Convergence of 3-D axis

[steelengineer]

I want to know if there is some work published on the convergence of dimensions. as in the axis coming into itself instead of going towards infinity.

I believe that the convergence of 3-D axis should be possible since, if we imagine a 2-D creature on top of a ball, and the creature may move in an apparent straight line, but end up at the same spot he started.

therefore, i think that if we realease a ray of light, and assume that it doesnt bend under gravitaional influences nor is blocked by an opaque object, the ray of light should come back to us- someday.

I have a mental model of what the dimensions under convergence may look like but i find it hard to put it on paper due to lack of sufficient mathematical knowledge, I'm working on it. however, i would like your views to its (im)possibility. any feedback will be appreciated - especially counterarguements :confused:

[Bo]

you can fiddle around a lot with geometry; and there might have been some mathematicians who did these kind of things.

your example of the sphere is used very often in general relativity. notice however that isn't realy what we normally mean by converging: converging means that from a certain point onwards, two quantities become equal; where on a sphere there are an infinite number of points where they are equal.

 

bo

[steelengineer]

Bo, I would like to know what you would call it when the +ve infinity axis starts turning into the -ve infinity axis. By the way is the geometry of doughnut shaped figures called 'topology' ????

[Bo]

Mathematically a topology is something very general; and mathworld (http://mathworld.wolfram.com/Topology.html) explains it better that I can :confused:

 

(for practical use if your not a mathematician: everything is a topology)

 

as for your question: I would call that a sphere with infinite radius.

 

Bo

[steelengineer]

I would call that a sphere with infinite radius.

 

Bo

That's the whole point!! that there is no such thing as infinity!! now, only if i could prove it

:confused:

By the way, thanx for the link, torus was the shape that i was looking for and they had a link for it that gave all the equations relating to it :confused:

[virtualmeet]

Hi,

I think that the best way to understand what would be a "convergent" axe is to consider it in an uper dimension (in our case, we have to look with the eye of a 4D creature). To explain that, suppose we were living in 2D world, we can easily imagine that if we were living in a "closed" surface (like one on the top of a sphere), then starting a ride from the top of the sphere will end, one day, at this starting point... Another possibility of our 2D world is an infinite surface (like a plane) and in this case, there is no way that one day we will reach our starting point.

The only way then is to live in a "closed" surface like a surface of a sphere...but there still a big probleme: what if this surface is not a static one and that it's radius is going up and up???

In this case, some mathematical calculation show that there is a limit which make it impossible to reach the starting point even if living in a closed surface : suppose we are moving with the speed of light C, then the distance D is :

D1 = C . t

the distance to cross on the top of the sphere is :

D2 = 2.Pi.R = 2.Pi.R(t) (if we consider R function on time).

Then :

D1 = D2 <==> C.t = 2.pi.R(t) <==> C = R(t)/t

(We have to consider a non static sphere because we are likely living in a non static 3D world)

This is not a direct response to your question but it's perhaps a new way to explore it.