Schild’s Ladder

"Here’s puzzle for you," his father said. "Suppose I draw an arrow here, on the ground in front of you, and tell you it’s the most important thing there is." He marked the globe as he spoke. "Wherever you go, wherever you travel, you’ll need to find a way to take this arrow with you."

This was too easy. "I’d use a compass," Tchicaya said. "And if I didn’t have a compass, I’d use the stars. Wherever I went, I could always find the same bearing."

"You think that’s the best way to carry a direction with you? Reproducing its compass bearing?"

"Yes."

His father drew a small arrow on the globe, close to the north pole, pointing due north. Then he drew another on the opposite side of the pole, also pointing due north. The two arrows shared the same compass bearing, but anyone could see that they were pointing in opposite directions.

Tchicaya scowled. He wanted to claim that this was just a perverse exception to an otherwise reasonable rule, but he wasn’t sure that was the case.

"Forget about north and south," his father said. "Forget about the stars. This arrow is your only compass; there is nothing else to steer by. You must take it with you. Now tell me how."

Tchicaya stared at the globe. He drew a path leading away from Baake. How could he duplicate the arrow as he moved? "I’d draw another arrow, each time I took a step. The same as the one before."

His father smiled. "Good. But how would you make each new one the same?"

"I’d make it the same length. And I’d make it parallel."

"How would you do that?" his father persisted. "How would you know that the new arrow was parallel to the old one?"

Tchicaya was unsure. The globe was curved, its geometry was complicated. Maybe it would be simpler to start with a flat surface, and then work his way up to the harder case. He summoned a translucent plane and drew an arrow in black. On command, his Mediator could duplicate the object faithfully, anywhere else on the plane, but it was up to him to understand the rules.

He drew a second arrow and contemplated its relationship with the first. "They’re parallel. So if you join the two bases and the two tips, they make a parallelogram."

"Yes. But how do you know that they make a parallelogram?" His father reached over and skewed the second arrow. "You can tell that I’ve ruined it, just by looking, but what is it that you’re looking for when you see that?"

"The distances aren’t the same anymore." Tchicaya traced them with his finger. "From base to base and tip to tip, it’s different now. So to make the second arrow a copy of the first, I have to make sure that it’s the same length, and that its tip is as far away from the first one’s tip as the bases are from each other."

"All right, that’s true," his father agreed. "Now suppose I make things more difficult. Suppose I say you have no ruler, no tape measure. You can’t measure a distance along one line and duplicate it on another one."

Tchicaya laughed. "That’s too hard! It’s impossible, then!"

"Wait. You can do this: you can compare distances along the same line. If you go straight from A to B to C, you can know if B is exactly half the journey."

Tchicaya gazed at the arrows. There was no half journey here, there was no bisected line in a parallelogram.

"Keep looking," his father urged him. "Look at the things you haven’t even drawn yet."

That clue gave it away. "The diagonals?"

"Yes."

The diagonals of the parallelogram ran from the base of the first arrow to the tip of the second, and vice versa. And the diagonals divided each other in two.

They worked through the construction together, pinning down the details, making them precise. You could duplicate an arrow by drawing a line from its tip to the base you’d chosen for the second arrow, bisecting that line, then drawing a line from the base of the first arrow, passing through the midpoint and continuing on as far again. The far end of that second diagonal told you where the tip of the duplicate arrow would be.

Tchicaya regarded their handiwork with pleasure.

His father said, "Now, how do you do the same thing on a sphere?" He passed the globe over to Tchicaya.

"You just do the same thing. You draw the same lines."

"Straight lines? Curved lines?"

"Straight." Tchicaya caught himself. Straight lines, on a globe? "Great circles. Arcs of great circles." Given any two points on a sphere, you could find a plane that passed through both of them, and also through the center of the sphere. The arc of the equatorsized circle formed where the plane cut through the surface of the sphere gave the shortest distance between the two points.

"Yes." His father gestured at the path Tchicaya had drawn, snaking away from their town. "Go ahead and try it. See how it looks."

Tchicaya copied the arrow once, a small distance along the path, using the parallelogram construction with arcs of great circles for the diagonals. Then he had his Mediator repeat the process automatically, all the way to the end of the path.