The lexicon consists of a set of word meanings and their semantic relationships. A systematic representation of the English lexicon based in psycholinguistic considerations has been put together in the database Wordnet in a long-term collaborative effort. We present here a quantitative study of the graph structure of Wordnet to understand the global organization of the lexicon. Semantic links follow power-law, scale-invariant behaviors typical of self-organizing networks. Polysemy (the ambiguity of an individual word) is one of the links in the semantic network, relating the different meanings of a common word. Polysemous links have a profound impact in the organization of the semantic graph, conforming it as a small world network, with clusters of high traffic (hubs) representing abstract concepts such as line, head, or circle. Our results show that: (i) Wordnet has global properties common to many self-organized systems, and (ii) polysemy organizes the semantic graph in a compact and categorical representation, in a way that may explain the ubiquity of polysemy across languages.