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Explore the power of diagrammatic and graphical reasoning in Category Theory and its application to various algebraic structures in this 33-minute lecture. Discover how hypergraph rewriting serves as a computational foundation for diagrammatic calculus, covering familiar concepts like monoid actions and exotic higher-arity structures such as semiheaps and Bhattacharya-Mesner algebras. Learn about the implementation of this approach as a Mathematica paclet, available in beta for hands-on experimentation.