Jon wrote:
> A sphere of large radius becomes a plane, reverses curvature and converges 
> on a new center.  It is shown that events measured from either center are 
> the same.  It is shown that a vector measured from either center is the same 
> vector.
> 
> http://mypeoplepc.com/members/jon8338/math/id24.html 
> 
> 

Hello Jon,

Please disregard the bad language and the objections against your ideas and their 
formulations. Some of these objections are indeed relevant, others are not: they only 
demonstrate great hair-splitting skills as regards terminology.

Well, to business now.

I guess you mean the 3D, 4D and higher-dimensional counterparts of the circles of 
Apollonius. See http://en.wikipedia.org/wiki/Circle_Geometry#Apollonius_circle in the 
Wikipedia article on circle geometry: http://en.wikipedia.org/wiki/Circle_Geometry .

Apparently you have two points in 3D space in mind, one point A from which a sphere 
emerges, and a second point B into which the sphere disappears.

Now consider the locus of points P that have a uniform ratio of distances from A and B:

PA:PB = lambda:mu, where lambda and mu are constants.

One proves easily with help of analytic geometry that this locus is a sphere
for lambda <> mu, and is the perpendicular bisector plane of AB if lambda = mu.

Next, imagine what happens with the sphere when one changes lambda:mu continuously from 
0:2 through 1:1 into 2:0 (or from 0:10 via 5:5 to 10:0, or whatever).

In my opinion it is exactly this what you mean:
the thing begins as a point at A, becomes an ever-increasing sphere as lambda:mu 
approaches 1:1, gets flat at 1:1, and becomes a sphere again when lambda:mu passes through 
1:1 on its way to 2:0. The sphere decreases and finally disappears into the point B.

It is an interesting exercise to find out how this transformation behaves as regards 
point-wise convergence, topological ruptures and related issues.

All this goes precisely this way in Euclidean spaces of any number of dimensions.

It remains to be seen whether the 3D hypersphere in 4D space moving in this manner from 
cradle to grave is a useful model of past, present and future of the universe.

Another interesting exercise: explore how the rotation group SO(4), which leaves the 
3-sphere S^3 unchanged as a whole, degenerates into the 3D displacement group
R^3 .x. SO(3) (semidirect product of translations and rotations) as the S^3 flattens out 
to Euclidean 3-D space.

Happy studies: Johan E. Mebius