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 $\mathbb{R}P^2$.

Boy surface


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.

  • $\begingroup$ Thanks! Many interesting pictures in the link. I have a hard time imagining what a mobius strip with circular boundary looks like. $\endgroup$ – Jiu Jan 12 at 12:23
  • $\begingroup$ first link is partly missing, looking forward to seeing it! (see my profile image) $\endgroup$ – uhoh Jan 12 at 14:30
  • $\begingroup$ @uhuh thanks, fixed! $\endgroup$ – 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 $RP^2$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.