I realized that the 504 triangle edges in the fundamental domain of the Klein quartic form 28 closed curves (each consisting of 18 triangle edges). Apparently they are called geodesics.
There are 42 pairs of these 28 geodesics that have two orthogonal intersections (in the middle of rhombi). The geodesics correspond to the vertices of the Coxeter graph, and the orthogonal pairs to its edges.
I would like to know how this would be described in correct mathematical terms, and if this is already described in some mathematical work. (A source would allow me mention this in the Wikipedia article.)
So what are these 28 closed curves in the Klein quartic? (Bourque 2021 mentions 28 closed geodesics, but the drawing does not match what I have drawn.)
Do they separate the surface in two halves? And BTW, what keeps them from being circles?
Can the two intersection points of an orthogonal pair be described as opposite points on the Klein quartic?
