Diagrams are Theoretical Physicist's best Friends

Quantum many-body theory and quantum field theory are among the most difficult mathematical constructs in existence. Historically it has been fortunate that these theories often allow diagrammatic treatments, which almost miraculously rationalize the calculation rules. The visualization of the underlying mathematics by using diagram expansions has led to new insight and clever summation rules allowing physically meaningful approximations to be introduced. In this paper I consider some of the basic features of diagram techniques and the conditions for their existence, starting from the simplest examples.