Summary: Spin networks, essentially labeled graphs, are ‘good quantum numbers’ for the quantum theory of geometry. These structures encompass a diverse range of techniques which may be used in the quantum mechanics of finite-dimensional systems, gauge theory, and knot theory. Though accessible to undergraduates, spin network techniques are buried in more complicated formulations. In this paper a diagrammatic method, simple but rich, is introduced through an association of \(2\times2\) matrices with diagrams. This spin network diagrammatic method offers new perspectives on the quantum mechanics of angular momentum, group theory, knot theory, and even quantum geometry. Examples in each of these areas are discussed.