sphere of large radius

[Jon]

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

[William Elliot]

Post by Jon
A sphere of large radius becomes a plane, reverses curvature and converges
on a new center.

If the center is not moved then the infinite radius sphere is the plane at
infinity, which though non-existing in Euclidean geometry, does exist in
projective geometry.

Since a naive person like you doesn't know about projective geometry but
only some vague notions of Euclidean geometry, you are forced to admit
that as the radius r, increases to infinity, the center will move. In
this event the center will be a point at infinity which again does not
exist in Euclidean geometry but only projective geometry.

What do you mean by reverses curvature? A plane has zero curvature.
Since the curvature went from 1/r to 0, to reverse that would mean that
the curvature went from -0 to -1/r? That is incorrect. Spheres do not
have negative curvature.

A plane can be considered as a sphere with zero curvature.
A point can be considered as a sphere with infinite curvature.
Is that the convergence of the centers that you expected?

Throw your trash away, it's not recreational nor does it show any
knowledge of math except for some worthless artsy-feely sort of pseudo
math. Are you an example of a new math educated student?

[pubkeybreaker]

Post by Jon
A sphere of large radius becomes a plane,

No it does not. A sphere and plane are not topologically
equivalent.

Now, a sphere WITH ONE POINT DELETED (i.e. the punctured sphere)
does become a plane as the radius --> oo.

Stop making stupid pronouncements about things you do not
understand.

Post by Jon
reverses curvature and converges
on a new center.  It is shown that events measured from either center are
the same.

This is meaningless word salad. Gibberish.

[JEMebius]

Post by Jon
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