There is an idea I've recently gotten interested in that doesn't seem to have a good agreed-upon name ("diagrammatic algebra?"). It centers around the use of two-dimensional diagrams of dots, possibly boxes, and arrows, and is related to (in no particular order) knot theory, braided monoidal categories, quantum groups and Hopf algebras, subfactors, planar algebras, and (topological) quantum field theory. However, it also has a more accessible aspect: it can be used as an elegant notation for working with $\text{Vect}$ (a particularly ubiquitous braided monoidal category; see question #6139), and at least one textbook has used a variant of it to develop the basics of Lie theory. There is also John Baez's Physics, Topology, Logic, and Computation: a Rosetta Stone, and another accessible introduction to some of these ideas is Kock's Frobenius Algebras and 2D Topological Quantum Field Theories. These ideas have also been used to understand quantum mechanics.

This is all pretty fascinating to me. These are elegant and beautiful ideas, and it seems to me that they are badly in need of unification and accessible exposition (something like Selinger's A survey of graphical languages for monoidal categories, but maybe with a more historical and/or expository bent). Beyond Baez's paper, does anyone know of any resources like that? Where can I learn more about what you can do with these diagrams that doesn't necessarily require a lot of background?

Related: how should I TeX these diagrams?