This might seem like a trivial question, but it may be more complicated than it seems. I'm wondering if it would be technically possible to visualize higher dimensional space. By that I mean seeing objects in say four dimension in one's imagination? Given that we cannot construct physical four dimensional objects, how would one go about imaging such an object?
I'm sceptical of any claims that anyone can directly visualise higher dimensional spaces. People who do work with them develop an intuition for them but this is because they understand what the important properties are and how they inter-relate.
I'm also sceptical that being able to visualise is sole key to understanding; this, on the face of it, seems wrong since when we see something we grasp it all at once advocated by the saying 'seeing is believing'; take for example, a square. If I show you this and say, how many corners does it have, you can immediately grasp what it is you are seeing and say, 'well, it has four corners'; then if I presented you with a 60-sided polygon and asked the same question, no-one will be able to grasp at once how many corners there are and the best that can be done is to count them. So here we see that being able to visualise, though important, is not the sole key to understanding.
When it comes to higher dimension because there is no possible direct visual representation then it solely symbolic representation here that matters. For example, the usual Pythagoras theorem is
x1*x1 + x2*x2 = y*y
This is in the plane, ie 2d. In 3d, it is
x1*x1 + x2*x2 + x3*x3 = y*y
It's easy then to guess that in 4d it will be
x1*x1 + x2*x2 + x3*x3 + x4*x4 = y*y
And the generalisation to n-dimensional spaces is straight-forward from here.
But what is this about 'guessing'? Does one 'guess' in mathematics! Surely that is an outrageous suggestion! Well, it's a form of inductive generalisation; which is justified post-hoc, that is by the richness of the theory built upon it and it's use elsewhere. For example, that this is the right generalisation of Pythagoras law is seen by the fact that the 4d version is what is used in Einsteins special relativity.
Is it possible to visualize higher dimensional space?
George Gamow discusses this question in One Two Three... Infinity (Bantam Books, New York, 1961). In "The World of Four Dimensions", Gamow writes that higher-dimensional figures can indeed be visualized in three-dimensional space, as the images of three-dimensional objects (as a sphere) can be projected onto a two-dimensional surface. That said, the observer must understand the limits of the resulting image: the projection has distorted the original object. Gamow goes on to discuss time as a fourth dimension.
You can go a long way towards visualizing higher dimensions if you think about how you see in three dimensions.
The first thing to realize is that there is a sense in which you don't see in three dimensions. Each retina is two-dimensional and your brain melds the images those retinas provide into a single picture. You can think of that picture as a 2D image with imperfect distance added. (Stereoscopic vision is just one of the ways your brain extracts that distance information.)
By analogy, a four-dimensional being might have three-dimensional retinas that would give them a 3-dimensional view combined with distance information. If they were looking at a wireframe 4D cube, then the image in their retina could be (depending on the angle) the classic cube-within-a-cube tesseract image. And the smaller cube would represent the more distance 3D face of the cube, just as the smaller square is the farther face with we view an ordinary wireframe cube.
In some sense, that is not true visualization; there is some abstract thought involved.
Ultimately, though, our brains have a great deal of plasticity. Our understanding of the 3D world comes from a lifetime of living in it. People with one eye and thus no stereo vision still perceive the world as 3D - they can move their head, their one good eye has to change focus for nearer objects, and so on. So having to project four or five dimensions down to the two of our retinas doesn't mean we can't visualize them. You could imagine an immersive VR-based video game where a 5D world is projected to 2D plus cues about distance in the z, w, and v directions. By leaning, you could move your head in our three normal dimensions, and you could control motion of the viewpoint in v and w using pedals, perhaps. Working together, your two arms would have more than enough degrees of freedom to manipulate a single arm in 5D space. At first, nothing you saw would make any sense. Things would seem to pass through each other and it would be hard to understand how to reach out and touch something.
But, when we look at one thing that's in front of another, the object in front effectively eats out a chunk of the one in back, from a naïve 2D point of view. It's our stereo vision and our infancy spent observing the world that teaches us that things can be behind other things. It is plausible that enough time spent playing in a simulated 5D world would give you a real sense that you can kind of see what's going on, so that you would be meaningfully be visualizing 5D objects using a brain built for 2D retinas.
Note that we are talking strictly about geometrical dimensions, not other things that may be considered a dimension.
Visualize: certainly no. You cannot view an N-dimensional object from a N-1 dimension. When you draw a cube on a paper you do not visualize the 3D cube, you visualize a 2D image and using extrapolation based on the fact that you recognize a 3D cube (because you have seen in in 3D before, so you have your extrapolation end point), you determine that your drawing is a representation of a 3D cube.
Extrapolate on how it would look: possibly.
To obtain a 2D square, you use 4x 1D lines perpendicular to each other (each perpendicular to 2 neighbor lines).
