How Does Riemann's Translucent Ball Illustrate Complex Plane Shadows?

[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

 

[HallsofIvy]

"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

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?

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.

 

[Organic]

Hi HallsofIvy,

Thank you for your reply.


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?

 

[HallsofIvy]

You first wrote

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.

and then you wrote

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.

You first wrote

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?

and then you wrote

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?

In what sense is the second post "clearer"?

 

[Organic]

My PDF

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

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



Organic