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 Rate This Message: http://www.cryonet.org/cgi-bin/rate.cgi?msg=19222