So now we know how to "see" objects of other dimensions, but why is it that we cannot completely understand them? Why can't we visualize a fourth or fifth or sixth dimension, and why can't we imagine living in only one or two dimensions?
The answer is actually fairly simple: We are 3D beings. We have been that way for our entire lives, and so has everything else we've ever been able to physically interact with. That kind of speaks for itself. We simply don't know, since there are no measurements for it, where a fourth dimension would go, spatially. And this gives us a vague idea of how 1D or 2D beings might feel about 3D, although ignorance of a third dimension seems impossible, not when it's so easy to visualize for us.
However, we have managed to simulate programs that move 4D things through our 3D slice of world (see Extradimensional Lenses) so that we can see them fully over time. Several of those will be linked below.
Anyway, 4D is definitely hardest to visualize so far, but 2D and 1D as well are a difficult concept to fully grasp. We're so used to 3D objects that it's hard to imagine a shape not having any depth whatsoever, hard to imagine not being aware of the second or third dimensions. To help: If you came across a floating 2D square, and walked behind it, it would appear to vanish. Going back to the 2D sphere manifold from before: as the manifold itself has no depth, the sphere simply has infinitely thin "walls". 2D is to 3D as 3D is to 4D. That is our best comparison at the moment.
When we talk about spheres in day-to-day life, we are referring to solid spheres. But in math spheres can be two-dimensional (a contorted closed-surface 2D manifold), and likewise, circles can be one-dimensional. To showcase this, first we have to define a one-dimensional manifold.
Points are not objects. Remember to think of them as places. They are proposed representatives of geographical locations in space, not things; they are written as coordinates specifying a location. Therefore, in mathematics, a line or line segment is an infinite amount of those points, making it a section of space(s). Thus, lines are also nothing more than places. One-dimensional manifolds are simply infinitely thin lines (and zero-dimensional manifolds are points), but according to topology, those lines can be contorted in any way to make new "shapes". Now, a two-dimensional "line", in the artistic sense (a very thin rectangle, but not infinitely thin), might form the shape of a French fry, and when contorted it could become a horseshoe. All 2D (without depth), of course.
Circles (contorted closed-surface 1D manifolds) are different from discs (open-surfaced, 2D, filled-in circles).
Hopefully, this visualization makes it easier to understand how 4D or even 5D manifolds and shapes and dimensions might come about.
Could something actually live in the first, second, or fourth dimensions? 1D creatures would exist along infinitely thin lines, so no ideas come to mind for their survival. 2D bodies would not be able to process waste without being sliced clean through — however, there is a species of anemone that produces waste and takes in nutrients through the same opening, so maybe. Since 3D beings are obviously able to function consistently just fine, maybe 3D and up is the way to go. Or would the 4th dimension get in the way? Is there a pattern to which numbers of dimensions are compatible with life?
Maybe the idea that things like organs and bodily storage and waste and nutrition are necessary for the survival of all life-forms is a narrow-minded one. After all, we've only ever had experience with three-dimensional organisms. Perhaps two-dimensional creatures would have ways of surviving without any of the things that are only possible in 3D, especially because new things might be possible in 2D. If the world were two-dimensional, the structure of biology, the most basic fundamentals of life as we know it, and every other aspect of existence involving molecular construction would be different. It's hard to know for sure what it would be like, but it's interesting to think about.
Being 3D makes our field of extradimensional vision disappointingly, and frustratingly, limited. Visualizing non-Euclidean geometries, however, is very much possible. And they might even be hypothetically survivable, if one were to be dropped within one. After all, as long as the geometry follows a set of rules consistently, living things can probably exist in it; their bodies, components of their world, and whatever resources they might need will simply follow those rules. Atomic structure may change, for example, rendering oxygen impossible. Or maybe it can still exist, just with a different shape. Or maybe, despite being warped, oxygen is fine just the way it is, since everything around it is warped too. How deeply would different geometric rules affect the foundations of a world? Down to the molecular level? Down to the subatomic level? Infinitely? Is it pointless to ask?
Since this is all hypothetical, let's go further. What about gravity? Do the laws of physics apply there? Or are "the laws of physics" only one set of many? What about multiple dimensions of time? 4D hyperbolic geometry?
Something noted by [MIT Tech Review] is that adding dimensions will add complexity, and not in the way you'd first think. Newton's laws of motion would become fragile and tremulous, since they would be highly sensitive to any perturbation.
Online 4D visualization programs: 4D Toys by Marc ten Bosh, Tesseract by Bartosz Ciechanowski, Interactive Models of 4D Objects by math.union.edu (includes several other pages on its site, accessible in the lower left, that I also recommend you check out).