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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>
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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.