From: utzoo!decvax!harpo!npoiv!hou5f!hou5b!hou5c!hou5e!mat Newsgroups: net.math Title: Re: Squares - (nf) Article-I.D.: hou5e.218 Posted: Thu Jan 27 12:19:40 1983 Received: Sun Jan 30 06:05:20 1983 References: uiucdcs.1389 Question is about measuring the length of a diagonal by zigzagging along lines parallel to the co-ordinate system ( sides of a square ). For some really good stuff along these lines, read B. Mandelbrot's book ``Fractals, Form, Chance, and Dimension''. But a quick description of what is happening: You are right in saying that the path length is always 2; it IS always 2. You are never actually measuring the diagonal. You are measuring a path which approximates the diagonal. The approximations are continually improving in their mean-square error, but their path lengths remain the same. -More than one way to measure the -goodness of fit of a cat skin. Mark Terribile