ABSTRACT
This chapter illustrates how people can construct mathematics by coordinating reversible and composable mental actions. It uses the simple example of reflective symmetry and reflecting the plane over a line. Coordinating pairs of such reflections produce rotations and translations as transformations of the plane. Furthermore, every isometry of the plane is the composition of, at most, three reflections. Such coordinations can also provide for intuitive understandings of formal propositions, including one about the sum of interior angles in a triangle. Thus, the chapter demonstrates the mathematical power that unfolds as one learns to coordinate even a single kind of mental action. By reflecting on reflections, the reader should begin to experience their personal mathematical power.
