Riemann Sphere

Robert P. Munafo, 2022 Oct 15.

The Riemann sphere is an extension of the (finite) complex numbers that includes all of the finite complex numbers plus a single additional point for "infinity". It has a number of useful features for complex analysis, including one very useful fact for the Mandelbrot iteration.

Consider the Mandelbrot (and Julia) recurrence relation:

Z₀ = 0

Z_n+1 = Z_n² + C

This iteration will either remain "bounded" forever (with Z never getting more than 2.0 away from the origin), or it will "diverge". In the latter case Z goes ever further away from the origin and never comes back.

If Z and C are real numbers, then when Z diverges it does so by going towards ever-larger positive values.

If Z and C are complex, then when Z diverges it does so by getting ever-farther away from the origin, but not in any single direction — the angle usually keeps changing. It is useful in this case to only consider the radius of the successive iterates, and ignore the fact that the angle is changing. In the Riemann sphere model of complex numbers, it is said that successive iterates Z_n get closer to "infinity", there being only a single infinity in the Riemann sphere.

For practical purposes, when doing the Mandelbrot iteration on complex numbers in the Riemann sphere, addition, subtraction, and multiplication can be done by the following rules:

If either of the inputs to a calculation is "infinity", then the result of the calculation is infinity.

Otherwise, perform the calculation in the normal way and see if it overflows. If it does not overflow, use the non-overflowed answer in the normal way.

If the calculation overflows, the answer is "infinity".

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