What is the Riemann Sphere?

Question

Reading from wikipedia I understood that Riemann Sphere is used to represent extended complex plane. But it would be great if someone could explain it in a less technical manner.

Answer 1

Imagine you're on an infinitely large, flat piece of paper with a transparent beach ball of diameter $1m$. Assume the ball is perfectly round. You have a lazer pen.

The paper is the complex plane. Place the beach ball down so that it rests on $0$, its centre is $0.5m$ off the ground and the point directly above $0$ is $1m$ off the ground.

Take your lazer pen, switch it on, and point it down at $0$ so that the pen is touching the point $1m$ above the ground. With your pen held in that place, you can move the little dot of light around the complex plane, highlighting any complex number you like. Your pen will also make a dot on the ball - a point on the Riemann Sphere! - and that dot will be unique to the complex number you are aiming at!

Further: If you point the pen such that the beam of light is parallel to the paper (yet still touching the ball at its highest point), you'll see that the dot disappears. This is where we extend the complex plane; we say the dot is at $\infty$ and take the highest point of the ball to be the unique point of the ball corresponding to $\infty$.

That's all there is to it really. I hope that helps $\ddot\smile$

Answer 2

The Riemann Sphere is a unit sphere--a sphere with radius $1$--with its south pole kissing the origin centered around the $z$ axis. Points on the sphere can be associated to points in the plane by projecting from the north pole $N$ through a point $P$ on the surface of the sphere onto the plane at a point $Q$ directly through what is called a line. This correspondence of points is called a stereographic projection. There are many good illustrations of them, as, for example in the below illustration take from Tristan Needham's treatise Visual Complex Analysis:

Answer 3

To undestand in what sense the sphere is an extended $\mathbb{C}$, you should first try to understand in what sense the circle is an extended $\mathbb{R}$. If you know about complex numbers, the easiest way of including $\mathbb{R}$ in a circle is via the map $t\mapsto \frac{t-i}{t+i}$ for all $t\in \mathbb{R}$, where $i=\sqrt{-1}$ and the circle is the unit circle in $\mathbb{C}$.

Answer 4

In topological sense the Riemann sphere is the “one point compactification” of $\mathbb{C}\simeq\mathbb{R}^2$.