What is the relation between Kanizsa's bias towards convexity and the Gestaltists' demonstrations that perceptual organization obeys a principle of pragnänz, or simplicity? Why should either kind of bias exist? Textbook accounts assign no functional role for these biases. Geon theory (Biederman 1987) proposes that we can understand these biases in terms of fundamental processes by which complex objects are decomposed into convex (or singly concave) regions at points of matched cusps according to the transversality regularity (Hoffman and Richards 1985). (...) Such decomposition yields simple, convex parts segmented between the concavities. A shape that contains concavities is generally regarded as complex insofar as it can be decomposed into the regions, or parts, between the concavities. It is these simple parts that are the stable elements of shape, not the whole object. In fact, geon theory leads to the expectation that shape recognition proceeds most efficiently when the parts are good (in the pragnänz sense) but the object is bad! (shrink)