12 Jan '10 11:57

Referring to Thread 124309 : "rational sphere" who put the simple question if a sphere made up of rational points is transparant or not...

I say the answer is "Yes, it is totally transparant!" The following problem hints why.

The new problem is:

Suppose you have a sphere of radius one, a unit sphere, where every point of the sphere is the set of (x;y;z) of R^3 and every point has the distance d=1 to the center of the sphere. (That's the definition of a sphere, isn't it?)

Question: How many of these points (x;y;z) have all of x, y, and z as rationals?

I know six trivial points (0;0;1), (0;1;0), (1;0;0) and their negative counterparts.

But are there more? Does it have infinitly more, or a some other number of points?

Can you rigourously prove your answer?

I say the answer is "Yes, it is totally transparant!" The following problem hints why.

The new problem is:

Suppose you have a sphere of radius one, a unit sphere, where every point of the sphere is the set of (x;y;z) of R^3 and every point has the distance d=1 to the center of the sphere. (That's the definition of a sphere, isn't it?)

Question: How many of these points (x;y;z) have all of x, y, and z as rationals?

I know six trivial points (0;0;1), (0;1;0), (1;0;0) and their negative counterparts.

But are there more? Does it have infinitly more, or a some other number of points?

Can you rigourously prove your answer?