Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Path: utzoo!watmath!clyde!bonnie!akgua!sdcsvax!sdcrdcf!hplabs!hao!seismo!harvard!wjh12!foxvax1!brunix!browngr!jfh From: jfh@browngr.UUCP (John "Spike" Hughes) Newsgroups: net.math Subject: Re: Strange Shapes Message-ID: <1634@browngr.UUCP> Date: Wed, 28-Nov-84 09:53:22 EST Article-I.D.: browngr.1634 Posted: Wed Nov 28 09:53:22 1984 Date-Received: Tue, 4-Dec-84 08:24:26 EST References: ihnet.177, <176@ihnet.UUCP> Lines: 14 There is a nice picture of a bounded 'shape' with infinite volume inn the book "Calculus on Mandifolds", by Michael Spivak, pub. Benjamin; the shape is best described as a failure to extend the usual definitionn of arclength (sorry about that bouncy 'n' key): one defines arclength by subdividing with points, and then taking the polygonal curve that passes through the points in the same order as the curve did. Then one computes the length of the polygonal path, and takes a limit as the "gap" between points goes to zero. If you try to do the same thing for surface area, eg for a cylinder, by inscribing polygos within it, it turns out that the limit need not exist: the polygons can get arbitrarily small without the sum of their areas tending toward a constant. In fact one can make this sum-of-areas go to infinity. There are more details (I think) in Spivak's Comprehensive Introduction to Differential Geometry, Vol. 1 or 2, Publish or Perish Press.