Kidney stone and sea shell formation, phyllotaxis (the arrangement of leaves or analogous elements such as seeds in a sunflower head), the formation of atmospheric aerosols, and a wide variety of patterns formed by the interaction of plant polymers with iodine all involve chemical reactions involving a nucleation step. I will start with a mathematical analysis of a common demonstration in introductory chemistry courses, which is supposed to allow one to compare the diffusion rates of two gases, but where understanding a nucleation step turns out to be crucial. The model will lead us to modifying this experiment (in live demonstrations) to produce mini tornados. Some key aspects of the model also apply to the patterns produced by nucleation and growth in biological systems. A theme in the mathematical analysis will be the development of techniques to measure the degree of mixing in a dynamical system.