# 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 $$\mathbb{R}P^2$$. ## 2 Answers

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.

• Thanks! Many interesting pictures in the link. I have a hard time imagining what a mobius strip with circular boundary looks like. – Jiu Jan 12 at 12:23
• first link is partly missing, looking forward to seeing it! (see my profile image) – uhoh Jan 12 at 14:30
• @uhuh thanks, fixed! – Alex J Best Jan 12 at 16:34

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 $$RP^2$$.