Riemann Sphere

Prerequisite: algebra 2

Remember this little prat?

I hated this guy (especially when it touched my roots). It seemed like a figment of imagination created by man. It was an ugly freak that did not belong in nature. Yuck yuck yuck the imposter..

If I had allowed myself to let go of the real plane for a little while, I might have come to appreciate the imaginary.

My favourite part in Zero by Charles Seife, ironically, was not the historical origin of zero, the development of the number zero, or the philosophical aspects of zero. The Riemann Sphere came into my spotlight.

The complex counterpart of our real unit circle looks like this:

Very neat eh. If the number is outside the circle, raising it to some exponent will make it grow further away from the circle. If the number is inside the circle, it will collapse toward the origin. Sorry for the nonstandard notation.

And here you have the actual sphere. Assign the complex plane at the equator, add infinity at the north and zero at the south. Place the south of the sphere on the origin of the plane.

The way Seife puts it, imagine that there is a light at the north of the sphere. Whatever is on the sphere will be projected onto the plane like a shadow.

Interestingly, circles that intersect the north of the sphere appear as lines on the plane. Seife's book Zero introduces infinity as zero's "twin", in which understanding one leads to a better understanding of the other. Really nice story, but that is for another post.

The sphere is cool because you can transform it. Say you transform a number x by i. Spin the sphere 90˚ then points on the sphere will correspond to their projected points on the plane, accordingly with the transformation.

For example:
1 on the plane = i on the transformed sphere
-i on the plane = 1 on the transformed sphere
-1 on the plane = -i on the transformed sphere
i on the plane = -1 on the transformed sphere

In other words, 1 = i after the transformation, and so on.

Seife was particularly interested in this transformation, regarding the division of zero and infinity.

The Riemann Sphere brought some kind of order to the chaos. I like how the imaginary terrain is mapped out, since it has always seemed an alternate ghostly realm to me. The neat visual tames my fear of the imaginary number. The best kind of math is one that is true, simple, yet aesthetic. The Riemann Sphere has achieved all those elements.

But still, I would rather not have i mess with my roots.