A mile south, east, and north

The scenario

You’re standing on the surface of the earth (as opposed to whatever your usual hangouts are), and walk a mile south, a mile east, and then a mile north and end up where you started. You see a bear. What color is the bear?

You’ve heard this one before, but just in case, the answer is: White. You are at the north pole and it was a polar bear. But that’s not the only point that it could be. The riddle is, can you describe the full set of solutions?

Solution

Pick any line of longitude where the length of the line is $1/n$ miles long, then go a mile north of that and you have a solution. In fact, this set is dense around the ring with circumference of 1 mile.