Riemann's Translucent ball

1. Nov 19, 2003

Organic

From "ZERO" - The Biography of a Dangerous Idea by Charles Seife:

"Riemann imagined a translucent ball sitting atop the complex plane, with the south pole of the ball touching zero. If there were a tiny light at the north pole of the ball, any figures that are marked on the ball would cast shadows on the plane below. The shadow of the equator would be a circle around the origin. Every point on the ball has a shadow on the complex plane. Every circle on the plane is the shadow of a circle on the ball, and a circle on the ball corresponds to a circle on the plane .... with one exception. a circle that goes through the north pole of the ball, the shadow is a line. The north pole is like the point at infinity."

http://www.nti.co.jp/~kobakan/contents/zero.html#chap5

Edit:

You can find Pdf file of mine on this sybject here:

http://www.geocities.com/complementarytheory/RiemannsBall.pdf

Organic

Last edited: Nov 22, 2003
2. Nov 19, 2003

HallsofIvy

Staff Emeritus
"1) Can we conclude that 1/x is symmetric to x/1 where
[0,oo] = {x : 0 <= x <= oo}"

What do you mean by "symmetric"? Symmetric with respect to what line or point?

I think what you mean is: Set up an x-y coordinate system in the plane and put a Riemann sphere with its "south pole" at the origin.
Now, look at the great circle on the sphere that is directly above the x-axis. Each point, x, on the positive x-axis corresponds to a point on that circle. If take the radius of the sphere to be R and let &phi; be the angle the line through the center of the sphere and the point x on the x-axis, then &phi;= tan-1(x/R). In particular, x=1 corresponds to the point with &phi= tan-1(1/R). Even more specifically, taking R=1, x= 1 corresponds to the point with &phi= tan-1(1)= 45 degrees. All x from 0 to 1 will correspond to points up to 45 degrees from the vertical. All x from 1 to &infinity; will correspond to points above 45 degrees.

I don't believe that x and 1/x will be "symmetric" in any reasonable sense.

In any case, your second question
makes no sense to me at all.

What do you mean by "the number system of x/1 "?

How could any number have "infinitely many digits at the left side of the floating point"?

The Riemann sphere is talking about a geometric representation of numbers- it has no relationship whatsoever with a decimal representation.

3. Nov 19, 2003

Organic

Hi HallsofIvy,

I wrote it again in a clearer way:

If we reduce Riemann's ball to a single circle that goas through south pole (= 0) and north pole (= oo) then 1 is the middle point (on the circle's line) between 0 and oo.

My questions are:

1) Can we conclude that 1/x is symmetric to x/1 where
[0,oo] = {x : 0 <= x <= oo}

2) If the answer to (1) is yes, then where is the number system of x/1 which is symmetric to 1/x (rational and irrational number systems) and has infinitely many digits at the left side of the floating point?

4. Nov 19, 2003

HallsofIvy

Staff Emeritus
You first wrote
and then you wrote
You first wrote
and then you wrote
In what sense is the second post "clearer"?

5. Nov 22, 2003

Organic

Last edited: Nov 22, 2003