by Greg Egan
“Set the pegs to the inverse-square law,” Inga instructed.
“I don’t understand,” Freya confessed.
“The first peg is the number one, for a distance of one. The inverse square of one is one, so make the height of the part that sticks out equal to one.”
“All right.” Freya did as she’d been told. “But the next peg isn’t ‘two’; there are six pegs before the next number.”
Inga said, “So the next peg is seven-sixths, and the inverse square of that is thirty-six forty-ninths. If you raise the peg almost halfway between four sixths and five sixths that will be close enough.”
“You expect my would-be investors to juggle fractions like this?” Freya joked.
“They won’t have to fret about the details,” Inga promised. “You just have to get things right yourself, and then show them a nice smooth curve along the tops of the pegs that passes through all the easy points that they can check themselves if they want to.”
Freya kept at it, until all the pegs were set—with the last one barely protruding at all.
“And this tells us…?” she wondered.
Inga handed her a flask full of ethane. “Fill it up,” she said.
Freya tipped ethane into the tank, into the space between one wall and the partition now shaped by her carefully positioned pegs.
“Now pour it all from there into here,” Inga suggested, passing Freya a much smaller, unadorned and unmodified tank, cubical in shape.
“Will it fit?” Freya wondered.
“Go ahead and see.”
Freya raised the larger tank and drained it carefully into the smaller one. There was no overflow; if anything, a narrow gap remained at the top of the second tank.
The two tanks had the same square cross-section, but though the first tank was eleven times longer in total, the volume that remained below the inverse-square curve of the partition was no more than that of the single cube.
Inga said, “So the weight of a mountain whose peak is twelve times farther from the center of the world than the ground is would actually weigh very slightly less than a mountain that only stretched from the ground to a height twice as far from the center of the world…if in the latter case, gravity didn’t vary with distance at all.”
Freya was confused for a moment, but then she understood. “People have no experience of gravity growing weaker. Our intuition comes from such a small range of heights that we can only imagine the weight of objects that obey the rules we actually see—where piling up a stack of rocks doesn’t mean the top few rocks weigh less than the bottom ones.”
“Exactly.” Inga took the small tank from her and placed it on the ground. “But now you still have to persuade them that a mountain of ice as tall as the center of the world is deep—with every portion weighing what it would at ground-level—might not be heavy enough to break the crust of ice beneath it.”
“How am I supposed to do that?”
Inga said, “Show them exactly how strong ice is, and how little it weighs. You can’t show them the mountain they need to imagine, but you can show them something smaller, supported by a smaller crust.”
“So…I just use two slabs of ice in the same proportion to each other? Like a map drawn to scale?”
Inga thought it over. “Not quite. If you shrink everything by the same amount, the weight of the load gets smaller faster than the area of the ice resisting that weight: the area of the vertical plane separating the adjacent regions in the crust that the load is trying to shear apart. So you actually need to exaggerate the size of the mountain compared to the depth of the crust, in order to make a fair demonstration of the necessary strength.”
Freya said, “Some people won’t understand that. It will look like I’m cheating.”
Inga was amused. “But not in favor of your own argument! If anything, it will look as if you’re being unfair on yourself, and trying too hard to break the crust.”
So she should try, not excessively hard, but fairly. Freya wanted to know the truth; if she’d been wasting her time that would be a painful thing to learn—but better to find out now than keep wandering from village to village when she ought to be helping her friends achieve their own saner and more modest goals.
“I don’t really know exactly how I should scale things,” she admitted.
“Let me think,” Inga replied. “I promise you, we get a good enough education in the science of forces and motion to keep our acrobats safe, but you’re calling on ideas I don’t use every day.” She closed her eyes, grimacing. “The height of the ground-level equivalent for an infinite mountain would be two hundred and fifty thousand strides—the same as the distance to the center of the world. Most people agree that the crust is at least one tenth as deep. If we want something smaller that gives a fair test of the strength of ice, we need to keep a constant ratio between the volume of ice in the load, and the area resisting the shear. So, whatever scales we apply to the height and width of the mountain, we need to apply both of them to the depth of the crust.”
“All right.” Freya was hanging on to the argument, just barely. “So for a start, we need to know the width of the mountain?”
“Yes.”
“The one my friends are considering would be a thousand strides across,” Freya recalled.
“So what’s a manageable mountain for an exhibit?” Inga mused. “It need not be as tall and skinny as the imaginary one. Maybe two strides tall, and one stride wide?”
“All right.” Freya didn’t want to think about the logistics of rolling such a cylinder from village to village, but if she pleaded to make the toy mountain any smaller, she feared the toy crust would end up thinner than a knife blade.
“So all in all, we need to scale the crust by one hundred and twenty-five million.” Inga laughed. “That brings it down to one five-thousandth of a stride.”
“That’s not going to work,” Freya concluded glumly.
“No, it’s impractical,” Inga agreed. “No lenses I’ve ever made are that thin. We need to multiply by five, at least: make the crust one thousandth of a stride deep, and the mountain ten strides tall.”
