At 21:11 20-11-00 -0500, you wrote:
>Alberto wrote:
>
>There are no boundaries if the big U is, spatially, a S3 [*]
>
>[*] the three dimensional surface that can be represented immersed
>in R4 by the equation x^4 + y^4+ z^4 + w^4 = 1
>
>Is that translatalbe to Laymanish?
>
>
>Doug
It helps to think of a model with one less dimension:
S2 is the surface of an ordinary sphere which can be embedded in ordinary
three-dimensional space R3. (The equation of this sphere would be x^2 +
y^2+ z^2 = r^2, FWIW.) If we were two-dimensional creatures living on the
surface of a large enough sphere, it would appear as if we were living on a
flat plane, a la the classic book _Flatland_. The space would be finite
but would have no boundaries: if you travel far enough in what looks to
you like a straight line, you would eventually come back to your starting
point. Such a space could also be expanding: think of the surface of a
balloon that is being blown up. In that case, all of space would be
expanding, also, there would be no point on the surface of the balloon (the
universe) that would be the center of the expansion.
(One way that our hypothetical Flatlanders could determine that they are
actually living on the surface of a huge sphere rather than on a flat
surface would be to lay out a big enough triangle and measure its angles
carefully. On a plane, the sum of the angles of a triangle is always
exactly 180 degrees. On the surface of a sphere, the sum of the angles of
a triangle is always greater than 180 degrees.)
S3, then is a hypersphere embedded in four-dimensional space R4. If it is
large enough, three-dimensional creatures living on S3 would see their
local surroundings as an uncurved three-dimensional space surrounding
them. Again, however, if you travel far enough in a straight line, you
would come back to your starting point. And the Big Bang has the universe
expanding in 4-space like the balloon above is expanding in 3-space, with
no point in the universe that is the center.
Any better? Note that this may or may not actually describe _our_ Universe
. . . as someone said, "The Universe is not only stranger than we imagine,
it is stranger than we _can_ imagine."
-- Ronn! :)
