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 abused limbs completely. She rose to her feet and took a step back, then rolled her shoulders and chirped with bliss as she rearranged half the flesh in her torso. Her two small wounds ended up at the sides of her back.
The guard said irritably, “Could you groom yourself outside?”
“With pleasure.” Yalda didn’t waste time asking him who’d paid her fine; Tullia would know which businesswoman from the Solo Club had taken pity on her, and would be able to advise her on the proper way to express her thanks.
Yalda moved slowly down the corridor toward the dazzling rectangle of light that marked the entrance to the barracks. She would never have lasted a year; she could admit that now. She would have been dead or insane in a dozen stints. She needed to visit the chemistry department at the first opportunity, and come back with something volatile enough to turn this whole abomination into rubble.
She stepped out beneath the sky, shivering, humming softly to herself. For a moment she was hurt that Tullia wasn’t waiting on the street to welcome her back to freedom, but that was petty; the world hadn’t come to a halt, Tullia still needed to earn a living. She extruded two fresh arms and shielded her eyes as she looked around, trying to orient herself.
“Yalda?” A male figure approached through the haze of brightness.
“Eusebio?” Yalda had lost count of all the lessons they’d missed. First her three stints on Mount Peerless, and now this unexplained absence. “I’m sorry, I couldn’t get word to you—”
He was close enough now for Yalda to read the embarrassment on his face. Of course he would have heard exactly what had happened to her.
“May I walk with you?” he said.
“Of course.” She let Eusebio lead the way; she had yet to recover her sense of direction, let alone decide where she wanted to go.
Eusebio remained silent for some time, his gaze directed at the ground. “If you choose to end our arrangement,” he said finally, “I’ll understand your decision. And I’ll pay you for the lessons to the end of the year.”
Yalda struggled to make sense of this strange invitation. Was he trying to tell her that he was so ashamed of her scandalous behavior that he no longer wished to be her student—but he expected her to act, sparing him the unpleasantness of having to dismiss her?
“Actually, I’d rather go on tutoring you,” she said coolly. If he wanted to be rid of her, he’d have to find the courage to spell it out.
Eusebio shuddered, emitting a hum that sounded more like shame than disgust. “I can’t believe you’re not angrier,” he said wonderingly. “It was my fault; I should have warned you.”
Yalda stopped walking. “What should you have warned me about?”
“Acilio, of course. All of them—but Acilio’s the worst.”
Yalda was utterly lost now. “How could you have known that Acilio would decide to throw a rock at me?” Unless the cosmos was spherical after all, and Eusebio had sat in his apartment one night reading the harmonics for the entire future.
“I couldn’t,” he replied. “And that might well have been sheer coincidence. But once he found out who you were, that you were connected to me…”
Yalda struggled to absorb this. “You mean, he asked for that huge reparation as a way of getting at you?”
Eusebio said, “Yes. Of course you humiliated him, so he didn’t care what harm he did to you, but the penalty was chosen for my edification.”
It was Eusebio who’d paid the fine and set her free. But she’d only faced the prospect of a lifetime in the cells in the first place because of some childish dispute between him and Acilio.
And she had been the last to know about any of this. When the sergeant had urged her to reconsider the resources at her disposal, he’d been hinting that he expected her to beg her wealthy employer to come to her aid.
“So what now?” she said bitterly. “You’ve bought me, you own me?”
Eusebio recoiled, wounded. “I was remiss in not warning you about my enemies, but I’ve never treated you with anything but respect.”
Yalda could not dispute that. “I’m sorry,” she said.
“Acilio is nothing to me!” Eusebio declared. “I don’t want to fight with him! But his grandfather and my grandfather are rivals. It’s all so tedious it would merely be a tired old joke if it didn’t damage other people’s lives. All I want to do is get an education and make something of myself. But I should have warned you that I have adversaries who’ll treat anyone close to me as fair game.”
“It might have been helpful,” Yalda agreed.
“I’ll give you their names, I’ll show you their portraits,” Eusebio promised. “Everyone you should avoid.”
“I probably shouldn’t injure… anyone, really,” Yalda decided.
Eusebio said, “These are people you don’t even want to bump in a queue.”
“I see.” Yalda contemplated the situation. “Is this over now? Or will Acilio have something more in mind for me?” She wasn’t keen on being shuttled in and out of prison until Eusebio was bankrupt. Couldn’t these idiots learn to ruin each other in pointless games of chance, instead?
“I don’t think he’ll repeat himself,” Eusebio said carefully. “And it’s one thing to exploit an opportunity, but using you to bludgeon me repeatedly would be seen as rather crass.”
“Well, that’s a relief. I’m so glad there are standards to be upheld.”
Eusebio met her gaze; he was still ashamed over what had happened, but he’d done all he could to make amends. “So what do you say about our lessons?”
“I want them to continue,” Yalda said. “Draw up a guide for me, on surviving the whims of Zeugma’s ruling class, and then we can get on with the things that matter.”
The prison guards hadn’t returned the coins the police had taken from her pocket, but Yalda still had some money in the bank. The clerk looked dubious when he compared the signature she made on her chest with a print of it on paper, and insisted on asking her three of her secret questions as well.