A Visit to Sphereland
A Visit to Sphereland
In [1] (example 1.4.2, p. 32), William Thurston describes an imaginary journey that takes place in the equally imaginary Sphereland. Instead of "Once upon a time…", Thurston begins, "Imagine creature A resting at the north pole and another creature B creeping away…" He then goes on to explain that line-of-sight in Sphereland follows the great circles known as geodesics. To creature A standing at the north pole these geodesics are equivalent to the lines of longitude that map our Earth. As creature B moves away from creature A it shrinks in size. However, when it crosses the equator an unusual thing happens: it actually increases in size until it reaches the south pole. At the south pole—where all the geodesics from the north pole converge—even more strangeness: creature A sees creature B everywhere, in every direction, albeit far off in the distance. Furthermore, had creature A turned around during creature B's journey, it would have seen a second, more distant image of creature B, as those lines-of-sight also intersect with creature B after passing through the south pole.
The ambulatory creature B is represented by the red line segment, and as Thurston tells the story: its bright red color becomes "slightly faded by the blue haze of the water vapor in the intervening air."