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From: jfh@browngr.UUCP (John "Spike" Hughes)
Newsgroups: net.math
Subject: Re: Strange Shapes
Message-ID: <1605@browngr.UUCP>
Date: Tue, 20-Nov-84 13:23:20 EST
Article-I.D.: browngr.1605
Posted: Tue Nov 20 13:23:20 1984
Date-Received: Thu, 22-Nov-84 07:17:39 EST
References: ihnet.176
Lines: 17

SPOILER (sort of) ****************************************************

The answer to the second part is that it's incorrect: the subset of space
consisting of all points whose distance from the origin is greater
than one is an example: its volume is infinite, but its surface area
is 4 pi.

By the way, the first problem seems to be a paradox unitl you consider the
following algorithm for painting the plane: take 1 gallon of paint. Use the
first half gallon to paint all the points whose distance from the origin
is less than one. Now use the next quarter gallon to paint the points
whose distance form the origin is between one and two. Now continue in
this manner (of course the paint is much thinner as you move away
from the origin).
   Nonetheless, you see that you can 'paint' and infinite area with
a finite quantity of paint. This is just what happens when you fill
the answer to part 1 with paint...