X-Message-Number: 19208
From: "Technotranscendence" <>
References: <>
Subject: Re: dimensions again
Date: Wed, 5 Jun 2002 20:53:44 -0400

On Wednesday, Tue, 4 Jun 2002 10:53:01 EDT Robert Ettinger
 wrote:
> Yvan Bozzonetti wrote that "curvature" in a space of N dimensions does
not
> require an extra dimension to support the curvature. As I think I
wrote
> previously, it seems to be a language problem, or perhaps a
perspective
> problem.
>
> Again, if a hypothetical being that only perceives one dimension could
live
> on the perimeter of a circle, he would say what Yvan said. His
universe (like
> Einstein's) is "finite but unbounded," and his single-coordinate
system,
> whether expressed as an angle or as a displacement from an origin, is
cyclic.
> From our point of view the circle necessarily exists in two
dimensions, and
> his "dimension" is curved in visible reality and not just as an
inference
> from cyclicality.
>
> Yvan does, however, appear to agree that there are problems with the
confused
> use of terms such as dimensions, coordinates, and degrees of freedom.

I'm with Yvan here and embrace his confusion, though I think it be
clarity.:)

In a mathematical sense, curvature is merely deviation from Euclidean
geometry.  This does not require higher dimensional support.  It merely
requires that, in the context, one can define the geometric properties
of the given space and how they differ from an Euclidean one.

That this is hard for humans to visualize for dimensions higher than two
leads people to adopt analogies, such as view a one dimensional
postively curved space as if it were a circle contained in a higher two
dimensional space.  From a strictly geometrical (or mathematical)
perspective, this is an analogy to help one visualize, but there need be
no higher dimension in which to embedd this "curved" space.  The same
goes for any finite dimensional or even infinite dimensional spaces.
(Heck, if you drop the geometric way of talking about it all together
(which I recommend, since it allows a lot more freedom to move around
here:), you can just see these things as sets with certain relations.
Euclidean geometry is a set with certain relations and other geometries
can be seen as other sets and curvature is merely a way of measuring
their similarities -- without a necessary spatial realization for it.)

A minor point, if you forgive me:  Even in the example of a hypothetical
one dimensional being living on a circle embedded in a higher space,
from his/her/its perspective this is no way to tell is the universe is
finite yet bounded or just an infinite series of the same one
dimensional stuff repeated over and over.  (Add changes over time and
such a being might think the universe is infinite because there's
constantly new stuff to run into -- if it's long enough.:)

Cheers!

Daniel Ust
http://uweb.superlink.net/neptune/

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