Obtaining the Fundamental Polygon of $\mathbb{R}P^2$

Question

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!

Answer 1

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).