This is to announce the availability of three papers giving a general theory of self-similarity. The first is an informal overview; the second two are the serious business. The seed was Peter Freyd's universal characterization of the real interval. In doing this research I've had a lot of help from people in response to questions that I've posted on this list, and of course this is also where Peter's original result appeared. I'd therefore like to express publicly my thanks to Bob Rosebrugh for his work in running it. Best wishes, Tom 1. "General self-similarity: an overview" Informal seminar notes explaining the ideas in (2) and (3). http://arxiv.org/abs/math.DS/0411343 2. "A general theory of self-similarity I" Consider a self-similar space X. A typical situation is that X looks like several copies of itself glued to several copies of another space Y, and Y looks like several copies of itself glued to several copies of X, or the same kind of thing with more than two spaces. Thus, the self-similarity of X is described by a system of simultaneous equations. Here I formalize this idea and the notion of a `universal solution' of such a system. I determine exactly when a system has a universal solution and, when one does exist, construct it. http://arxiv.org/abs/math.DS/0411344 3. "A general theory of self-similarity II: recognition" This paper concerns the self-similarity of topological spaces, in the sense defined in (2). I show how to recognize self-similar spaces, or more precisely, universal solutions of self-similarity systems. Examples include the standard simplices (self-similar by barycentric subdivision) and solutions of iterated function systems. Perhaps surprisingly, every compact metrizable space is self-similar in at least one way. From this follow the classical results on the role of the Cantor set among compact metrizable spaces. http://arxiv.org/abs/math.DS/0411345 19-Nov-2004 08:32:04 -0400,1231;000000000000-00000000