Inga was fiddling with some contraption from her crate. “Every child learns the inverse-square law, though?”
“Yes,” Freya agreed. “But the law itself is one thing; all its consequences are another.”
“So you need to make this consequence visible,” Inga replied. She took the gadget and offered it to Freya. It was a long tank with a square cross-section, and a flexible partition inside. The partition was connected to a series of pegs that protruded through holes in the side of the tank and emerged along an attached board, on which a grid had been drawn, its intervals marked with numbers.
“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.”