Obtaining the Fundamental Polygon of $\mathbb{R}P^2$ On this page, Wikipedia shows, under the "Examples" heading, the fundamental polygons of the Sphere and the Real Projective Plane. Can we obtain the latter diagram from the former? I thought that this might be possible since $\mathbb{R}P^2$ is defined as the quotient of the sphere under the equivalence relation that identifies antipodal points, but I couldn't find a way to make it work. Any insight is appreciated!

It suffices to convince yourself the diagram below represents the antipodal map, with the shaded triangle as fundamental domain.

On the subject, there's a nice way to visualize antipodal identification of a cuboctahedron (a polyhedral model of a sphere) yielding a tetrahemihexahedron (a polyhedral model of a real projective plane).
