Why does the real projective plane / Boy surface look like this? 
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 $\mathbb{R}P^2$.


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 $RP^2$.
