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On 18 Aug 2007 05:57:35 -0400, Bran <...@none.org> wrote:
Hi!
Bob Day wrote:
> I'm trying to figure out whether gravity in Tuomo Suntola's
> Dynamic Universe Theory could obey the inverse square law.
> In conjunction with that, knowing the surface area of a 3-D
> sphere of radius r, embedded in the surface of a 4-D sphere
> of radius R appears to be relevant. But I'm pretty baffled as
> to how residing on a 4-D sphere would affect the area of a
> 3-D sphere. Does anyone know?
Not knowing anything about Suntolas's theory I try to offer simple
geometrical answer.
The volume of "3-D sphere" (which I presume means S_3, that is a
3-manifold, and not an S_2, as a border of 3D ball) in flat space
is
2 \pi^2 r^3 . (*)
If we embed such a sphere S_3 in an S_4 space with radius R
then the volume would be
2 \pi^2 R^3 \sin^3(r/R) (**)
(the radius r of S_3 is defined as the distance along the S_4 manifold
from the center of S_3 to any of S_3 points).
In the limit of small r (r << R) the (**) behaves like (*).
With further increasing r the rate of growth of (**) slowes and
at (r=\Pi R/2) stops. At this point sphere S_3 is an "equator" of
S_4 and further increase of r leads to decrease of Vol[S_3]. Finally at
r -> \Pi R the sphere shrinks to a point.
Similar results can be received in other dimension. For instance
* S_1 (circle) embedded in S_2 has perimeter
2 \Pi R \sin(r/R)
* S_2 embedded in S_3 has area
4 \Pi R^2 \sin^2(r/R)
Hope this helps,
--
B.
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On 18 Aug 2007 05:57:35 -0400, ol...@webtv.net (Oscar Lanzi III) wrote:
Go one dimension lower. You'll see that the circumference of a small
circle on the surface of a 3-dimensional ball is not pi times the radius
when measured on the spherical surface. A similar thing happens in your
four-dimensional problem.
But you di obtain the familiar Eucliean result as a limit when the
radius is small, for any dimensionality.
--OL
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