The Riemann Sphere

Complex Analysis · Axiom Academy

A geometric model for the extended complex plane — where infinity becomes just another point on a sphere. Sit the unit sphere S^2 so its south pole touches the origin of the complex plane. To map a point z in the plane onto the sphere, draw the straight line from the north pole N through z . Where that line pierces the sphere is the image of z . Every z gets exactly one sphere point, and every sphere point (except N ) comes from exactly one z — a perfect bijection . 2. The Formula and Its Landmarks Writing z = x + iy , the line from N through z hits the unit sphere at an explicit point. The three coordinates are a clean rational function of x , y , and |z|^2 = x^2 + y^2 : always lands on the unit sphere: the three coordinates square-sum to 1 Three landmark inputs pin down the geometry. The animation walks z through each one and shows where it lands on the sphere: The origin maps to (0,0,-1) . The bottom of the sphere is "zero." The unit circle maps to the equator, where the height coordinate is 0 . As z runs off to infinity, its image climbs to N = (0,0,1) , which we call . |z| < 1 fills the southern hemisphere; |z| > 1 fills the northern one. Take z = 1 . Then |z|^2 = 1 , so the image is — a point on the equator, exactly as the |z| = 1 rule promises.

This is the written version of the interactive lesson above. See the full Complex Analysis course.