X-Message-Number: 19222
From: "Technotranscendence" <>
References: <>
Subject: Re: Ust (curved space)
Date: Fri, 7 Jun 2002 07:12:58 -0400

On Thu, 6 Jun 2002 09:57:30 EDT Robert Ettinger  wrote:
> "In a mathematical sense, curvature is merely deviation from Euclidean
> geometry."
>
> My point precisely. A language problem. It's a mistake to talk about
"curved
> space," unless there really is another dimension to support or define
the
> curvature.

Not really.  In the mathematical sense, a space S(n) -- where n is the
dimension of that space -- can be curved in the sense that it deviates
from E(n) -- where E is an Euclidean space of the same dimension.
There's no need for a comparison with S(n+1) or E(n+1).

For example, in a hyperbolic space of dimension two, one would only need
compare such things as the sums angles of trianlges to the expected
Euclidean sum for the same.  (The Euclidean would be, of course, two
right angles, while the hyperbolic would always be less than two right
angles for any triangle with nonzero sides.)  Again, there's no need to
posit a high dimenstion to "support or define the curvature" -- even a
higher Euclidean one.

In fact, my point was that use of a higher dimension -- or a lower
dimension analogues in a higher dimensional space (such as a football in
our perceptual space) -- is merely a means to help one visualize the
concept.  In a strict mathematical sense, these are analogies -- not
isometries.

I didn't read the beginning of this thread, so I don't know how this
relates to cryonics...

Cheers!

Dan
http://uweb.superlink.net/neptune/
    For a list of my works see:
http://uweb.superlink.net/neptune/MyWorksBySubject.html

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