Imagine a world where our basic intuition about points, lines, and planes no longer applies—a world where three points in a three-dimensional linear space need not lie in any plane. This is the tropical world. In this talk, I will tell a story about tropical linear spaces, where many familiar incidence properties of classical linear geometry fail in surprising ways. I will also discuss how both the fulfillment and the violation of these properties resonate across other areas of mathematics, including algebraic curves, Lorentzian polynomials, and matroid combinatorics.