosmos. This was the offer Tullia had made —“for one moment in time, there are no secrets from you anywhere”—no longer rendered useless by exponential growth.

In a toroidal cosmos, predictions became reasonable: by knowing about your immediate surroundings, you could predict what would happen with sufficiently slow waves in the immediate future. You’d be neither helpless nor absurdly omniscient. A wave faster than anything you’d prepared for could always come along and surprise you—like a Hurtler appearing out of nowhere—but if it didn’t, things proceeded as expected.

Replace the torus with its four-dimensional equivalent, and light that followed these hypothetical rules started to behave just as it did in the real world.

Yalda lowered her head and tried to rest her arms again, but her shoulders were burning with fatigue. She couldn’t replenish those muscles, either; all of the moves that would have achieved that required her to separate her arms completely.

At least she now knew how to word her message to Tullia. “If you can’t help me pay this fine,” she’d say, “I only ask you to think carefully about the shape my body is in.”

On the eleventh day, two guards with lamps entered Yalda’s cell and unclamped her chain from the wall. She didn’t question what was happening; if the sergeant had rescheduled her appearance for a few days earlier, so much the better.

Upstairs, she was almost blinded by the glare. She didn’t realize she’d been taken to a different room than before until one of the guards made her kneel and held something in front of her face. As he turned the object, her eyes were stabbed by a glimmer of reflected sunlight.

“Are you ready?” he demanded impatiently.

“For what?” she asked, alarmed and confused.

“Someone paid your fine,” he said. “We’re cutting you free.”

Yalda tightened the skin between her arms, shrinking it down to a thumb-sized stub. She’d had fantasies about making the cut herself, or even using her teeth to tear through the skin, but at least there was still no flesh to be severed.

The guard had her place her arms on a wooden bench. The process was swift, and if it wasn’t exactly painless it hurt far less than the original melding. When the guard slipped the chain off her arm, Yalda resorbed the