Much of the present article serves as an introduction to a set of ideas familiar in dynamical systems theory. No familiarity with these ideas is assumed on the part of the reader. The ideas are then combined in simple, geometric arguments to prove Sarkowskii’s theorem. This theorem is important in the study of one‐dimensional, deterministic, dissipative dynamical systems. It provides a unified framework for the occurrences of both orderly and chaotic motions. The relationship of the mathematical models discussed here to real physical and biological systems is discussed briefly and the reader is referred to the literature for descriptions of diverse, beautiful, relevant experiments.