By drawing a cube on a paper or by seeing it on a screen (a 2D surface - see Figure below), we can sort of visualise how a 3D cube would look like.
I was wondering whether we will be able to visualise the tessaract (4D cube) if we build a 3D model of it? However, if we are able to, it will contradict the common belief that one (as a 3D organism) cannot visualise the 4th Dimension. This leads to the question below.
Question: What is the fundamental difference between seeing the 4th dimensions in 3-dimensions and seeing the 3rd dimension in 2D view that stops us from being able to extend our understanding of the 3D to 4D?
Well, some thoughts so far is that to visualize depth in 3D, we use light/shading (for example a 3D object looks lighter nearer to the edge and darker in the centre). The 4th dimension is time/duration, so a question would whether we can make use of time (e.g. moving the object through time???) to help us visualise the 4th dimension better??? If so, how and if not, why not?
It is all based on our imagination.
As you say, we are $3$D organisms, and as such, we have visual memories and experiences from a $3$D world. This $2$D drawing of a cube is not similar at all to a real cube, but the image created in our brains while watching it resembles the one we get when watching an actual $3$D cube. Thus, as far as we can tell (with our limited senses), the $2$D cube and the real one are similar to one another.
Unfortunately, we don't meet $4$D objects in our daily life, thus we don't have any visual memories of them. So no picture we see can remind us of a $4$D cube, simply because we don't bear in our minds any image of it.
Yes, we could use time for viewing the $4$th dimension. For example, one can think of $S^3$ as a $2$-sphere that starts from a point, gets bigger and bigger, then smaller and smaller again, until it is reduced again to a point.