Bos-Karshirl, a young female engineer Novato had requested some time ago from Capital City, arrived by boat on a foggy even-day. The two of them stood on the pebbly beach and looked up at the massive blue pyramid and the tower rising out of its apex. The tower disappeared after a short distance into the gray gloom.
“Incredible,” said Karshirl. She turned and bowed to Novato. “I agree: this is a fascinating thing for an engineer to study. Thank you for requesting me—although I’ll admit I’m surprised you asked for me. I’m young, after all; there are much older and more experienced engineers who would enjoy a chance to examine this.”
“You’re not that young, Karshirl,” said Novato. “You’re eighteen or so; I was just eleven, an apprentice glassworker, when I invented the far-seer.”
“Still…” said Karshirl, then, evidently deciding not to press her good fortune, “Thank you very much. I do appreciate the opportunity.” The younger female leaned back on her tail and looked up at the tower, lost in the fog. “How tall is the tower?” she asked.
“I have no idea,” said Novato.
Karshirl clicked her teeth. “Good Novato, have you forgotten your trigonometry? All you have to do is move a known distance from the tower’s base—a hundred paces, say—then note the angle between the ground and the top of the tower. Any good set of math tables will give you the height.”
“Of course,” said Novato. “But that’s predicated on the assumption that one can see the top of the tower. But we can’t, even on the clearest of days. The tower simply goes up and up, straight to the zenith. I’ve seen it pierce right through clouds, with the cloud looking like a gobbet of meat skewered on a fingerclaw. The tower is sufficiently narrow that it fades from view before its summit is reached. The best time to view it is on clear mornings just before dawn, when the tower itself is already illuminated by the sun, but the sky is still dark. Still, I can’t make out its apex. I’ve looked at its upper reaches with a far-seer, and, again, it fades from view rather than coming to a discrete end.”
“That’s incredible,” said Karshirl.
“Indeed.”
“But wait—there’s another way to measure it. You’ve said there is a vehicle of some kind moving up its interior?”
“Several, as it turns out. We call them lifeboats.”
“Well, all you have to do is mark one of the lifeboats, so you can be sure to recognize the same one later. Measure the distance between two of the tower’s rungs—you can do that, at least, with trig, even if you can’t actually reach the rungs. Choose a couple of rungs that are a good distance apart and also are a good ways up the shaft so that the lifeboat will be up to speed by the time it passes them. Then simply see how long it takes for a lifeboat to traverse that distance. That will give you its traveling speed. After that, all you have to do is wait for the lifeboat to make a round trip up and down the shaft. Assuming the lifeboats do indeed go all the way to the top, and assuming they travel at a constant speed, you’ll be able to calculate the tower’s height by dividing half the total elapsed time by the lifeboat’s known speed.”
If Karshirl had been looking at Novato, instead of tipping her muzzle up at the tower, she’d have stopped before getting to the end of her explanation, since Novato’s face made it clear that she’d already thought of all this. “We tried that, of course,” said Novato. “The lifeboats accelerate quickly at first, but almost immediately seem to reach their maximum velocity. It seems the lifeboats are moving at something like one hundred and thirty kilopaces per day-tenth.”
“Good God!” said Karshirl, her eyelids strobing up and down. “That’s faster than even a runningbeast can manage.”
“Twice as fast, to be precise,” said Novato. “And it takes the lifeboats—wait for this—twenty days to make the round trip. Now, granted, there’s a lot of room for error—these are just back-of-a-sash calculations—but if you do the math, that would imply that the tower is on the order of thirteen thousand kilopaces tall.”
“But, good Novato, our entire world has a diameter of only twelve thousand kilopaces,” said Karshirl. “You can’t be seriously suggesting that the tower is taller than our world is wide. Something must be going on that we can’t see. The lifeboats must stop at the top for days on end, or else slow down once they get out of sight.”
Novato felt slightly surprised. She’d selected Karshirl for her own reasons, but was beginning to regret the choice. “Surely you wouldn’t discard data just because it doesn’t fit your expectations.”
“Oh, indeed,” said Karshirl, somewhat piqued. “I’m a good little scientist, too. However, I am also a structural engineer, which is something you are not. And I tell you, Novato, based on well-established engineering principles, that the tower cannot be as tall as you say. Look: stability is a real concern when building towers. You know the old story of the Tower of Howlee, told in the—the fiftieth, I think—sacred scroll? That was a tower that would reach up to the sky so that one could touch the other moons.” Novato nodded.
“But Howlee’s Tower is utterly impossible,” said Karshirl. “A sufficiently long, narrow object will buckle if it is held straight up.” She raised a hand. “Now, I know you’ve said that this tower is made out of stuff that’s harder than diamond. That’s irrelevant. No matter how great its compressive strength, such a tower will buckle if the ratio of its length to width goes above a certain value. In the old scroll, which was written long before we knew just how far away the other moons were, Howlee’s Tower was said to be twenty-five kilopaces tall, and had a base fifty paces on a side. You couldn’t build a tower like that out of any material. In fact, one can’t even build a scale replica of Howlee’s Tower, at any scale. It will buckle and collapse.”
“Because of the buffeting of the wind?” asked Novato.
“No, it’s not that. You can’t even build a scale model of Howlee’s Tower inside a sealed glass vessel, in which there are no air currents at all.”
“Why not?” said Novato.
Karshirl looked around vaguely, as if wishing she had something to draw a picture on. Failing to find anything, she simply turned back and faced Novato. “Let’s say you build a tower that’s a hundred paces tall and has a base of, oh, one square centipace.”
Novato’s tail swished in acceptance. “All right.”
“Well, visualize the top of this structure: it’s a flat roof, one square centipace in area.”
“Yes.”
“Consider the corners of that roof. There’s no way they will be perfectly even. One of them is bound to be a small fraction lower than the others. Even if they are all even originally, as the ground shifts even infinitesimally under the tower’s weight, one corner will end up lower than the others.”
“Ah, I see: the tower, of course, will lean toward the lowest corner, even if only very slightly.”
“Right. And when the tower does lean, that makes the lowest corner even lower, and the tower will lean some more, and on and on until the whole thing is leaning over like a tree in a storm—no matter how strong the building material is.”
“So the tower can’t be thirteen thousand kilopaces high,” said Novato.
“That’s right. Indeed, it can’t be anywhere near that high.”
Novato leaned back on her tail. “Obviously the pyramidal base gives the tower some stability, but the actual tower itself is only fourteen paces wide. How high could a tower that wide be?”
“Oh, I’m no Afsan,” said Karshirl. “I’d need to sit down with ink and writing leather to figure that out.”