How important was visual thinking to the discoveries leading to geometric group theory? Visual thinking was needed to discover Cayley graphs as a means of representing finitely generated groups. This is not the triviality it might seem: Cayley graphs must be distinguished from the diagrams we use to present them visually. A Cayley graph is a mathematical representation of a generated group, not a visual representation. It consists of the following components: a set \(V\) (“vertices”), a set \(E\) of ordered pairs of members of \(V\) (“directed edges”) and a partition of \(E\) into distinguished subsets, (“colours”, each one for representing right multiplication by a particular generator). The Cayley graph of a generated group of infinite order cannot be fully represented by a diagram given the usual conventions of representation for diagrams of graphs, and distinct diagrams may visually represent the same Cayley graph: both diagrams in Figure 14 can be labelled so that under the usual conventions they represent the Cayley graph of \((S_{3}, \cdot, \{f, r\})\), already illustrated by Figure 10. So the Cayley graph cannot be a diagram.