South-Pointing Chariot

by James Buckland

The South-Pointing Chariot was an ancient Chinese design for a compass, a chariot with a pointing statue on it which would face south at all times, achieved via the use of precision differential gears connected to the wheels. That is to say, if the chariot were facing south, and it were turned 90˚ to face east, the statue on it would turn 90˚ in the other direction to compensate, so that it would always remain facing south. There are many arguments over the engineering possibility of such a construction; whether it would work, whether it would remain precise; how it was calibrated. For the purposes of this essay, we’ll assume the machine existed to within a certain level of accuracy.

An interesting side effect of the South-Pointing Chariot is as follows: even assuming it works as precisely as possible, it will still tend to drift over time. This is due to the curvature of the Earth, and a property called holonomy. This property is best described with a thought experiment.

You are an intrepid explorer, able to cross great distances on foot. You are also a salesman, holding one of those large plastic arrows, used to advertise the location of a shop. You are at the equator — perhaps somewhere in South America, maybe Brazil. You are holding the arrow due north. Now walk a quarter of the Earth’s circumference north, until you’re at the north pole. For this whole time, the arrow has still been facing north, until the very instant you hit the pole. Now it’s technically facing south, but really it’s tangent to the surface of the Earth, pointing off into space.

That’s fine. Turn right, ninety degrees, but compensate with the arrow, so that it’s still facing the same direction it was. That is — if you turn clockwise by 90˚, you rotate it counterclockwise within your hands by 90˚, to compensate. Now, take a step off the pole. The arrow is facing East.

Don’t believe me? Walk south, to the equator again. You’re no longer Brazil-bound — you’ll end up somewhere in Africa, a quarter of the Earth’s circumference around the equator from Brazil. Your arrow is facing east. You walk back along the equator to Brazil (a long but convenient chain of archipelagos provide a path) and the arrow is still facing east.

Complete the entire cycle again, and the arrow will now be facing south. Every time you trace out this strange triangle, where every turn is a right angle, the arrow will rotate another 90˚ clockwise. If your triangle had a left turn at the north pole, it would have rotated 90˚ counterclockwise instead.

This is called holonomy — loosely, it is the rotation of an arrow during parallel transport along the surface of a curved surface, such as the Earth. Our particular path traces out a spherical triangle, but holonomy would hold for any shaped path, of any size — simply to a lesser degree.

Back to the South-Facing Chariot. The more of mainland china this chariot might have traced, the more it would have drifted from its original southern bearing. A few experimental journeys could have proven this information very powerful — based on the knowledge that a full trip from equator to pole to equator would turn the machine a full right angle, they might have calculated the radius of the earth from a smaller trip, by proportion. Incredibly powerful stuff.

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