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)