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/ Rate This Message: http://www.cryonet.org/cgi-bin/rate.cgi?msg=19208