“Ten!” Freya didn’t think she’d ever seen an artificial structure ten strides tall; even the largest of the fair’s tents probably fell short of that by a stride or two. “It was kind of you to give this so much thought, but what you’re describing is completely beyond me.”
Inga said, “You want to build a bridge between worlds, but not a tower ten strides tall?”
“I don’t have the tools,” Freya replied. “I don’t even have a cart.”
Inga stood silently in the lamplight for a while, weighing up the problem. “This is important, isn’t it?” she said finally. “One way or the other. I’ve seen what some of the farms are like. We need new geysers, or we need to reach Tvíburi.”
“That’s what I believe.”
“Then you’d better come and meet the others,” Inga decided. “And see if we can talk them into it.”
Freya said, “Talk them into what?”
Inga smiled. “Letting you join the fair, along with a new exhibit: either a tower of ice that weighs so much it might as well be infinite, supported by a sheet of ice a fraction of its height…or a tower of ice that shatters that support, and shows us the way to make geysers. But whichever it is, it will be in aid of something marvelous: a bridge to Tvíburi, or the pick to end all picks, rising up from the ice field into the sky.”
Freya watched from the ground as Inga and Naja fitted the final level to the scaffolding. She hated standing by when other people were working, but her new friends had spent half their lives performing similar tasks, and she’d already slowed down the construction too much by getting in the way and trying to learn everything at once.
“Done!” Naja called down, before jumping off the edge of the platform. Inga took the ladder, and she had advised Freya to do the same. “Falling eight or ten strides is painless—and mildly entertaining the first few times—
but your older self will thank you for protecting your knees from unnecessary jolts.”
Inga and Freya rolled the last piece of the column over the ice field to the edge of the structure, and maneuvered it into the sling. Freya’s hands were still tender from the dozens of small cuts she’d acquired while learning to carve a roughly hewn slab into a cylinder like this. But the version in front of them now had been rolled back and forth over five different grades of abrasive sand, to the point where it was almost smooth. Inga had promised that if they went ahead and took the exhibition to the public, she’d show Freya how to give the cylinders a near-optical polish, so they gleamed in the sunlight.
While the two of them hauled on the pulley rope together, Naja climbed the ladder beside the ascending cylinder, steadying it and making sure it didn’t slip or start swinging. They were lucky: there was almost no wind today.
When the cylinder reached the top platform, Naja adjusted the ropes connected to the sling so that Freya and Inga’s next few pulls would make the thing vertical. Then they clamped the rope and rested for a moment, before taking the ladder to join Naja.
“This is it!” Inga marveled. “Once we add this piece, the tower will be as good as infinite.”
Freya remained silent; she didn’t know what she was hoping for any more. The work had proved so exhausting that by the time they were raising the third piece, she’d been desperately willing the surrogate crust to shatter, just to spare her any more labor. And if it had happened then, or with the addition of the fourth, the result might have been convincing enough to persuade her that Gro’s project was the only one to back.
But if it happened now? Nobody knew exactly how thick the real crust was, or what other errors their imperfect model might contain. If their mock-infinite tower broke its supporting base, that would be enough to put an end to any hope of proceeding with the bridge, but Freya would be dead long before it was certain that geysers could be summoned this way.
The three of them worked together, unhooking the cylinder from the vertical rope that had raised it and attaching it to the horizontal loop that would carry it for the last stage of its journey. Freya imagined a crowd of spectators below, some of them still hoping to win prizes for guessing the correct breaking point, or wagering on the crust’s invincibility.
“Won’t word of the result spread between the villages?” she’d asked Inga.
“It will, but no one will believe what they haven’t seen with their own eyes. People will bet on their own instincts about the forces at play here, not someone else’s claims that contradict those feelings.”
Freya tugged on the rope, while Inga and Naja walked beside the cylinder toward the hole in the center of the platform, from which a stubby portion of the existing tower could be seen protruding. When the new piece was hanging right over its four predecessors—each one a little narrower than the last—Freya joined them. If something went wrong in this final stage, the more hands there were to steady the tower, the better their chances would be to keep it from toppling.
“Ready?” Inga asked, kneeling down at one side of the cylinder while Naja kneeled at the opposite point.
“Yes,” Freya replied.
Inga spoke a brief, guttural word in the fair’s dialect that had no precise translation, though its present usage was transparent: as she finished uttering it, she and Naja simultaneously sliced through two cords at the bottom of the sling, which were holding its two halves together. The tensed, elastic structure snapped apart and the cylinder began to descend.
The final piece of the column had barely a thumb’s breadth to fall. It landed squarely in place with a thud, and with the sling clear of the impact. Freya waited, arms spread, ready to respond as Inga and Naja climbed to their feet. But nothing had skewed the release, and as far as she could tell the cylinder wasn’t shaking or swaying at all. Let alone falling.
They’d placed the thin sheet of ice that was supporting the whole structure on top of two square blocks half a stride high, and two strides apart. If that crust cracked and the base of the tower dropped to the ground, the effect would not be subtle. But so far, it was holding.
