How does one go about research in mathematics? More importantly, what is the point? In this talk, I will share my perspective on these questions, motivated by some examples from my own experience and research activities. In particular, I will attempt to articulate my thoughts on the use of metaphorical and figurative reasoning in the pursuit of mathematical understanding. Once established, we will put these ideas into practice by examining a result from mathematical music theory related to the study of musical gestures and their topological characterization in a convenient category of spaces.