To obtain a 3D cube, you use 6x 2D squares perpendicular to each other (each perpendicular to 4 neighbor squares)
To obtain a 4D hypercube, you use 8x 3D cubes perpendicular to each other (each perpendicular to 6 neighbor cubes)
The above is a 4D object represented on 2D. Quite the loss of visual information here. A 3D holographic projection could offer way more clues as to how a hypercube may look like. In that case. you would a 4D object represented on 3D. Much more visual info to get from that.
But what humans lack in this case is the initial image of how the 4D hypercube looks in 4D. Due to that, there is no end point for the extrapolation to work, so in most cases humans will not be able to extrapolate at all a 4D object.
I would argue that a good familiar grasp of the mathematics, exactly allows such visualisation. Children do not look at triangles and know pythagoras' theorem by a kind of subitism. Even our supposedly intuitive grasp of 3D objects took a huge amount of learning and interaction to aquire, and is limited to the domains of our experience. Say fluid dynamics, we have some experience from swimming or whatever, but grasping methane seas or cloud dynamics takes educated contemplation. We just take tge educated contemplation we did as babies and toddlers for granted.
Consider mathematical savantism. People describe associating complex textures and synaesthetic characters to numbers, which allows the recall of mathematical connections like factorials quickly. This is reusing visual processing and identity recognition for mathematical purposes. Old brain region for new jobs, just like when people lose sight or a limb, areas can be repurposed.
I suggest that if by educated contemplation, we can interact meaningfully with higher dimensions, then yes we can visualise them. Like say the many 2D surfaces used to explain special and general relativity, which allow the build up of a mental picture of lorentz transforms, time dilation and so on. We can learn about topolgy, properties of klein bottles and 3spheres.
And we can go on to imagine a holographic universe, where we are a 4D surface in a 5D space - a major cosmological model and potential way to explain black holes and the information paradox. 11 dimensions of string theory, trickier. But we can look at the half integer spin of an electron and note a moebius strip has that same symmetry (turn it twice to get back where you started). We try to find heuristics, feed our intuitions, and of course do the math.
Now, can we visualise higher dimendions easily, or with the facility we apply to our 'native' experiences? Different question, and of course, the answer is no. But we shouldn't just throw our hands up either.
According to a certain definition of "n-dimensional space," the geometry of a thing provides three dimensions, and the color is a fourth, its mass a fifth, its temperature a sixth, and perhaps even the six degrees of freedom in motion are six more, not to mention its time-history which is yet another dimension.
But you are probably referring to four-dimensional space: three physical dimensions + time. I think a neat visualization is one of those 3D shapes transforming. When you see it changing over time, if you try to imagine that the thing is all of the images in the sequence, then that may help to "visualize higher dimensional space."
But one difficulty I think you will have is converting the thought that "this is how the thing is changing over time" into a thought like "all of this is showing me what we have here."
If you have already decided whether to believe the "A theory of time", i.e. that things really do change, then you will think differently than if you adopt the "B theory of time", i.e. all time together is reality in four dimensions and therefore, nothing truly changes ever.
The key to the question would be the definition of "visualize." As an example, we see the hypercube visualized in several of these answers, but that may or may not be the concept of "visualization" that you are interested in.
One might argue that we can't "visualize" anything higher than 3 dimensions because our eyes (vision) see a 3d world (or rather, see into a 3d world, via a pair of 2d projections).
On the other hand, it's currently believed that the neurons of the neocortex are laid out in a rather regular 2d pattern. If one can claim visualisation is something doable with the neocortex (rather than the cerebellum or other parts), one would argue that visualization must naturally contain mappings from 3d to 2d, suggesting a definition of visualization may include mapping from higher dimensions to lower. And if you make the assumption that visualization is entirely described via the firing of neurons, one must argue that visualization includes mapping a continuous space into a discrete space of neuron firings. Arguably that is a much more extraordinary mapping than a mere 4d to 3d continuous mapping.
Also perhaps useful is to look at kinethestics. Modern physics will model the movement of a rigid body in 6 or 9 dimensions, accounting for not only positions but velocities and even accelerations. Movement of the spine may involve dozens of these, so visualizing spinal movement might call for modeling 100 dimensions or more, depending on how you frame the problem.
So in all, this answer may not give you an easy yes or no, but it does point to the reality that the concept of "visualization" is not so firm that yes or no answers become easy.
Four-dimensional space is fine, that is just a three-dimensional object viewed over time.
A five-dimensional object would take a special kind of reasoning. If you consider that space-time scales with the mass in effect then you can possibly imagine in a fifth-dimension. Directly observing the fifth-dimension is probably less obvious to the outside observer and is usually shown by inference, such as gravitational lensing but, may be more noticeable to the in-situ observer. So far this is coordinates t,t-1,x,y,z.
For a sixth-dimension, it is only necessary to extend some facet(s) of your imaginary device onto an additional plane where they may be presented possiby, from this perspective, disordered and without any regular order that we can understand. Imagining and accurately imagining are not the same thing.