Fundamental operators arising in Euclidean harmonic analysis are often associated with sets of directions. What are these operators? Why are they important? How do direction sets enter the picture and influence their behaviour? The central issues underpinning these questions are rooted in analysis and geometry, but recently tools from extremal combinatorics and algebraic geometry have also played an important role in their study. We will choose an example of this genre of operators and discuss some recent work on it, joint in part with Jongchon Kim, Izabella Laba and Alessandro Marinelli.