exupero's blog

# Compass directions on a toroidal world

In the previous post I described how with a large enough sun the day/night cycle in the interior of our toroidal planet switches back and forth as travelers' latitude increases toward the inner equator, but I only illustrated the shadows on a toroidal cross section. Here's a simplified top-down view:

Anyone native to the inner equator would find the idea of a round sun crossing the sky very alien. In the deepest interior, daylight comes from two quasi-elliptical segments that grow bigger all morning and taper away through the evening. Creation myths endemic to those latitudes might involve a celestial fish and its reflection leaping out of the sea and diving back in.

Not only do night and day swap a couple times on the trip to the inner equator, but possibly the direction of daylight switches too.

On the torus's exterior, the sun rises in the east. At the same time and longitude on the interior, those fishy segments of reflected light have risen halfway up the western sky. If a moving picture helps:

At least, that's the way it seems from this top-down view, but it's due to our choice of east/west convention. Notice how the fish-suns spin around the axis clockwise, the same direction as the real sun. How, then, can their rising and setting be the reverse of the real sun's? The answer is: they aren't. For someone facing north on the exterior, the rising sun is to their right, and for someone facing north on the interior, the rising fish-suns are also on their right. To keep east and west on consistent sides when facing north, we have to flip the compass whenever we cross a polar rim:

It's counterintuitive to swap east and west at the polar rim. The sun that was in the east and to the right at 89°N is still to the right at 91°N, so why should it now be considered to be in the west? The dilemma comes from something we take for granted on Earth, that crossing the north pole does two things at once: 1) it switches a traveler from facing north to facing south, and 2) it moves the traveler across the planetary axis to the opposite side of the world relative to the sun. On a torus, those two things never happen at the same time. A traveler switches from north to south as they cross the 90°N polar rim, but they're still on the same side of the torus. One transition has become two, and now we have to decide which determines the orientation of east and west.

If we don't like flipping the compass and want to preserve east and west in crossings of the polar rim, then travelers have to know whether they're on the inside or the outside to know whether east on their left or on their right. That seems confusing, but I suppose one could get used to it, especially given that for anyone who (for some reason) didn't know which side they were on, figuring it out is as easy as looking at the sky. I like to think some practical consideration would tip the scales for which convention to follow, but so far I haven't figured out what it is. If you have an idea, feel free to email me.

Perhaps it's our north-south convention that needs adjustment. So far I've considered north to be up, toward the top polar rim, and south to be toward the bottom polar rim, but maybe we should think of the directions north and south the same way we've thought of latitudes north and south, that is, as particular directions moving away from the outer equator. If we do, then even though travelers cross polar rim and start "down," they continue north.

This has the added advantage of mimicking compass behavior. If our torus has a magnetic field, it would loop around the ring and pass through the empty middle, and compass needles in the interior will always point toward the bottom polar rim. Compasses don't reverse direction when crossing the top polar rim, so neither should north and south. If north and south don't switch when crossing the polar rim, there's no reason to swap east and west, and east will always be to the right when facing north. That means, at least by our chosen convention, daylight in the interior does indeed rise in the west.

In the final post of this series, we'll take a look at the challenges facing mapmakers.