# 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 $$RP^2$$.