File:MappingInfinities.png

English: Kind of like Gerardus Mercator's projection of continents on earth, or even moreso a stereographic projection of directions (magenta arrows) in a crystal lattice, in reverse this plot maps integer-points from -10 to +10 along the black real-axis manifold-line to points on a blue circle, with positive and negative infinities joining up at the green point on top.

Here with a circle of radius 1, the manifold-points at black-line real-number values of ±2 map to points on the blue circle half-way up. Stereographic projection of directions, conversely, maps directions 90-degrees from center onto a flat horizontal circle of radius 2.

This plot also shows how the ordering of reals is cyclic, like branch-cuts in the complex plane, with points counter-clockwise always greater and points clockwise always less. We normally choose to break the cycle at ±∞, although this is not always the best choice e.g. as in the case of absolute temperature T, where T=0 is a better place to break.