I've never before looked at Mandelbrot's original paper on self- similar geometric objects. Damned if he doesn't have as his first formal example the interval. He points out that it the same as the result of taking N copies, diminishing each by a factor of N, and then stringing them together. So when N = 2 he was pointing out that the closed interval was an invariant object of the functor I used. (No, of course, he didn't point out that it was a final co-algebra. And he was thought he was talking about the half-open interval, not the closed interval.) He goes on to tell how to make other self-similar curves: Choose a polygonal curve with N straight pieces, all the same length. Then: "Replace each of its N legs by a curve deduced from the whole..... through similarity of a fixed ratio. One is left with a curve made of N^2 legs;.....the desired self-similar curve is approaced by an infinite sequence of these steps." It can all be found on JSTORE: How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension Benoit Mandelbrot Science, New Series, Vol. 156, No. 3775. (May 5, 1967), pp. 636-638. ("Statistical" because the coast of Britain is not self-similar in the strict formal sense.) 26-Nov-2004 14:26:21 -0400,1904;000000000001-00000000