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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)

enter image description here

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Why «can't we imagine a 4d object»?! –  Mariano Suárez-Alvarez Jul 7 '13 at 9:44
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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
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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
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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
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closed as off-topic by Steven Landsburg, Theo Johnson-Freyd, Andres Caicedo, Chris Godsil, Ramiro de la Vega Jul 7 '13 at 22:06

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2 Answers

You could slice the 4D object with a 3D 3-flat that varies in the 4th dimension. Here are snapshots of such a slicing of the hypercube down a diagonal:
     Hypercube Slices
     (Image above from Fleischfilm.)
Here is a video of a (much!) more complex object, a "4D quaternion Julia set," being sliced by a moving 3-flat: video link. And here is a still from the animation:
Julia set
(Image above from Creative Applications Network.)

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There is a polyhedral model of the Projective Plane due to Brehm, Ulrich: How to Build Minimal Polyhedral Models of the Boy Surface, The Mathematical Intelligencer Vol. 12, 51-55 (1990). See a partial account of this here, and also other articles on the Brehm model available on the web. John Robinson transformed Brehm's polyhedral model of the Mobius Band into a sculpture he called Journey.

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