Details

In the center of this Demonstration is the globe of Sphereland that appears as viewed by an off-world, 3D visitor. Creature A's more indigenous 2D, 360° view is schematically represented by the narrow ring with tick marks surrounding the globe. The tick marks correspond exactly to the 16 great circles passing through the north pole. When a great circle, representing creature A's line-of-sight, intersects the red line, it turns yellow.

As the caption states, if the light in Sphereland did not fade as a function of distance, the object at the creature's antipodal point—the point exactly opposite on the globe—would form the backdrop—in every direction—of its world view. This is so because every line-of-sight geodesic drawn through the creature's position passes through that other point as well. In this Demonstration, the south pole is the north pole's antipodal point and vice versa.

Many other strange things happen in Sphereland. When engaged in conversation, two creatures can see each other's backs simply by turning around. In fact, if the appropriate great circles are unobstructed, a creature would see its own back directly in front of itself, with a left-right reversal.

For more challenges to your imagination both in Sphereland as well as higher-dimensional spaces like a 3-torus, elliptic space, or the Poincaré dodecahedral space, see [1].

Reference

[1] W. P. Thurston, Three-Dimensional Geometry and Topology, Volume 1, Princeton: Princeton University Press, 1997.