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From: eklhad@ihnet.UUCP (K. A. Dahlke)
Newsgroups: net.math
Subject: Re: Strange Shapes
Message-ID: <177@ihnet.UUCP>
Date: Sun, 25-Nov-84 10:00:20 EST
Article-I.D.: ihnet.177
Posted: Sun Nov 25 10:00:20 1984
Date-Received: Mon, 26-Nov-84 07:53:56 EST
References: <176@ihnet.UUCP>
Organization: AT&T Bell Labs, Naperville, IL
Lines: 32

Thanks for the many responses to "strange shapes".
If you were not familiar with infinite-area finite-volume shapes,
here is one example.
Take the graph  y = 1/x  and rotate it around the x axis.
Let x run from 1 to infinity, and close the shape at x = 1 with a unit disc.
This hyperbola of revolution works because volume drops as the cube of R,
and surface area drops as the square of R.
The difference is just enough to produce a finite volume
and an infinite surface area.  Try the integrals yourself.

One interesting response stated the shapes need not be unbounded.
A snowflake (fractle) kind of shape might work.
I guess I never said the shapes had to be smooth or continuous.
I am a little skeptical, and very interested.
Can anyone give an analytic definition for a bounded
shape with these properties.

As for the converse, I was a bit embarrassed by the appropriate response
"all points whose distance from the origin >= 1"
This shape indeed has infinite volume and finite surface area, but it
wasn't quite what I had in mind.
I better learn to phrase questions unambiguously.
I wanted the infinite volume to remain inside the shape.
Now define inside (aargggg).
Anyways, I didn't think much about this part, too busy eating.
I guess one could flatten non-convex portions, decreasing area and
increasing volume.
The shape must then be convex, and therefore bounded or infinite, and...
(wave wave wave, it's intuitively obvious, won't bore you with details).
-- 

Karl Dahlke    ihnp4!ihnet!eklhad