File:Cosmic Topology fig6b.jpg

The circle-in-the-sky method. The method is illustrated here in a 2D torus space. The fundamental polyhedron is a square (with a dotted outline), all of the red points are copies of the same observer. The two large circles (which are normally spheres in a three-dimensional space) represent the last scattering surfaces (lss) centered on two copies of the same observer. One is in position (0, 0), its copy is in position (3,1) in the universal covering space. The intersection of the circles is made up of the two points A and B (in three dimensions, this intersection is a circle). The observers (0,0) and (3, 1), who see the two points (A, B) from two ``opposite directions, are equivalent to a unique observer at (0, 0) who sees two identical pairs (A, B) and (AÕ,BÕ) in different directions. In three dimensions, the pairs of points (A, B) and (AÕ,BÕ) become a pair of identical circles, whose radius $r_31$ depends on the size of the fundamental domain and the topology.