In my experience, the discrete version of the Gauss-Bonnet theorem is a really great conversation starter to talk about the kinds of things mathematicians like. Everyone knows that the exterior angles in a polygon has to sum up to 2 pi, but did you know that you can define "exterior angles" on the vertices of a polyhedron that have to sum up to 4 pi? Can you think of a single line of reasoning that explains both facts? (Remember that the circumference of the unit circle is 2 pi and the area of the unit sphere is 4 pi...)

Without opening the article I swore I thought they were going to talk about the concave curved surfaces of spiked, icosahedron-shaped "Artifact" of Baldur's Gate 3.

I can’t remember the last time I saw someone talk about the Gauss-Bonnet theorem without talking about soccer balls