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Path: utzoo!linus!faron!pws
From: pws@faron.UUCP (Phillip W. Servita)
Newsgroups: net.math
Subject: Re: Strange Shapes
Message-ID: <169@faron.UUCP>
Date: Tue, 20-Nov-84 16:18:01 EST
Article-I.D.: faron.169
Posted: Tue Nov 20 16:18:01 1984
Date-Received: Wed, 21-Nov-84 04:40:16 EST
Reply-To: pws@faron.UUCP (Phillip W. Servita)
Organization: MITRE Corp., Bedford, Ma.
Lines: 40
Keywords: SPOILER-SPOILER-SPOILER


In article <ihnet.176> eklhad@ihnet.UUCP (K. A. Dahlke) writes:
>Here is a question which some of you have probably heard before.
>What 3-dimensional shape has a finite volume, but an infinite 
>surface area?
>You can fill it up with paint, but you can't paint it.
>I will give the answer in a week if it has not been spoiled.
>The second part is harder.
>Prove that no shape has a finite surface area 
>and an infinite volume.
>I might give the answer to this part too,
>if I get the chance to think about it during turkey (chomp chomp).
>Enjoy continuously!
>
>Karl Dahlke    ihnp4!ihnet!eklhad


as for the first part, consider the curve y = 1/x (x >= 1)
rotate this around the x-axis. the resulting shape has volume
PI, but its surface area integral:

     /~\ infinity
     |
     |
     |   (1 + 1/x^4)^.5                diverges by inspection. (always 
2*PI |  ---------------- dx            below the curve y = 1/x) 
     |         x
     |
   \_/ 1

BUT, neither you, nor i, nor the Pentagon could possibly keep 
one of these things in their bedroom. so here is something a 
bit smaller: construct a "cylinder" whose cross section is 
a snowflake curve. such a cylinder can be constructed so as 
to fit inside of a normal cylinder of arbitrarily small diameter 
and height, but STILL would have infinite surface area.

                                       happy munching,
                                             -phil