zlacker

That's amazing. It just goes on and on with beautiful plots. About halfway down, there's a section on understanding the Riemann sphere:

> From what I can tell, one of the settings used to deal with division by 0 is the so-called Riemann sphere, which is where we take a space shuttle and use it to fly over and drop a cow on top of a biodome, and then have the cow indiscriminately fire laser beams at the grass inside and around the biodome. That's my intuitive understanding of it anyway.

And there's even animation (with Mathematica source) for this one. However:

> (Note the cow cannot be spherical or it will roll off).

>>hansjo+(OP)
(I am not a mathematician) but if you want an actual explanation of the Riemann sphere, I really like the one given in "Visual Complex Analysis" by Tristan Needham. It's really very cool. Google books may have it.

This site is amazing though. It just keeps going...

>>hansjo+(OP)
I'll attempt a layman's explanation of a Riemann sphere for anyone who has no idea what this is about:

The Riemann sphere lets you deal with dividing by 0 by adding one more point to the complex plane: ∞[1]. Imagine putting it in the air above the origin, 0, and folding the midpoints of the four sides of your graph paper to meet there. (Yup, put all 4 points of the arrows in the same spot!) That's the Reimann sphere[2].

Going from a point the sphere back to the plane is a little weird.

1. Put a cow (point) on the top of the biodome (sphere) at infinity.

2. Pick the point on the sphere that you want to give a home on the complex plane.

3. Have the cow fire a laser beam (draw a line from infinity) through the point.

4. Follow that line back to wherever it hits the plane. That's the equivalent point on your complex plane.

Play with this a bit. Points near the top of the sphere (near the cow, at infinity) will shoot laser beams way off into the distance. Points near the bottom of the sphere will burn the grass right nearby on the plane itself.

You can then reverse the process ("If my cow were to shoot a laser at this grass, what part of the the biodome will get hit?"), to go from the plane to the sphere, because "folding up the sides of paper" doesn't accurately model what happened to make the sphere.

[1] Math with ∞ is what you'd expect: 3/0 = ∞. 3/∞ = 0.

[2] Since there is an infinite number of points on a sphere, this is entirely possible, and only mildly unsettling.

>>pflats+Jb
That sounds similar to the projective space used in computer vision.

>>Camill+cu
Maybe it's because Riemann sphere is nothing else but a complex projective line, but I guess the reason you feel they're related is that both of these are manifolds which are created by performing certain identifications of points inside flat prototypes.

>>pflats+Jb
the geometric rationale - the axiom of parallel lines.

On classic, Euclid, plane a pair of parallel lines would never intersect. Given a line and a point outside of the line, there can be only one line through the point which would be parallel to the first line.

If one modify the axiom to allow for many such lines through the point (the lines which would never intersect the first line) - that would result in hyperbolic plane geometry.

If one modify the axiom to state that no such line through the point is possible (i.e. any line through the point would intersect the original line) - that would result in the Riemann sphere geometry.