Why does the real projective plane / Boy surface look like this?

Question

In geometry, Boy's surface is an immersion of the real projective plane in 3-dimensional space found by Werner Boy in 1901

My question is, you can see that the Boy surface is made up of three identical parts. But how does the number $3$ come up? I cannot see it in the definition of $\mathbb{R}P^2$.

Answer 1

There are probably many ways to answer this question, and I'm totally unqualified to do so but here is one thing: it is possible to construct versions of Boy's surface with 5-fold (and larger odd numbered) symmetry, (there are some lovely illustrations in Models of the Real Projective Plane: Computer Graphics of Steiner and Boy Surfaces - Francois Apery, you may have access to this at https://link.springer.com/content/pdf/bbm%3A978-3-322-89569-1%2F1.pdf)

So the question could instead be why does the rotational symmetry of a Boy's type immersion have to be odd, it's probably something to do with the non-orientability of the surface.

Answer 2

3 occurs in the usual definition of $RP^2$ as the set of lines in $R^3$. That is, the quotient space of $R^3-0$ that identifies $x\sim cx$ for all nonzero $x\in R^3$ and nonzero real $c$. The homeomorphism $(x_1,x_2,x_3)\to(x_2,x_3,x_1)$ for example induces a threefold symmetry of $RP^2$.