The three of them stood motionless for a while, their eyes fixed on the top of the tower. Then Inga said, “Let’s go down and see what’s happening.”
As they walked across the platform, Freya kept looking back, expecting the cylinder to plummet at any moment. She was last onto the ladder, and she gripped the side rails tightly, prepared for the scaffolding to lurch wildly if the tower collapsed, with the five pieces tumbling as they fell, crashing into the beams around them.
Back on the ground, as they approached the sheet of ice, Freya would have sworn she could discern a subtle change in its shape. But when it came to her turn to squat down and inspect the graduated rods beside it, there was no sign that it had sagged, buckled or bent. No cracks or flaws had appeared. The strain of holding up the completed tower was not enough to deform it to any measurable degree, let alone tear it apart.
Freya remained as she was, too unsteady to rise, but Inga reached down and clasped her shoulder. She said, “It’s starting to look as if my great-grandchildren might just get a chance to visit Tvíburi.”
4
Gro finished her inspection of the double doors at the top of the stairs. “One down, twenty-six to go,” she said.
Freya glanced over her shoulder at the next platform they’d be visiting. The slanted legs of the would-be tripod had come a long way toward their meeting point, but though the straight-line distance between the tops of the columns was barely fifty strides now, the trip down to the ground then up again would take at least half a day. “I won’t be happy until the ice has grown around the frames, and we can check that the gap can be made airtight,” she replied. All they’d shown so far was that the doors themselves, and the box between them, wouldn’t leak.
Gro gathered up her tools into her pack, and motioned for Freya to go before her. There was something dreamlike about stepping through a door whose frame was surrounded by thin air, into a dark wooden box, and emerging through a second unwalled doorway to confront a flight of stairs so long that its lower reaches were impossible to perceive, shrunken by perspective into a crack in the side of the tripod’s leg.
Freya hooked her safety ropes onto the stair rail and waited for Gro to join her. There was plenty of room for two people to descend side-by-side, and Freya hated having someone behind her, an invisible presence constantly threatening to stumble and send her sprawling.
“Do you want something to eat?” Gro asked, attaching her own ropes to the right-hand rail.
“No.” Freya never brought food of her own on these trips; the idea of eating on the platforms, let alone the stairs, made her uneasy. Anything that needlessly distracted her, lowered her guard or occupied her hands, could be dangerous. And whether or not that fear was justified, it was more than enough to guarantee poor digestion.
Gro reached back and managed to pull a small loaf out of her pack without unstrapping it, then started chewing as they began their descent. “Here’s a hypothetical to ponder,” she said, her words muffled by the contents of her mouth. “If a dozen new geysers appeared tomorrow, what would you do? Walk away from all this, or keep going?”
Freya wasn’t sure that her own choice would really matter, now. “Wouldn’t it make more sense to ask one of the youngsters?”
Gro didn’t dispute that, but she’d already decided to have her say. “If it were up to me, I’d keep going.”
“Really?”
“The geysers will always come and go; we could never be sure we wouldn’t face the same problem again. And having come this far, we’d be foolish to waste it—to throw away all that expertise, in the hope that we could start again from nothing if we had to.”
“Hmm.” Freya was inclined to agree, though she doubted that the farmers would be willing to keep feeding them. “You haven’t heard rumors, have you?” If there’d been an eruption nearby, half the farmers in t
he area would already have departed to stake their claims over the fresh soil. But if a geyser had appeared on the other side of the world, with no hope of any of the locals benefiting, word of it would come far more slowly, and take much longer to be confirmed.
“No, I’m just thinking out loud.”
In the silence that followed, Freya’s thoughts drifted, only to settle on her unnatural surroundings. The stairs were set deeply enough in the slanted column that she suffered no fear of toppling sideways, even if that offered the most direct route to death: if she somehow ended up on the cylindrical surface, she would rapidly slide around it, plummet through the air, and crash into the plain below. But it was the unobstructed descent ahead of her that always seemed most perilous, beckoning her forward, inviting her to trip and fall. And however confident she was, intellectually, that any such fall would drag her no farther than the next supporting post for the safety rail, some part of her mind refused to accept that: the slender ropes from her wrist to the rail felt like ineffectual talismans, utterly useless against the power of the stairway’s vertiginous gradient.
Everything about the construction so far was a triumph of fact over intuition. She still couldn’t stare at the slanted tripod legs without expecting them to topple over. The counterweights that rose vertically from each foot, lessening the risk that the torque would tear the base from the ground, just looked like a joke, a gesture, as manifestly inadequate as the ropes. She wouldn’t be happy until the columns actually met up, visibly supporting each other. But then the whole construction would need to be repeated for the next level, even more precariously, with the legs of the new, wider tripods rising up from the tops of the old ones.
“Do you think we’ll live to see the second level complete?” she asked Gro.
“We might. If the Yggdrasil keeps pumping the same amount of water in total, those columns should rise three times faster.”