The complex plane has no point at infinity — every direction stretches forever. The Riemann sphere stitches infinity to a single point at the north pole. Stereographic projection: lay a sphere on the complex plane, draw a line from the north pole through any point on the sphere down to the plane. The map is a one-to-one correspondence between the sphere and ℂ ∪ {∞}.
why ∞ becomes a point
Place the unit sphere centered at origin in 3-space. Identify ℂ with the equatorial plane (z = 0). For any point P = (x, y, z) on the sphere (other than the north pole N = (0, 0, 1)), the stereographic projection π(P) is the point in the equatorial plane on the line from N through P:
π(P) = (x/(1−z), y/(1−z))
The north pole has no image in ℂ — its projection "goes to infinity." Add a single point ∞ to ℂ and identify it with N. The result is the Riemann sphere — a compact, connected, simply connected 2D manifold homeomorphic to S² that is also a Riemann surface (the unique compact Riemann surface of genus 0).
Properties of stereographic projection:
• Conformal: angles are preserved exactly. Tiny crosses stay perpendicular under projection.
• Circles → circles: every circle on the sphere maps to either a circle in the plane (if it doesn't pass through N) or a straight line (if it does).
• NOT distance-preserving: a circle near N stretches enormously when projected.
Why this matters in mathematics:
• Complex analysis: rational functions f: ℂ → ℂ extend naturally to f: ℂ∪{∞} → ℂ∪{∞}. Poles become regular points on the sphere.
• Möbius transformations: f(z) = (az+b)/(cz+d) are exactly the conformal automorphisms of the Riemann sphere. They form the group PSL(2, ℂ).
• Topology: every simply connected non-trivial Riemann surface is the Riemann sphere, the disk, or the upper half-plane — the uniformization theorem.
• String theory / quantum gravity: the Riemann sphere is the "worldsheet" of a closed string at tree level.