Although the study of clustering is centered around an intuitively compelling goal, it has been very difficult to develop a unified framework for reasoning about it at a technical level, and pro- foundly diverse approaches to clustering abound in the research community. Here we suggest a formal perspective on the difficulty in finding such a unification, in the form of an impossibility theo- rem: for a set of three simple properties, we show that there is no clustering function satisfying all three. Relaxations of these prop- erties expose some of the interesting (and unavoidable) trade-offs at work in well-studied clustering techniques such as single-linkage, sum-of-pairs, k -means, and k -median. - anand