Category theory is primarily a language for expressing ideas across many different fields of mathematics, but categories are interesting mathematical objects in their own right. One way of defining a category is as a graph equipped with algebraic structure, and this algebraic structure can be equivalently expressed as adding, to the vertices and edges in the graph, triangles and higher simplices to form a special type of simplicial set. This principle applies to all sorts of "composable cell shapes", from arrows (giving categories) to squares (giving double categories) to certain planar trees (giving multicategories). Algebraic operations on "pasting" diagrams of any of these shapes give rise to a corresponding geometric description, where new shapes are added that express the same information just as triangles represent composition of arrows. Without assuming any background in categories or simplicial sets, I will describe this correspondence and show off some of the shapes that come out of it.