From: utzoo!decvax!harpo!ihnp4!ixn5c!inuxc!pur-ee!uiucdcs!grunwald Newsgroups: net.math Title: Squares - (nf) Article-I.D.: uiucdcs.1389 Posted: Wed Jan 26 18:27:51 1983 Received: Sun Jan 30 04:28:58 1983 #N:uiucdcs:28200003:000:1448 uiucdcs!grunwald Jan 26 18:20:00 1983 I've got this little problem which I have been unable to put away for awhile, and yet I realise that there must be a trivial solution to the thing. Assume you have a measuring stick which can only measure the edges of a square. Now, you want to find the length of the diagonal of a unit square. So, for the first approximation, you measure the edges, sum them and say "length is 2". Well, to get a "better approximation" you divide the square into 4 even parts. You measure the inside path from the lower left corner to the upper right corner. Again, you get a length of 2. Now, you continue subdividing the unit square in to n sqaured sub-squares, and taking a path from lower-left to upper right. If we let the n -> infinity, then we would expect to eventally get a value of sqrt(2) for the length. But, as best as I have been able to determine, the limit is still 2. What's the flaw in the above argument? (the following diagrams may or may not make the problem easier to understand) 1: ------- square of unit area | X Path marked by X is of length 2 | X (well, not in this picture, but in real life...) XXXXXXX 2: ------- 4 squares on 1/4 unit area | | X path marked X is of length 2 ---XXXX | X | XXXX--- 4: ----xxx 16 squares, 1/16 unit area | | x | path marked x is of length 2 ----x-- | | x | --xxx-- | x | | --x---- | x | | xxx---- Carry this on till the area of each small square approachs 0.