# Visualization of the real projective plane [closed]

Consider a closed (compact and without boundary) and non-orientable 2-manifold $M$. By Whitney embedding theorem, one can embed $M$ in $\mathbb{R}^4$. $M$ cannot be embeded in $\mathbb{R}^3$ and just can be immersed in $\mathbb{R}^3$.

We can not imagine a four-dimensional object. Can we use different figures of immersed $M$ in $\mathbb{R}^3$ and obtain imagination of $M$ in $\mathbb{R}^4$ (even a bit)? Can knowing of different figures of immersed $M$ help to imagine embedded $M$ in higher dimension? For example, the real projective plane is a closed and non-orientable 2-manifold, so it can be embedded in $\mathbb{R}^4$ and we cannot imagine this. But different figures of immersed real projective plane in $\mathbb{R}^3$ are within our reach. (following figures, from left to right respectively: cross-cap, Roman's surface, Boy's surface)

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## closed as off-topic by Steven Landsburg, Theo Johnson-Freyd, Andres Caicedo, Chris Godsil, Ramiro de la VegaJul 7 '13 at 22:06

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question does not appear to be about research level mathematics within the scope defined in the help center." – Steven Landsburg, Theo Johnson-Freyd, Andres Caicedo, Chris Godsil, Ramiro de la Vega
If this question can be reworded to fit the rules in the help center, please edit the question.

Why «can't we imagine a 4d object»?! –  Mariano Suárez-Alvarez Jul 7 '13 at 9:44
What is the question? "Can we use different figures...?" That must really have a positive answer, just add color to your graphics, and you have one way to represent the 4:th dimension. –  Per Alexandersson Jul 7 '13 at 9:57
@PerAlexandersson Could you explain that more? I do not understand what you mean by "adding colors". –  Sepideh Bakhoda Jul 7 '13 at 10:21
Immersions of the projective plane, in my opinion, is one of the least-gratifying ways of visualizing this object. It's far easier to think of the plane as the $SO_3$-homogeneous space where the point stabilizers are $O_2$. This immediately gives you the picture of a disc with antipodal points on the boundary identified. –  Ryan Budney Jul 7 '13 at 10:36
I'd like to suggest, if the OP agrees, to change the question to "what's the nicest visualization you know of the real projective plane?", to add the tags "big-list" and to make it CW. Then the goal of the question would be well-defined, and I might even have a few things to say (^_^) –  Daniel Moskovich Jul 8 '13 at 3:18