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