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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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closed as off-topic by Steven Landsburg, Theo Johnson-Freyd, Andrés E. Caicedo, Chris Godsil, Ramiro de la Vega Jul 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, Andrés E. 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
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
ℝℙ¹ is the space of lines through the origin so I just imagine a line nailed into the origin spinning around. (Reminding myself when it reaches noon it's already been there at 6:00 since unlike a ray the line is bi-directional hence $a=-a$.) – isomorphismes Mar 24 '15 at 19:21

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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