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.