From: utzoo!decvax!yale-com!leichter
Newsgroups: net.math
Title: Re: Squares - (nf)
Article-I.D.: yale-com.742
Posted: Thu Jan 27 21:12:00 1983
Received: Sun Jan 30 09:11:43 1983
References: uiucdcs.1389

There is nothing "wrong" with the process described, when you get right down
to it, except that it gives the "wrong" answer!  As soon as you go to any
infinite or limiting process, you have no a priori reason to believe your
intuition is correct.  You've made an implicit claim here:  The "limit", in
some sense of the word, of your smaller and smaller subdivisions is the
diagonal AND this notion of "limit"  fits correctly with the notion of length.
Well, unfortunately, this claim is incorrect - as your example shows.  One
has to be more subtle in defining "limit" for a series of geometric operations
like this if one expects the limit of the areas to be the area of the limits.

(In fact, if you want to define area for surfaces in three-space by the limit
of smaller and smaller triangular patches covering the surface - and obvious
notion - one finds that this just will not work at all - it's easy to find
example of surfaces that have a finite area if you do the sub-division one
way, but an infinite area if you do the subdivision another way.  It MAY be
possible - I don't remember, you should be able to find it in any good
calculus text - to construct a figure for which one can get ANY answer above
some lower limit as the area by an appropriate subdivision.)

Another demonstration of why your intuition is just plain wrong for infinite
cases:  Consider the hyperbola y = 1/x for x>0.  Spin it around the x axis
to form an infinite horn.  If you sit down and work it out - simple calculus -
you will find that the VOLUME of the horn is finite - but the surface area
is infinite!  Alternatively:  You can fill the horn with a finite amount
of paint, but no amount of paint will cover the outside.

Strange but true.

							-- Jerry